Twists in Topological Codes Héctor Bombín Perimeter Institute Outline Motivation Anyons and twists Twists in toric codes Twists in topological subsystem codes Quantum error correction attempts to create faithful quantum channels from noisy ones Typically this involves encoding in a subspace Motivation Noisy channel En co di ng D ecoding Noiseless channel The code subspace can be defined in terms of commuting observables: check operators (CO) Errors typically change CO values: Motivation CO measurement → error syndrome → → compute most probable error →correct Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons (a) (b) (c) j jp a b j jp a b e m (d) a b c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- formation on fusion channels and the topological nature of braiding makes TQC naturally robust against local perturbations, providing a complement to fault-tolerant quantum computation [? ? ]. In condensed matter, anyons emerge as excitations in systems that exhibit topological order [? ]. A possible way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not computationally powerful. In this paper we address an strategy to recover compu- tationally interesting anyon-like behavior from systems with very simple anyonic statistics. Our starting point are the symmetries that anyons may exhibit. Anyon models have three main ingredients: (i) a set of labels that identify the superselection sectors or topological charges, (ii) fusion/splitting rules that dictate the charges of composite systems, and (iii) braiding rules that dictate the effect of particle exchanges. A symme- try is a label permutation that leaves braiding and fusion rules unchanged—for a recent survey, see [? ]—. Given a symmetry s, we can imagine cutting the system along an open curve, as in Fig. ??(d), and then gluing it again “up to s”. Ideally the location of the cut itself is unphysical, Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- formation on fusion channels and the topological nature of braiding makes TQC naturally robust against local perturbations, providing a complement to fault-tolerant quantum computation [? ? ]. In condensed matter, anyons emerge as excitations in systems that exhibit topological order [? ]. A possible (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not computationally powerful. In this paper we address an strategy to recover compu- tationally interesting anyon-like behavior from systems with very simple anyonic statistics. Our starting point are the symmetries that anyons may exhibit. Anyon models have three main ingredients: (i) a set of labels that identify the superselection sectors or topological charges, (ii) fusion/splitting rules that dictate Motivation In some settings locality is crucial Topological codes have geometrically local COs Topological subsystem codes (TSC): Just 2-local measurements! Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd 〈Ci〉 = ci Z1Z2 Y2X3 Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are c mplex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of qu ntum in- formation on fusion channels and the topological nature of braiding makes TQC naturally robust against local perturbat ons, providi g a complement to fault-tolerant quantum computation [? ? ]. In condensed matter, anyons emerge as excitations in sy tems that exhibit topological orde [? ]. A possible way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not computationally powerful. In this paper we address an strategy to recover compu- tationally interesting anyon-like behavior from systems with very simple anyonic statistics. r r (c) (d) (e) (a) (b) 1 2 3 X Y ZI r r r r r r r r b b b b b bb b b b b gg gg g g g gg g gg g Our starting point are the symmetries that anyons may exhibit. Anyon models have three main ingredients: (i) a set of labels that identify the superselection sectors or topological charges, (ii) fusion/splitting rules that dictate the charges of composite systems, and (iii) braiding rules that dictate the effect of particle exchanges. A symme- try is a label permutation that leaves braiding and fusion rules unchanged—for a recent survey, see [? ]—. Given a symmetry s, we can imagine cutting the system along an open curve, as in Fig. ??(d), and then gluing it again “up to s”. Ideally the location of the cut itself is unphysical, only its endpoints have a measurable effect. In partic- ular, transporting an anyon around one end of the line changes the charge of the anyon according to the action of s. Our aim is to explore to which extent these topolog- ical defects, that we call twists for short, can be “treated as anyons” and used in TQC. Twists are being indepen- dently studied by Kong and Kitaev [? ]. An interesting precedent are the Alice strings appearing in some gauge models [? ], which can cause charge conjugation under monodromy, whereas the twists that we will discuss here exchange electric and magnetic charges. Rather than trying a general, abstract approach, we will focus on a well-known spin model, the toric code model, and address twists constructively. In this model anyons have no computational power, but we will show that twists behave as Ising anyons [? ], which are com- putationally interesting. In fact, they do not directly allow universal computation, but there exist strategies to overcome this difficulty [? ? ? ]. In [? ], Wootton et al. also try to mimic the non-abelian behavior in an abelian system, using an entirely different approach and philosophy. We remark that, although the discussion will mainly be in terms of topological order, it has direct application in the closely related context of topological codes [? ? ? ? ]. Anyon models— Anyon models are mathematically characterized by modular tensor categories, but we will not need such generalities (for an introduction, see for example [? ]). Instead, we will illustrate the content of anyon models with an example: Ising anyons. The first element of an anyon model is a set of la- Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodr my, transform anyons according to a symmetry. We study the realization of such defects in the t ric code m del, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These id as can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd 〈Ci〉 = ci Z1Z2 Y2X3 Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The onlocal encoding of quantum in- formation on fusion channels and the topological nature of braiding makes TQC naturally robust against local perturbatio s, providi a complement to faul -tolerant quantum computation [? ? ]. In condensed matter, anyons emerge as excitations in sy tems that exhibit opological ord r [? ]. A possible way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not computationally powerful. In this paper we address an strategy to recover compu- tationally interesting anyon-like behavior from systems with very simple anyonic statistics. r r (c) (d) (e) (a) (b) 1 2 3 X Y ZI r r r r r r r r b b b b b bb b b b b gg gg g g g gg g gg g Our starting point are the symmetries that anyons may exhibit. Anyon models have three main ingredients: (i) a set of labels that identify the superselection sectors or topological charges, (ii) fusion/splitting rules that dictate the charges of composite systems, and (iii) braiding rules that dictate the effect of particle exchanges. A symme- try is a label permutation that leaves braiding and fusion rules unchanged—for a recent survey, see [? ]—. Given a symmetry s, we can imagine cutting the system along an open curve, as in Fig. ??(d), and then gluing it again “up to s”. Ideally the location of the cut itself is unphysical, only its endpoints have a measurable effect. In partic- ular, transporting an anyon around one end of the line changes the charge of the anyon according to the action of s. Our aim is to explore to which extent these topolog- ical defects, that we call twists for short, can be “treated as anyons” and used in TQC. Twists are being indepen- dently studied by Kong and Kitaev [? ]. An interesting precedent are the Alice strings appearing in some gauge models [? ], which can cause charge conjugation under monodromy, whereas the twists that we will discuss here exchange electric and magnetic charges. Rather than trying a general, abstract approach, we will focus on a well-known spin model, the toric code model, and address twists constructively. In this model anyons have no computational power, but we will show that twists behave as Ising anyons [? ], which are com- putationally interesting. In fact, they do not directly allow universal computation, but there exist strategies to overcome this difficulty [? ? ? ]. In [? ], Wootton et al. also try to mimic the non-abelian behavior in an abelian system, using an entirely different approach and philosophy. We remark that, although the discussion will mainly be in terms of topological order, it has direct application in the closely related context of topological codes [? ? ? ? ]. Anyon models— Anyon models are mathematically characterized by modular tensor categories, but we will not need such generalities (for an introduction, see for example [? ]). Instead, we will illustrate the content of anyon models with an example: Ising anyons. The first element of an anyon model is a set of la- Motivation Fault-tolerance in topological memories: repeatedly measure error syndrome → → keep track of errors How to compute? Transversal gates (color codes) Boundaries & code deformation NOT possible for TSCs!!! Motivation In 2D excitations are very special: ANYONS. Using anyon symmetries → All CLIFFORD gates by code deformation on TSCs!!! Topological codes VS Topological order Code subspace ↔ Ground subspace Error correction ↔ Energy gap Error syndrome ↔ Excitation configuration Anyons 2D: statistics beyond bosons and fermions Ingredients of an anyon model: Top. charges Fusion rules Braiding rules Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd 〈Ci〉 = ci Z1Z2 Y2X3 #= Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- formation on fusion channels and the topological nature of braiding makes TQC naturally robust against local perturbations, providing a complement to fault-tolerant quantum computation [? ? ]. In condensed matter, anyons emerge as excitations in systems that exhibit topological order [? ]. A possible way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not computationally powerful. (a) (b) (c) j jp a b j jp a b e m (d) a b c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g In this paper we address an strategy to recover compu- tationally interesting anyon-like behavior from systems with very simple anyonic statistics. Our starting point are the symmetries that anyons may exhibit. Anyon models have three main ingredients: (i) a set of labels that identify the superselection sectors or topological charges, (ii) fusion/splitting rules that dictate the charges of composite systems, and (iii) braiding rules that dictate the effect of particle exchanges. A symme- try is a label permutation that leaves braiding and fusion rules unchanged—for a recent survey, see [? ]—. Given a symmetry s, we can imagine cutting the system along an open curve, as in Fig. ??(d), and then gluing it again “up to s”. Ideally the location of the cut itself is unphysical, only its endpoints have a measurable effect. In partic- ular, transporting an anyon around one end of the line changes the charge of the anyon according to the action of s. Our aim is to explore to which extent these topolog- ical defects, that we call twists for short, can be “treated as anyons” and used in TQC. Twists are being indepen- dently studied by Kong and Kitaev [? ]. An interesting precedent are the Alice strings appearing in some gauge models [? ], which can cause charge conjugation under monodromy, whereas the twists that we will discuss here exchange electric and magnetic charges. Rather than trying a general, abstract approach, we will focus on a well-known spin model, the toric code model, and address twists constructively. In this model anyons have no computational power, but we will show that twists behave as Ising anyons [? ], which are com- putationally interesting. In fact, they do not directly Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also b applied i the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd 〈Ci〉 = ci Z1Z2 Y2X3 #= Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- formation on fusion channels and the topological nature of braiding makes TQC naturally robust against local perturbations, providing a complement to fault-tolerant quantum computation [? ? ]. In condensed matter, anyons emerge as excitations in systems that exhibit topological order [? ]. A possible way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not computationally powerful. (a) (b) (c) j jp a b j jp a b e m (d) a b c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g In this paper we address an strategy to recover compu- tationally interesting anyon-like behavior from systems with very simple anyonic statistics. Our starting point are the symmetries that anyons may exhibit. Anyon models have three main ingredients: (i) a set of labels that identify the superselection sectors or topological charges, (ii) fusion/splitting rules that dictate the charges of composite systems, and (iii) braiding rules that dictate the effect of particle exchanges. A symme- try is a label permutation that leaves braiding and fusion rules unchanged—for a recent survey, see [? ]—. Given a symmetry s, we can imagine cutting the system along an open curve, as in Fig. ??(d), and then gluing it again “up to s”. Ideally the location of the cut itself is unphysical, only its endpoints have a measurable effect. In partic- ular, transporting an anyon around one end of the line changes the charge of the anyon according to the action of s. Our aim is to explore to which extent these topolog- ical defects, that we call twists for short, can be “treated as anyons” and used in TQC. Twists are being indepen- dently studied by Kong and Kitaev [? ]. An interesting precedent are the Alice strings appearing in some gauge models [? ], which can cause charge conjugation under monodromy, whereas the twists that we will discuss here exchange electric and magnetic charges. Rather than trying a general, abstract approach, we will focus on a well-known spin model, the toric code model, and address twists constructively. In this model anyons have no computational power, but we will show that twists behave as Ising anyons [? ], which are com- putationally interesting. In fact, they do not directly Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric u der some permutations of their topological charges. One can then conceive topol gical defects hat, under monodrom , transform anyons according to a symmetry. We study the realization of such defects in the toric code m del, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd 〈Ci〉 = ci σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 $= Q ∈ {a, b, . . . } Particle statistics are articularly ich in two spatial di- mensions, where beyond the usual fermions and bosons there e ist more generally anyons (se [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- formation on fusion channels and the topological nature of braiding makes TQC naturally robust against local perturbations, providing a complement to fault-tolera t quantum computation [? ? ]. In condensed matter, anyons emerge as excitations in systems that exhibit topological order [? ]. A possible way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? (a) (b) (c) j jp a b j jp a b e m (d) a b c r g b b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b rb b g br r g g b g ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not computationally powerful. In this paper we address an strategy to recover compu- tationally interesting anyon-like behavior from systems with very simple anyonic statistics. Our starting point are the symmetries that anyons may exhibit. Anyon models have three main ingredients: (i) a set of labels that identify the superselection sectors or topological charges, (ii) fusion/splitting rules that dictate the charges of composite systems, and (iii) braiding rules that dictate the effect of particle exchanges. A symme- try is a label permutation that leaves braiding and fusion rules unchanged—for a recent survey, see [? ]—. Given a symmetry s, we can imagine cutting the system along an open curve, as in Fig. ??(d), and then gluing it again “up to s”. Ideally the location of the cut itself is unphysical, only its endpoints have a measurable effect. In partic- ular, transporting an anyon around one end of the line changes the charge of the anyon according to the action of s. Our aim is to explore to which extent these topolog- ical defects, that we call twists for short, can be “treated as anyons” and used in TQC. Twists are being indepen- dently studied by Kong and Kitaev [? ]. An interesting precedent are the Alice strings appearing in some gauge models [? ], which can cause charge conjugation under monodromy, whereas the twists that we will discuss here exchange electric and magnetic charges. Rather than trying a general, abstract approach, we will focus on a well-known spin model, the toric code model, and address twists constructively. In this model Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topol gical codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd 〈Ci〉 = ci σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 $= Q ∈ {a, b, . . . } Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and boso s there exist more generally anyo (see [? ] for a com- pilation of the basic references). Anyonic statistics are co plex enough to give rise to the notion of topological qua m computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ?(a-c). The nonlocal e c di g of quantum in- form tion on fusion channels and the topologic l nature of braiding makes TQC naturally robust against local pe turbations, providing a complement to fault-tolerant quantum computati n [? ? ]. In condensed matter, anyons emerge as excitations in sys ems that exhibi top logical order [? ]. A possible way to obtain these exotic phases is by engineeri g suit- able Hamiltonians on lattice spin systems [? ? ? ? ? (a) (b) (c) j jp a b j jp a b e m (d) a b c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g ]. I deed, implementations on optical lattices have been propos d [? ]. Unfortu at ly, the anyon models that ap- pear in simple models are not computationally powerful. In th s paper we address an strat gy to recover compu- tationally interesting anyon-like behavior from systems with very simple anyonic statistics. Our starting point are the symmetries that anyons may exhibit. Anyon models have three main ingredients: (i) a set of labels that identify the superselection sectors or topological charges, (ii) f sion/splitting rules that dictate the charges of composite systems, and (iii) braiding rules that dictate the effect of particle exchanges. A symme- try is a label p rmutation tha leaves braiding and fusion rules unchanged—for a recent survey, see [? ]—. Given a symmetry s, we can imagine cutting the system along an open curve, as in Fig. ??(d), and then gluin it again “up to s”. Ideally the location of the cut itself is unphysical, only its endpoints have a measurable effect. In partic- ular, transporting an anyon around one end of the line changes the charge of the anyon according to the action of s. Our aim is to explore to which extent these topolog- ical defec s, that we all twists for short, can be “treated a anyons” and used in TQC. Twists are being indepen- dently studied by Kong and Kitaev [? ]. An interesting precede are the Alice strings appearing in some gauge models [? ], which can cause charge conjugation under monodromy, whereas the twists that we will discuss here exchange electric and magnetic charges. Rather than trying a general, abstract approach, we will focus on a well-known spin model, the toric code model, and address twists constructively. In this model Topological Order with a Twis : Ising Anyons fr m an Abelian M del H. B mbin Perimeter Institut for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a sy m try. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd 〈Ci〉 = ci σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 $= Q ∈ {a, b, . . . } Q′ Particle statistics re particularly rich in two spati l di- mensi ns, wh re beyond th usual fe mi ns and bosons there exist more generally anyons (see [? ] for c m- pilation of the basic references). Anyonic statistics ar co plex en ugh t give rise to the n tion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braidi d fusing nyons, see Fig. ??( -c). The n nlocal e oding of quan um i - formation on fusion channels and e topological nature of braiding makes TQC naturally robust against local perturb ti s, p ovidi g a complement to fault- olera t quantum computation [? ? ]. (a) (b) (c) j jp a b j jp a b e m (d) b c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g In condensed matter, anyons emerge as excitations in systems that exhibit topological order [? ]. A possible way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, mple entations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not computationally powerful. In this paper we address an strategy to recover compu- tationally interesting anyon-like behavior from systems with very simple anyonic statistics. Our starting point are the symmetries that anyons may exhibit. Anyon models have three main ingredients: (i) a set of l bels hat identify the superselection sectors or topological charges, (ii) fusion/splitting rules that dictate the charges of composite systems, and (iii) braiding rules that dictate the effect of particle exchanges. A symme- try is a label permutation that leaves braiding and fusion rules unchanged—for a recent survey, see [? ]—. Given a symmetry s, we can imagine cutting the system along an open curve, as in Fig. ??(d), and then gluing it again “up to s”. Ideally the location of the cut itself is unphysical, only its ndpoints have a measurable effect. In partic- ular, transporting an anyon around one end of the line changes the charge of the anyon according to the action of s. Our aim is to explore to which extent these topolog- ical defects, that we call twists for short, can be “treated as anyons” and used in TQC. Twists are being indepen- dently studied by Kong and Kitaev [? ]. An interesting precedent are the Alice strings appearing in some gauge models [? ], which can cause charge conjugation under monodromy, whereas the twists that we will discuss here Topological Order wi h a Twist: Isin Anyo s from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Carolin St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutati ns f their topological charges. One can then con eive topological defects that, unde monodromy, transform a yons ccording to a symmetry. We study the realization of such defects in the toric c de model, showing that a process where defects are braided and fused has the s me outcome as f they were Ising anyons. These ideas can also be appli d i the context f topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd 〈Ci〉 = i σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 $= Q ∈ {a, b, . . . } Particl statistics are particularly rich in two s atial di- mensions, where beyond th usual fermions and bosons th re exist more ge erally a yons (see [? ] for a com- pilation of the basic refere e ). Anyonic statistics are complex enough to give rise t the notion of top logical qua tum computation (TQC) [? ? ? ], where compu- tat ons are carried out by braiding and fusing anyons, see Fig. ?(a-c). The no local enco ing of quant m in- formation on fusion cha nels and the topological nature of braiding m kes TQC naturally robust against local perturbations, providing a c mplement to fault-tolerant quantum computation [? ? ]. In condensed matter, anyons emerge as excitations in systems hat exhibit topological order [? ]. A possible way to obta n t e exotic phases is by engine ring suit- able Hamiltonians on lattice spin systems [? ? ? ? ? (a) (b) (c) j jp a b j jp a b e m (d) a b c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g ]. Indeed, implementations on optical lattices have been propose [? ]. Unfortunately, th anyon models that ap- pear in imple models are n t com utationally powerful. In this paper we address an strategy to recover compu- tationally interesting anyon-like behavior from systems with very simple anyonic statistics. Our starting point are the s mmetries that anyons may exhibit. Anyon mod ls have th e main ingredients: (i) a set of labels t at identify the superselection sectors or topological charge , (ii) fusion/split ing rul s that dictate the charges of co posite systems, a d (iii) braiding rules at di ate the eff ct of particle exchanges. A symme- ry i a l bel permutation that leaves braiding and fusion rules unchang d—for a rec nt survey, see [? ]—. Given a symmetry s, we can imagine cutting the system along an pen curve, as in Fig. ??(d), and then gluing it again “up to s”. Ideally the location of the cut itself is unphysical, only its endpoints have a measurable effect. In partic- ular, tra sporting an anyon rou d one end of the line changes the charge of the a yon according to the action of s. Our aim is to explore to which extent these topolog- ic l defect , tha we call twists for short, can be “treated as a yons” and used in TQC. Twists are being indepen- ently studied by Kong and Kitaev [? ]. An interesting precedent are the Alic strings appearing in some gauge models [? ], which can cause charge conjugation under monodromy, hereas the twists that we will discuss here exch ge electric and magnetic charges. Rather than t ying a general, abstract approach, we will focus n a well-known spin model, the toric code model, and addre s twists constructively. In this model Topological Order with a Twist: Isi g Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric u er me permutations of their topological cha ges. One can then conceive topol gical defects th t, under monodromy, tra sform anyons acc ding to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they wer Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd 〈Ci〉 = ci σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 $= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . Particle statistics a e parti ularly rich in two spat al di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the b sic eferences). Anyonic statistics ar complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- formation on fusion channels and the topological nature of braidi g makes TQC natura ly robust against local (a) (b) (c) j jp a b j jp a b e m (d) a b c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g perturbations, providing a complement to fault-tolerant quantum computation [? ? ]. In co densed mat er, anyons emerge as excitatio s in systems that exhibit topological order [? ]. A possible way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not com utationally powerful. In this paper we address an strategy to recover compu- tationally interesting anyon-like behavior from systems with very simple anyonic statistics. Our starting point are the symmetries that anyons may exhibit. Anyon models have three mai ingredients: (i) a s t of labels that identify the superselection sect rs or topological charges, (ii) fusion/splitting rules that dictate the charges of composite syst ms, n (iii) braiding rules that dictate the effe t of article exchanges. A symme- try is a label permutation that leaves braiding and fusion rules unchanged—for a recent surv y, see [? ]—. Given a symmetry s, we can imagine cutting the system along an open curve, as in Fig. ??(d), and then gluing it again “up to s”. Ideally the location of the cut itself is unphysical, only its endpoints have a measurable effect. In pa tic- ular, ra spor ing an anyon arou d one end of the line changes the ch rge of the anyon according to the action of s. Our aim is to explore to which extent thes topolog- ical defects, that we call twists for short, can be “treated as anyons” nd used in TQC. Twists are being indepen- dently studied by Kong and Kitaev [? ]. An interesting precedent are the Alice strings appearing in some gauge t Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We stu y the realiz tion f such defects in the toric code m del, sh wing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd 〈Ci〉 = ci σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 $= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? Particle statistics are particu arly rich in two spatial di- mensions, where beyond the usual fermions and bosons there xist more generally anyons (see [? ] for a com- pilation of e basic references). Anyonic statistics are complex enough to give rise to the notion of topological (a) (b) (c) j jp a b j jp a b e m (d) a b c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g quantum computation (TQC) [? ? ? ], where compu- tatio are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- for ation on fusio channels and the topological nature of braiding makes TQC naturally robust against local perturbations, providing a complement to fault-tolerant quantum comput ion [? ? ]. In condensed matter, anyons emerge as excitations in systems tha exhibit topol gica order [? ]. A possible way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, i plementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pe r in sim le od ls are ot co putationall powerful. In this paper we address an strategy to recover compu- tationally interesting anyon-like be avior from systems with very simple anyonic statistics. Our starting point are the symmetries that anyons may exhibit. Anyon models have three main ingredients: (i) a set of labels that identify the superselection sectors or topological charges, (ii) fusion/splitting rules that dictate the charges of composite systems, and (iii) braiding rules that dictate the effect of particle xchanges. A symme- try is a lab l permutation that l aves braiding and fusion r les u change —for a rece t survey, see [? ]—. Giv n a symme ry s, we can imagine cutting the system along an open curve, as in Fig. ??(d), and then gluing it again “up to s”. Id ally t e location of the cut its lf is unphysic l, only its endpoints have a measurable effect. In partic- ular, transporti g an anyo around one e d of the line changes the charge of the anyon according to the action Topological QC Qubits encoded on fusion channels BUT: In abelian models a x b = c → trivial code Compute = Braid Measure = Fuse Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd 〈Ci〉 = ci σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 $= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons (a) (b) (c) j jp a b j jp a b e m (d) a b c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- formation on fusion channels and the topological nature of braiding makes TQC naturally robust against local perturbations, providing a complement to fault-tolerant quantum computation [? ? ]. In condensed matter, anyons emerge as excitations in systems that exhibit topological order [? ]. A possible way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not computationally powerful. In this paper we address an strategy to recover compu- tationally interesting anyon-like behavior from systems with very simple anyonic statistics. Our starting point are the symmetries that anyons may exhibit. Anyon models have three main ingredients: (i) a set of labels that identify the superselection sectors or topological charges, (ii) fusion/splitting rules that dictate the charges of composite systems, and (iii) braiding rules that dictate the effect of particle exchanges. A symme- try is a label permutation that leaves braiding and fusion rules unchanged—for a recent survey, see [? ]—. Given a symmetry s, we can imagine cutting the system along an open curve, as in Fig. ??(d), and then gluing it again “up to s”. Ideally the location of the cut itself is unphysical, Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physi s, 31 Caroline St. N., Waterloo, Ontari N2L 2Y5, Canada Anyon models can b symmetric und r some permutations of their t pological charges. One can then conceive t pological defects that, under mono ro y, transform a yons according to a symmetry. We s udy the realization of such defects in th tor c code model, showing that a process where defects are braided and fused has th same outco e as if they were Ising anyons. These ideas can also be applied in the context of topological c des. PACS numbers: 05.30.Pr, 03.67.Lx, 7 .43.Cd 〈Ci〉 = ci σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 $= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 Particle st tistics are particularly r h in two spatial di- mensions, where beyond th usual f rmions and bo ons (a) (b) (c) j jp a b j jp a b e m (d) a b c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g there exist more generally a yons (see [? ] for a com- pilation of the basic r feren es). A yonic statist s are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c) The nonlocal encoding of quantum in- formation on fusion channels and the topological nature of braiding m kes TQC naturally robust against loc l perturbations, providing a complement to fault-toleran quantum computation [? ? ]. In condensed matter, anyons emerge as excitations in systems tha exhibit topological rder [? ]. A possible way to obtain these exotic phases is by eng neering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices h ve been proposed [? ]. Unfortunately, the an on models that ap- pear in simple models are not computationally powerful. In this paper we dd ess an strategy to recover compu- tationally interesting anyon-like behavior from systems with very simple anyonic statist s. Our starting poin are the symmetries that anyons may exhibit. Anyon models have three main ingredients: (i) a set of l b ls that identify th superselection sectors or topological charges, (ii) fusion/splitting ru es that dictate the charges of composite systems, and (iii) braid ng rules that dictate the effect of particle exchanges. A symme- try is a label permutation that leaves braiding and fusion rules unchanged—for a recent su vey, see [? ]—. Given a symmetry s, w can imagine cutting the system along an open curve, as in Fig. ??(d), and then gluing it again “up to s”. Ideally the location of the cut i self is unphysical, Topological Order w th a Twist: Ising Anyons fr m an Abelian Model H. Bombin Perimeter Institute for Th oretical Physi s, 31 Caroline St. N., W terloo, Ontario N2L 2Y5, Canada Anyon models can be sy metric u der o e perm tation of h i topol gical charges. One can then conceive topological d fects that, under mo od o y, transform y ns according to a symmetry. We study the realization of such de ects i the oric code model, showing t at a process where defects are braid d and fused has th same outco as if they were Ising anyons. Th se ide s can also be applied i the context of topological codes. PACS numbers: 05.30.Pr, 3 67 Lx, 7 .43.Cd 〈Ci〉 = ci σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 $= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 Particle st tistic are particularly rich in two spatial di- mensions, where b yond the usual fermions and boso s (a) (b) (c) j jp a b j jp a b e m (d) a b c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b r r g b r g b b g br r g g b g there exist more gen rally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistic re complex enough t giv rise to the notion of pol gical quantum computation (TQC) [? ? ? ], where compu- tations are carried out by br id ng an fusing anyo s, see Fig. ??(a-c). The nonloc l e c ding of quantum in- formation on fusi n chan els and the topological nature of braiding makes TQC atur lly robust again local perturbations, providing a complement to faul - lera quantum computation [? ? ]. In conde sed matt r, anyons e erg as excitations n systems that exhibi topological rder [? ]. A possible way to obtain these ex tic phases i by ngineeri uit- able Hami toni ns on l ttice spin systems [? ? ? ]. Indeed, impl mentations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not compu ationally p werful. In this paper we address n strategy o r cover compu- tationally interesting anyo -like behavior from syste s with very simple anyonic statistic . Our starting point a e he symmetries tha anyons ma exhibit. Anyon models have three main ingredients: (i) a set of lab ls th t identify the supers lection sectors or topological charges, (ii) fusion/splitti g rules that dictate the charges of comp site sy tems, and (iii) braid ng rules that dic ate the eff c of particle xchang s. A symme- try is a label permutation th t l aves braiding and fu io rules unchanged—for a rece t surv y, see [? ]—. Given a symmetr s, we can i agine cutting the syst m along an open curve, as in Fig. ??(d), and then gluing it again “up to s”. Ideally the loca ion f the cut its lf is unphy ical, Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric u der some permutations of their topological charges. O can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the conte t of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2 3 Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- formation n fusion chann ls and he topolo ical ature of braiding makes TQC naturally robust against local perturbations, providing a complement to fault-tolerant quantum computation [? ? ]. In condensed matter, a yo s emerge as excitati s in systems that exhibit topological order [? ]. A possible way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not computationally powerful. In this paper we address an strategy to recover compu- ationally interesting any n-l ke beh vi r from systems with very simple anyonic statistics. Our starting point are the symmetries that anyons may exhi it. Anyon models have three main ingredients: (i) a set of labels that identify the superselection sectors or topological charges, (ii) fusion/splitti g rules that dictate the charges of composite systems, and (iii) braiding rules that dictate the effect of particle exchanges. A symme- try is a label permutation that leaves braiding and fusion rules unchanged—for a r cent survey, see [? ]—. Given a symmetry s, we can imagine cutting the system along an open curve, as in Fig. ??(d), and then gluing it again “up to s”. Ideally the location of th cut itself is unphysical, Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd i|ψ〉 = i|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 Particle statistics are pa ticularly rich in two spatial di- mensions, where beyond the usual fermions and bosons (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise t the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The onlocal encoding of quantum in- f rmati on fusion chann ls and the topological nature of braiding makes TQC nat rally robust against local perturbations, providing a complemen to fault-tolerant quantum computation [? ? ]. In cond ns d matter, ny ns emerge as ex it tio s in syst ms tha exhib t topological order [? ]. A possible way o obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, impleme tati ns on ptical lattic s have been proposed [? ]. Unfortun tely, the anyon models that ap- pear in simple models are no computationally powerful. In this paper we address an st ategy to recover compu- ta ionally in r sting anyon-like b havior from systems with very simple nyonic statistics. Our starting point are the symmetries that anyons may exhibit. Anyon models have three mai ingredients: (i) a set of labels tha identify th superselection sectors or top logical charges, (ii) fusion/splitting rules that dictate the charges of composite systems, an (iii) braiding rules hat dictate the effect of particl exchanges. A symme- try is a label permu ation that leaves braiding and fusion rules uncha ed—for a r ce t survey, see [? ]—. Given a symmetry s, we can imagine cutting the system along an open curve, as in Fig. ??(d), and then luing it again “up to s”. Ideally th locatio of the cut itself is u physical, t Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- formation on fusion channels and the topological nature of braiding makes TQC naturally robust against local perturbations, providing a complement to fault-tolerant quantum computation [? ? ]. In condensed matter, anyons emerge as excitations i systems that exhibit topological order [? ]. A possible way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? (a) (b) (c) e m (d) a c r g b b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b r r g b r g b b g br r g g b g ]. In , imple t io s n optical lattic s have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not computationally powerful. I this p per w addr ss n str tegy to recover compu- t tionally eresting a yon-like behavior from systems with very simple anyonic statistics. Our s arting point are th ymm ries that anyons may exhibit. Anyon models have three main ingredients: (i) set of labels that identify the supers lection sectors or to ologic l c arges, (ii) fusion/splitti rules that dictate the charges of composite systems, and (iii) braiding rules that dictate the effect of p r icle exchanges. A symme- try is a label permutation that leaves braiding and fusion rules unchanged—for a recent survey, see [? ]—. Given a symmetry s, we can imagine cutting the system along an Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- formation on fusion channels and the topological nature of braiding makes TQC naturally robust against local perturbations, providing a complement to fault-tolerant quantum computation [? ? ]. In condensed matter, anyons emerge as excitations in systems that exhibit topological order [? ]. A possible (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not computationally powerful. In this paper we address an strategy to recover compu- tationally interesting anyon-like behavior from systems with very simple anyonic statistics. Our starting point are the symmetries that anyons may exhibit. Anyon models have three main ingredients: (i) a set of labels that identify the superselection sectors or topological charges, (ii) fusion/splitting rules that dictate Anyon symmetry: charge permutation producing an equivalent anyon model Imagine ‘cutting’ the anyons’ 2D world and gluing it again up to a symmetry Symmetries Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- formation on fusion channels and the topological nature of braiding makes TQC naturally ro ust against local perturbations, providing a complement to fault-tolerant quantum computation [? ? ]. In condensed matter, anyons emerge as excitations in systems that exhibit topological order [? ]. A possible (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not computationally powerful. In this paper we address an strategy to recover compu- tationally interesting anyon-like behavior from systems with very simple anyonic statistics. Our starting point are the symmetries that anyons may exhibit. Anyon models have three main ingredients: (i) a set of labels that identify the superselection sectors or topological charges, (ii) fusion/splitting rules that dictate the charges of composite systems, and (iii) braiding rules that dictate the effect of particle exchanges. A symme- Topological Order with a Twist: Ising A yons from an Abelian Model H. Bombin Perimet r Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under s me permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the r alizat on of such defects in the t ric code model, showing that a process where defects are braid d nd fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05. 0.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Particle statistics are particul rly rich in two spatial di- mensions, where b yond the usual fermions and bosons there exist m re generally anyons (see [? ] for a co - p lation of the basic reference ). Anyonic statistics are complex noug to give rise to the notion of topological qua tum computation (TQC) [? ? ? ], where com u- tations are carried out by braiding and fusing an o s, see Fig. ??(a-c). The n nloc l encoding of quantum in- form tion on fusion channels nd the topological n ture of braiding makes TQC naturally robust against local perturb ti ns, providing a comp ment to fault-to erant qua tum computation [? ? ]. In condensed matter, anyons emerge as excitation in systems that exhibit topological order [? ]. A possible (a) b (c) e m (d) a c r g b g b r g (c) d (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g way to obtain these exotic phases is by engineering suit- able Hamilton ans on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- p ar in simple odels are not computationally powerful. In thi paper we address an strategy to recover compu- tationally int resting anyon-like b havior from systems with very simple anyonic statistics. Our starting point are the s mmetries that anyons may exhibit. Anyon models hav three main ingredients: (i) a set of labels that identify th superselection sectors or topological charges, (ii) fu ion/splitting rules that dictate the charges of composite systems, and (iii) braiding rules tha dict te the effect of particle exchanges. A symme- Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal e c ding of quantum in- formation o fusion channels and the topological nature of braiding akes TQ naturally robust against local perturbations, providing a comple e t to fault-tolerant quantum computation [? ? ]. In conde sed mat er, any ns emerge as ex ita i n in systems that exhi i t pological or r [? ]. A possible (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not computationally powerful. In this paper we address an strategy to recover compu- tationally interesting anyon-like behavior from systems with very simple anyonic statistics. Our starting point are the symmetries that anyons may exhibit. Anyon models have three main ingredients: (i) a set of labels that identify the superselection sectors or topological charges, (ii) fusion/splitting rules that dictate th charges of composite systems, and (iii) braiding rules tha d ctate he effect of particle exchanges. A symme- try is a label permutation that leaves braiding and fusion Twists Across the cut, charges change: Topologically, the cut location is unphysical. Endpoints are meaningful: under monodromy they permute charges → TWISTS Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- formation on fusion channels and the topological nature of braiding makes TQC naturally robust against local perturbations, providing a complement to fault-tolerant quantum computation [? ? ]. In condensed matter, anyons emerge as excitations in systems that exhibit topological order [? ]. A possible (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not computationally powerful. In this paper we address an strategy to recover compu- tationally interesting anyon-like behavior from systems with very simple anyonic statistics. Our starting point are the symmetries that anyons may exhibit. Anyon models have three main ingredients: (i) a set of labels that identify the superselection sectors or topological charges, (ii) fusion/splitting rules that dictate Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- formation on fusion channels and the topological nature of braiding makes TQC naturally robust against local perturbations, providing a complement to fault-tolerant quantum computation [? ? ]. In condensed matter, anyons emerge as excitations in systems that exhibit topological order [? ]. A possible (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not computationally powerful. In this paper we address an strategy to recover compu- tationally interesting anyon-like behavior from systems with very simple anyonic statistics. Our starting point are the symmetries that anyons may exhibit. Anyon models have three main ingredients: (i) a set of labels that identify the superselection sectors or topological charges, (ii) fusion/splitting rules that dictate Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for heoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the co text of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- formation on fusion channels and the topological nature of braiding makes TQC naturally robust against local perturbations, providing a complement to fault-tolerant quantum computation [? ? ]. In condensed matter, anyons emerge as excitations in systems that exhibit topological order [? ]. A possible (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models a e not computationally powerful. In this paper we address an strategy to recover compu- tationally interesting anyon-like behavior from systems with very simple anyonic statistics. Our starting point are the symmetries that anyons may exhibit. Anyon models have three main ingredients: (i) a set of labels that identify the superselection sectors or topological charges, (ii) fusion/splitting rules that dictate Topological O der with a Twist: Ising Anyons from an Abelian Model H. Bombin P rimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, C nada Anyon models can be sy metric under some permutati ns of their t pological charges. One can the conceive t pological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process wher defects are braide and fused as the same utcome as if th y were Isi g anyons. These ideas can al o be applie in the contex of t pological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ ×Q′ = q1 + q2 + . . . |ψ〉 −→ ? 1q2q3 . . . q −→ pi(q) Particle statistics are particularly rich in two spatial di- me ions, where beyond the usual fermio s and bosons th re exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to giv rise to the notion of t pological q antum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of q antum in- formation on fusion channels and he t pologic l nature of braiding makes TQC n turally robust against local perturbations, providing a complement to fault-tolerant q antum computation [? ? ]. I condensed matter, anyons emerge as excitations in systems that exhibit t pological order [? ]. A possible (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g way to obtain these exotic pha es is by engineering suit- able Hamiltonia s on lattice spin systems [? ? ? ? ? ]. Indeed, implementati s on optical lattices hav been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are n t computationally powerful. In this pap r we address an strategy to r cover compu- tationally interesting anyon-like behavior from systems with very simple anyonic statistics. Ou starting point are the symmetries that anyons may exhibit. Anyon models have three main ingredients: (i) a set of labels that iden ify the superselection sectors or t pological charges, (ii) fusion/splitting rules tha dictate Topological Order with a Twist: Ising Anyons fr m an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The non ocal enc ding of quantum in- formation on fusion channels and the topological nature of braiding makes TQC naturally robust against local perturbations, providing a complement to fault-tolerant quantum computation [? ? ]. In condensed matter, anyons emerge as ex itations in systems that exhibit topological order [? ]. A possible (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not computationally powerful. In this paper we address an strategy to recover compu- tationally interesting anyon-like behavior from systems with very simple anyonic statistics. Our starting point are the symmetries that anyons may exhibit. Anyon models have three main ingredients: (i) a set of labels that identify the supersel ction sectors or topological charges, (ii) fusion/splitting rules that dictate Topologi al Order with a Twis : Ising An ns rom an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada nyo models can be symmetric under some permutatio s of their topological charges. O e can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showi g that a process where defects are braide and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ { , b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Particle statistics are particularly rich in two spatial di- mensions, where beyond the usu l fermions and bosons there exist m re g nerally anyo s (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carrie out by br iding and fusing any ns, see Fig. ??(a-c). The nonlocal ncoding of quantum in- formation on fusion channels a d the topological n ture of braiding makes TQC naturally robust against oca perturb tions, providing a complement to fault-tolerant quantum computation [? ? ]. In conden ed matter, any ns emerge as excitations in systems that exhibit topological order [? ]. A possible (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g r r g g b g way to obtain these exotic phases i by engineering suit- able Hamiltonians on lat ice spin systems [? ? ? ? ? ]. Indeed, implementati ns on optical l ttices hav been propos d [? ]. Unfortunately, the anyon models that ap- pear i simple models are not computationally pow rful. In this paper we address an strategy to recover c pu- tationally interesting anyon-like behavior from systems with very simple anyonic statistic . Our starting point are the symmetri s th t anyons may exhibit. Anyon models have three main ingr dients: (i) a set of labels that identify the superselection s ctors or topological charges, (ii) fusion/splitting rules that dictate Topological Order with a Twist: Ising Anyons fro an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- formation on fusion channels and the topological nature of braiding makes TQC naturally robust against local perturbations, providing a complement to fault-tolerant quantum computation [? ? ]. In condensed matter, anyons emerge as excitations in systems that exhibit topological order [? ]. A possible (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not computationally powerful. In this paper we address an strategy to recover compu- tationally interesting anyon-like behavior from systems with very simple anyonic statistics. Our starting point are the symmetries that anyons may exhibit. Anyon models have three main ingredients: (i) a set of labels that identify the superselection sectors or topological charges, (ii) fusion/splitting rules that dictate Twists Example: quantum double of Z2 (toric code) Charges: Fusion: Braiding: Nontrivial symmetry: realization? Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g see Fig. ??(a-c). The nonlocal encoding of quantum in- formation on fusion channels and the topological nature of braiding makes TQC naturally robust against local perturbations, providing a complement to fault-tolerant quantum computation [? ? ]. In condensed matter, anyons emerge as excitations in systems that exhibit topological order [? ]. A possible way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not computationally powerful. Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) Particle statistics are particularly rich in two spatial di- mensions, where b yond the usual fermions and bosons there exist more generally anyo s (see [? ] for a com- pilation of the basic references). Anyonic statistics are (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g complex enough to give rise to the notion of topological qua tum computatio (TQC) [? ? ? ], where compu- tations are carried out by br iding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- formation on fusion channels and the topological nature of braiding makes TQC naturally robust against local perturbations, providing a complement to fault-tolerant quantum computation [? ? ]. In condensed matter, anyons emerge as excitations in systems th t exhibit opological order [? ]. A possible way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ bosons (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basi references). Anyonic statistics are co lex e ough to give rise to the notion of topological quantum comp t tion (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- for ation on fusion channels and the topological nature of braiding makes TQC naturally robust against local perturb tions, providin complement to fault-tolerant quantum computation [? ? ]. Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon mod l can be symmetric under som permutations of their t pological charges. One can then conceiv topological defects that, under mon dromy, transform anyons according to a symmetry. We study the r alization f such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q× ′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e, , $} e $ e× $ = m m× $ = e e e = m m = $× $ = 1 (2) e,m→ bosons $→ fermi (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of th basic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- formation on fusion channels and he topological n ture of braiding makes TQC naturally robust against local perturbations, providing a complement to fault-tolerant quantum computation [? ? ]. Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according t a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if th y were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- formation on fusion channels and the topological nature Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some per utations of th ir topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric co e model, showing that a process wh r defects are braided and fused has th same outcome as if th y were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.3 .Pr, 03.6 .Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are co plex enough to give rise to the notion of topological qu ntum comp ation (TQC) [? ? ? ], where compu- tations are carried out by bra ding and fusing anyons, see Fig. ??(a-c). Th no local encoding of quantum in- formation on fusion channels and the topological nature Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g = − Particle statisti s ar particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be sym etric under some permutations of their t pological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused h s the same outc me as if they were Ising anyons. These ideas can also be applied in he context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemR e = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g = − 1 #= q #= q′ #= 1 Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons th e exist more g nerally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex en ugh to give rise to the notion of topological Topological Order with a Twist: Ising Anyons from a Abelia Model H. Bombin Perimeter Institute for Theor tical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. O e can then conceive topological def cts that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing hat a process where defects are braided and fuse has he same outcome as if hey were Ising any ns. These id as can also be applied in th c ntext f topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g = − 1 #= q #= q′ #= 1 Particle statistics are particularly rich in tw spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of t pological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.C Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bos ns $→ fermion (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g = − 1 # q # q′ # 1 Particl stati i i l rly rich in two spatial di- mensions, er s al fer ions and bosons there exist r s (see [? ] for a com- pilation of t e ). nyonic statistics are complex eno t e notion of topological Topological Order ith a Tw st: Isi g Anyons from n Abelian Model H. Bombin Perim ter Institute for Theoretical Physics, 31 Caroli e St. N., W terloo, Ontario N2L 2Y5, Canada An o models can be symmetri under some permutations f their opological charges. One can then conceive topological def cts that, under monodro y, tr sform anyons ac rding a symmetry. We study the realization f such defects in the toric code model, sh wing that a process where defects re brai d and fused has the sam outcom as if they were Ising anyons. These ideas can also be applied in the context of topol gical codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 em me = Re!R!e Rm!R!m = −1 (1) {1, e,m, $} e×m $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g = − 1 #= q #= q′ #= 1 Particle statistics particularly rich i tw spatial di- m nsions, wh r be ond the u ual fermi ns and bosons the exist more g rally anyons (see [? ] for a com- pilation of the basic r ferences). Anyonic s atistics are co plex enough to giv rise to the noti n of topological Topological Order with a Twist: Isi g Anyons from an Abelian Model H. Bombin Perim ter Institut for T eoretical Physics, 31 Caroline St. N., Waterloo, Ontari N2L 2Y5, Canada Anyon models can be symmetric under some permutations f their topological charges. One can then c nceive topological defects that, under monodr y, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects re braided nd fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of t pological codes. PACS numbers: 05.30.Pr, 0 .67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 Rem me = Re! !e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e e = m m = $× $ = 1 (2) e,m→ bos ns $→ fermion (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g = − 1 # q # q′ # 1 Particl statis i ar i l rly ric in two spatial di- m nsions, re s al fer ions and bosons there exist r s (se [? ] fo a com- pilation of t e ). nyonic statistics are c plex eno t e n ti n of topological Topological Order with a Twist: Ising Anyons fro an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. O e can then conceive topological defects that, under monodromy, transfo m anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a pr cess where def cts are br ided nd fused has e same outcome as if the were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1 {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g = − 1 #= q #= q′ #= 1 e↔ m Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons Toric code Qubits form a square lattice 4-local check operators at plaquettes Hamiltonian: Excitations live in plaquettes 2 (a) (b) y X X X X X X X Z Z Z Z Z Z Z Z Z there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- formation on fusion channels and the topological nature of braiding makes TQC naturally robust against local perturbations, providing a complement to fault-tolerant quantum computation [? ? ]. In condensed matter, anyons emerge as excitations in systems that exhibit topological order [? ]. A possible way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not computationally powerful. In this paper we address an strategy to recover compu- tationally interesting anyon-like behavior from systems with very simple anyonic statistics. Our starting point are the symmetries that anyons may exhibit. Anyon models have three main ingredients: (i) a set of labels that identify the superselection sectors or topological charges, (ii) fusion/splitting rules that dictate the charges of composite systems, and (iii) braiding rules that dictate the effect of particle exchanges. A symme- try is a label permutation that leaves braiding and fusion rules unchanged —for a recent survey, see [? ]—. Given a symmetry s, we can imagine cutting the system along an open curve, as in Fig. ??(d), and then gluing it again “up to s”. Ideally the location of the cut itself is unphysical, only its endpoints have a measurable effect. In partic- ular, transporting an anyon around one end of the line changes the charge of the anyon according to the action of s. Our aim is to explore to which extent these topolog- ical defects, that we call twists for short, can be “treated as anyons” and used in TQC. Twists are being indepen- dently studied by Kong and Kitaev [? ]. An interesting precedent are the Alice strings appearing in some gauge models [? ], which can cause charge conjugation under monodromy, whereas the twists that we will discuss here exchange electric and magnetic charges. Rather than trying a general, abstract approach, we will focus on a well-known spin model, the toric code model, and address twists constructively. In this model anyons have no computational power, but we will show that twists behave as Ising anyons [? ], which are com- putationally interesting. In fact, they do not directly allow universal computation, but there exist strategies to overcome this difficulty [? ? ? ]. In [? ], Wootton et al. also try to mimic the non-abelian behavior in an abelian system, using an entirely different approach and philosophy. We remark that, although the discussion will mainly be in terms of topological order, it has direct application in the closely related context of topological codes [? ? ? ? ]. Anyon models— Anyon models are mathematically characterized by modular tensor categories, but we will not need such generalities (for an introduction, see for example [? ]). Instead, we will illustrate the content of anyon models with an example: Ising anyons. The first element of an anyon model is a set of la- bels that identify the superselection sectors or topological charges of the model. For Ising anyons there are three: 1, σ and ψ. Any given anyon carries such a charge, which cannot be changed locally. We can also attach a charge to a set of anyons or a to a given region. A region without anyons has trivial charge 1. Next we need a set of fusion rules that specify the pos- sible values of the total charge in a composite system. In terms of anyon processes, fusion rules specify the possible outcomes of the fusion of two anyons, see Fig. ??(c). For Ising anyons fusion rules take the form σ × σ = 1+ ψ, σ × ψ = σ, ψ × ψ = 1. (3) That is, a pair of σ-s may fuse into the vacuum or produce a ψ, a σ and a ψ always fuse into σ and two ψ-s into the vacuum. Fusion rules are commutative and 1× a = a. When two σ anyons are far apart, their total charge, which might be 1 or ψ, becomes a non-local degree of freedom. This is indeed an example of a topologically protected qubit, since there are two possible global states. We can measure this qubit in the charge basis by fusing the two σ-s and checking the output. In general for any set of anyons with given charges there is a fusion space that describes the non-local degrees of freedom related to fusion outcomes. For example, for 2n σ-s with indefinite total charge the fusion space has dimension 2n. Braiding operations as those in Fig. ??(a,b) act on the fusion space in a topologically protected way. This action is in general described by braiding rules, but in the case of Ising anyons it is possible to characterize braiding with Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c r g b g b r g ( ) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g = − 1 #= q #= q′ #= 1 e↔ m X Z Ak := XkZk+iXk+i+jZk+j. Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g = − 1 #= q #= q′ #= 1 e↔ m X Z Ak := XkZk+iXk+i+jZk+j. Topological Order with a T st: Ising Anyo s from an Abeli Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ont rio N2L 2Y5, Canada Anyon models ca be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transfor anyo according to a symmetry. We study the r alization of such defects in the toric code model, showing that a process where defects are braided and fuse has the same outcome as if they w re Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! −1 RemRme = !R!e = Rm!R!m −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g r r g g b g = − 1 #= q #= q′ #= 1 e↔ m X Z Ak := XkZk+iXk+i+j j. Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterlo , Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons ac ording to a symmetry. We study the realization of such defects in the toric code model, showing that a proces where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be ap lied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . } Q′ Q×Q′ = q1 + q2 + . |ψ〉 −→ ? q1q2q3 . q −→ pi(q) Re = Rmm = 1 R! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b rb b g br r g g b g = − 1 #= q #= q′ #= 1 e↔ m X Z Ak := XkZk+iXk+i+jZk+j. Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g = − 1 #= q #= q′ #= 1 e↔ m X Z Ck := XkZk+iXk+i+jZk+j. 2 (a) (b) y X X X X X X X Z Z Z Z Z Z Z Z Z H := − ∑ k Ck, Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- formatio on fusion channels and the topol gical natur of braiding makes TQC naturally robust against local perturbations, providing a complement to fault-tolerant quantum computation [? ? ]. In condensed matter, anyons emerge as excitations in systems that exhibit topological order [? ]. A possible way to obtain hese ex tic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortu tely, the a y n models that ap- pear in simple models are not computationally powerful. In this paper we address an strategy to recover compu- tationally interesting anyon-like behavior from systems with very simple anyonic statistics. Our starting point are the symmetries that anyons may exhibi . Anyon models have three main ingredients: (i) a set of labels that identify the superselection sectors or topological charges, (ii) fusion/splitting rules that dictate the charges of composite systems, and (iii) braiding rul s that dictate the effect of particle exchanges. A symme- try is a label permutation that leaves braiding and fusion rules u changed —for a recent survey, see [? ]—. Given a symmetry s, we can im gi e c tting the system along an open curve, as in Fig. ??(d), and then gluing it again “up to s”. Ideally the location of the cut itself is unphysical, only it endpoi ts have a measur ble effect. In partic- ular, transporting an anyon around one end of the line changes the charge of the anyon according to the action of s. Our aim is to explore to which extent these topolog- ical defects, that we call twists for short, can be “treated as anyons” and used in TQC. Twists are being indepen- dently studied by Kong and Kitaev [? ]. An interesting precedent are the Alice strings appearing in some gauge models [? ], which can cause charge conjugation under monodromy, whereas the twists that we will discuss here exchange electric and magnetic charges. Rather than trying a general, abstract approach, we will focus on a well-known spin model, the toric code model, and address twists constructively. In this model anyons have no computational power, but we will show that twists behave as Ising anyons [? ], which are com- putationally inter sti g. I fact, they do not directly allow universal computation, but there exist strategies to overcome this difficulty [? ? ? ]. In [? ], Wootton et al. also try to mimic the n n-abelian behavior in an abelian system, using an entirely different approach and philosophy. W remark tha , although the discussion will mainly be in terms of topological order, it has direct application in the closely related context of topological codes [? ? ? ? ]. Anyon models— Anyon models are mathematically characterized by modular tensor categories, but we will not need such generalities (for an introduction, see for example [? ]). Instead, we will illustrate the content of anyon models with an example: Ising anyons. The first element of an anyon model is a set of la- bels th t identify the superselection sectors or topological charges of the model. For Ising anyons there are three: 1, σ and ψ. Any given anyon carries such a charge, which cannot be changed locally. We can also attach a charge to a set of anyons or a to a given region. A region without anyons has trivial charge 1. Next w need a set of fusion rules that specify the pos- sible values of the total charge in a composite system. In terms of anyon processes, fusion rules specify the possible outcomes of the fusion of two anyons, see Fig. ??(c). For Ising anyons fusion rules take the form σ × σ = 1+ ψ, σ × ψ = σ, ψ × ψ = 1. (3) That is, a pair of σ-s may fuse into the vacuum or produce a ψ, a σ and a ψ alw ys fuse into σ and two ψ-s into the vacuum. Fusion rules are commutative and 1× a = a. When two σ anyons are far apart, their total charge, which might be 1 or ψ, becomes a non-local degree of freedom. This is indeed an example of a topologically protected qubit, since there are two possible global states. We ca measu this qubit in the charge basis by fusing the two σ-s and checking the output. In general for any set of anyons with given charges there is a fusion space th t describes the non-local degrees of freedom related to fusion outcomes. For exampl , for 2n σ-s with indefinite total charge the fusion space has dimension 2n. Toric code String operators create/destroy excitations at their endpoints Two types of excitations: e (light) and m (dark) 2 (a) (b) y X X X X Z Z Z Z H := − ∑ k Ck, Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- formation on fusion channels and the topological nature of braiding makes TQC naturally robust against local perturbations, providing a complement to fault-tolerant quantum computation [? ? ]. In condensed matter, anyons emerge as excitations in systems that exhibit topological order [? ]. A possible way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not computationally powerful. In this paper we address an strategy to recover compu- tationally interesting anyon-like behavior from systems with very simple anyonic statistics. Our starting point are the symmetries that anyons may exhibit. Anyon models have three main ingredients: (i) a set of labels that identify the superselection sectors or topological charges, (ii) fusion/splitting rules that dictate the charges of composite systems, and (iii) braiding rules that dictate the effect of particle exchanges. A symme- try is a label permutation that leaves braiding and fusion rules unchanged —for a recent survey, see [? ]—. Given a symmetry s, we can imagine cutting the system along an open curve, as in Fig. ??(d), and then gluing it again “up to s”. Ideally the location of the cut itself is unphysical, only its endpoints have a measurable effect. In partic- ular, transporting an anyon around one end of the line changes the charge of the anyon according to the action of s. Our aim is to explore to which extent these topolog- ical defects, that we call twists for short, can be “treated as anyons” and used in TQC. Twists are being indepen- dently studied by Kong and Kitaev [? ]. An interesting precedent are the Alice strings appearing in some gauge models [? ], which can cause charge conjugation under monodromy, whereas the twists that we will discuss here exchange electric and magnetic charges. Rather than trying a general, abstract approach, we will focus on a well-known spin model, the toric code model, and address twists constructively. In this model anyons have no computational power, but we will show that twists behave as Ising anyons [? ], which are com- putationally interesting. In fact, they do not directly allow universal computation, but there exist strategies to overcome this difficulty [? ? ? ]. In [? ], Wootton et al. also try to mimic the non-abelian behavior in an abelian system, using an entirely different approach and philosophy. We remark that, although the discussion will mainly be in terms of topological order, it has direct application in the closely related context of topological codes [? ? ? ? ]. Anyon models— Anyon models are mathematically characterized by modular tensor categories, but we will not need such generalities (for an introduction, see for example [? ]). Instead, we will illustrate the content of anyon models with an example: Ising anyons. The first element of an anyon model is a set of la- bels that identify the superselection sectors or topological charges of the model. For Ising anyons there are three: 1, σ and ψ. Any given anyon carries such a charge, which cannot be changed locally. We can also attach a charge to a set of anyons or a to a given region. A region without anyons has trivial charge 1. Next we need a set of fusion rules that specify the pos- sible values of the total charge in a composite system. In terms of anyon processes, fusion rules specify the possible outcomes of the fusion of two anyons, see Fig. ??(c). For Ising anyons fusion rules take the form σ × σ = 1+ ψ, σ × ψ = σ, ψ × ψ = 1. (3) That is, a pair of σ-s may fuse into the vacuum or produce a ψ, a σ and a ψ always fuse into σ and two ψ-s into the vacuum. Fusion rules are commutative and 1× a = a. When two σ anyons are far apart, their total charge, which might be 1 or ψ, becomes a non-local degree of freedom. This is indeed an example of a topologically protected qubit, since there are two possible global states. We can measure this qubit in the charge basis by fusing the two σ-s and checking the output. In general for any set of anyons with given charges there is a fusion space that describes the non-local degrees of freedom related to 2 (a) (b) y X X X X X X X Z Z Z Z Z Z Z Z Z there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- formation on fusion channels and the topological nature of braiding makes TQC naturally robust against local perturbations, providing a complement to fault-tolerant quantum computation [? ? ]. In condensed matter, anyons emerge as excitations in systems that exhibit topological order [? ]. A possible way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are no computati nally p werful. In this paper we address an strategy to recover compu- tationally interesting anyon-like behavior from systems with ve y simple anyonic statist cs. Our starting point are the symmetries that anyons may exhibit. Anyon models have three main ingredients: (i) a set of labels that identify the superselection sectors or topological charges, (ii) fusion/splitting rules that dictate the charges of composite systems, and (iii) braiding rules that dictate the effect of particle exchanges. A symme- try is a label permutation that leaves braiding and fusion rules unchanged —for a recent survey, see [? ]—. Given a symmetry s, we can imagine cutting the system along an open curve, as n Fig. ??(d), and then gluing it again “up to s”. Ideally the location of the cut itself is unphysical, only its endpoints have a measurable effect. In partic- ular, transporting an anyon arou d one end of the line changes the charge of the anyon according to the action of s. Our aim is to explore to which extent these topolog- ical defects, that we call twists for short, can be “treated as anyons” and used in TQC. Twists are being indepen- dently studied by Kong and Kitaev [? ]. An interesting precedent are the Alice strings appearing in some gauge models [? ], which can cause charge conjugation under monodromy, whereas the twists that we will discuss here exchange electric and magnetic charges. Rather than trying a general, abstract approach, we will focus on a well-known spin model, the toric code model, and address twists constructively. In this model anyons have no computational power, but we will show that twists behave as Ising anyons [? ], which are com- putationally interesting. In fact, they do not directly allow universal computation, but there exist strategies to overcome this difficulty [? ? ? ]. In [? ], Wootton et al. also try to mimic the non-abelian behavior in an abelian system, using an entirely different approach and philosophy. We remark that, although the discussion will mainly be in terms of topological order, it has direct application in the closely related context of topological codes [? ? ? ? ]. Anyon models— Anyon models are mathematically characterized by modular tensor categories, but we will not need such generalities (for an introduction, see for example [? ]). Instead, we will illustrate the content of anyon models with an example: Ising anyons. The first element of an anyon model is a set of la- bels that identify the superselection sectors or topological charges of the model. For Ising anyons there are three: 1, σ and ψ. Any given anyon carries such a charge, which cannot be changed locally. We can also attach a charge to a set of anyons or a to a given region. A region without anyons has trivial charge 1. Next we need a set of fusion rules that specify the pos- sible values of the total charge in a composite system. In terms of anyon processes, fusion rules specify the possible utcomes of the fusion of two anyons, see Fig. ??(c). For Ising anyons fusion rules take the form σ × σ = 1+ ψ, σ × ψ = σ, ψ × ψ = 1. (3) That is, a pair of σ-s may fuse into the vacuum or produce a ψ, a σ and a ψ always fuse into σ and two ψ-s into the vacuum. Fusion rules are commutative and 1× a = a. When two σ anyons are far apart, their total charge, which might be 1 or ψ, becomes a non-local degree of freedom. This is indeed an example of a topologically protected qubit, since there are two possible global states. We can measure this qubit in the charge basis by fusing the two σ-s and checking the output. In general for any set of anyons with given charges there is a fusion space that describes the non-local degrees of freedom related to fusion outcomes. For example, for 2n σ-s with indefinite total charge the fusion space has dimension 2n. Braiding operations as those in Fig. ??(a,b) act on the fusion space in a topologically protected way. This action is in general described by braiding rules, but in the case of Ising anyons it is possible to characterize braiding with Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g = − 1 #= q #= q′ #= 1 e↔ m X Z Ak := XkZk+iXk+i+jZk+j. Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g = − 1 #= q #= q′ #= 1 e↔ m X Z Ak := XkZk+iXk+i+jZk+j. Topological Order with a T st: Ising Anyo s from an Abeli Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ont rio N2L 2Y5, Canada Anyon models ca be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transfor anyo according to a symmetry. We study the r alization of such defects in the toric code model, showing that a process where defects are braided and fuse has the same outcome as if they w re Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! −1 RemRme = !R!e = Rm!R!m −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g r r g g b g = − 1 #= q #= q′ #= 1 e↔ m X Z Ak := XkZk+iXk+i+j j. Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterlo , Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons ac ording to a symmetry. We study the realization of such defects in the toric code model, showing that a proces where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be ap lied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . } Q′ Q×Q′ = q1 + q2 + . |ψ〉 −→ ? q1q2q3 . q −→ pi(q) Re = Rmm = 1 R! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b rb b g br r g g b g = − 1 #= q #= q′ #= 1 e↔ m X Z Ak := XkZk+iXk+i+jZk+j. Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σ2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e = $ e× $ = m × $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g = − 1 #= q #= q′ #= 1 e↔ m X Z Ak := XkZk+iXk+i+jZk+j. Topological Order th a Twist: Isi g Anyons from Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyo models can be symmetric under some permutations of their topological charges. One can then conceive topologic l defects that, under monodro y, tra form anyons according to a symmetry. W study the realization of suc defects in the toric code model, showing that a process where defects are brai ed and fused has the same outcom as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm 1 R!! = −1 mRme = Re!R!e Rm!R!m = −1 (1) {1, e,m, $} e× = $ e× $ = m m× $ = e e× e = ×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g g = − 1 #= q #= q′ #= 1 e↔ m X Z Ak := Xk iXk+i+jZk+j. Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the re lization of such defects in the toric code model, sho ing that process where defects re braid d and fused as the same utcome s if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) ee = Rmm = 1 R!! −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g g b r g gr b b r r g b r g b b g br r g g b g = − 1 #= q #= q′ #= 1 e↔ m X Z Ak := XkZk+iXk+i+jZk+j. Topological Order with a Twist: Ising Anyons from an Abelian Model H. B mbin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e m = $ e× $ = m $ e× e = m×m = $× $ 1 e,m→ bosons $ fer io (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g = − 1 #= q #= q′ #= 1 e↔ m Z Ak := XkZk+iXk+i+jZk+j. 2r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g (a) (b) complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- formation on fusion channels and the topological nature of braiding makes TQC naturally robust against local perturbations, providing a complement to fault-tolerant quantum computation [? ? ]. In condensed matter, anyons emerge as excitations in systems that exhibit topological order [? ]. A possible way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not computationally powerful. In this paper we address an strategy to recover compu- tationally interesting anyon-like behavior from systems with very simple anyonic statistics. Our starting point are the symmetries that anyons may exhibit. Anyon models have three main ingredients: (i) a set of labels that identify the superselection sectors or topological charges, (ii) fusion/splitting rules that dictate the charges of composite systems, and (iii) braiding rules that dictate the effect of particle exchanges. A symme- try is a label permutation that leaves braiding and fusion rules unchanged—for a recent survey, see [? ]—. Given a symmetry s, we can imagine cutting the system along an open curve, as in Fig. ??(d), and then gluing it again “up to s”. Ideally the location of the cut itself is unphysical, only its endpoints have a measurable effect. In partic- ular, transporting an anyon around one end of the line changes the charge of the anyon according to the action of s. Our aim is to explore to which extent these topolog- ical defects, that we call twists for short, can be “treated as anyons” and used in TQC. Twists are being indepen- dently studied by Kong and Kitaev [? ]. An interesting precedent are the Alice strings appearing in some gauge models [? ], which can cause charge conjugation under monodromy, whereas the twists that we will discuss here exchange electric and magnetic charges. Rather than trying a general, abstract approach, we will focus on a well-known spin model, the toric code model, and address twists constructively. In this model anyons have no computational power, but we will show that twists behave as Ising anyons [? ], which are com- putationally interesting. In fact, they do not directly allow universal computation, but there exist strategies to overcome this difficulty [? ? ? ]. In [? ], Wootton et al. also try to mimic the non-abelian behavior in an abelian system, using an entirely different approach and philosophy. We remark that, although the discussion will mainly be in terms of topological order, it has direct application in the closely related context of topological codes [? ? ? ? ]. Anyon models— Anyon models are mathematically characterized by modular tensor categories, but we will not need such generalities (for an introduction, see for example [? ]). Instead, we will illustrate the content of anyon models with an example: Ising anyons. The first element of an anyon model is a set of la- bels that identify the superselection sectors or topological charges of the model. For Ising anyons there are three: 1, σ and ψ. Any given anyon carries such a charge, which cannot be changed locally. We can also attach a charge to a set of anyons or a to a given region. A region without anyons has trivial charge 1. Next we need a set of fusion rules that specify the pos- sible values of the total charge in a composite system. In terms of anyon processes, fusion rules specify the possible outcomes of the fusion of two anyons, see Fig. ??(c). For Ising anyons fusion rules take the form σ × σ = 1+ ψ, σ × ψ = σ, ψ × ψ = 1. (3) That is, a pair of σ-s may fuse into the vacuum or produce a ψ, a σ and a ψ always fuse into σ and two ψ-s into the vacuum. Fusion rules are commutative and 1× a = a. When two σ anyons are far apart, their total charge, which might be 1 or ψ, becomes a non-local degree of freedom. This is indeed an example of a topologically protected qubit, since there are two possible global states. Toric code twists To get twists, we simply add dislocations Twists can be locally created in PAIRS only Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g = − 1 #= q #= q′ #= 1 e↔ m X Z Ak := XkZk+iXk+i+jZk+j. Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada A yon models can be symmetric under som permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the r alization of such defects in the t ric code model, showing that a process where def cts are braided and fu ed has the same outco e as if they were Isi g ny ns. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g = − 1 #= q #= q′ #= 1 e↔ m X Z Ak := XkZk+iXk+i+jZk+j. Topologic l Order w th a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institut for Theoretical Physics, 31 Carolin St. N., Waterloo, O tario N2L 2Y5, Canada Anyon models can be symm tric und r some p r utat ons of th ir topol gical charges. One can then conceiv topological defec s that, under onodromy, transform anyon according to a symme ry. We study he r aliz tion of such efects in the toric code model, howing that a process where defects are braided and fused h the same outcome as if they were Ising anyons. These ideas can als b applied in the context of topological codes. ACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ b sons $→ fermion (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g = − 1 #= q #= q′ #= 1 e↔ m Z Ak := XkZk+iXk+i+jZk+j. Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada A yon models an be sym etric under some permu ations of their topological charges. O e can then conceive topological defects that, unde monodrom , transf m any ns according to a symmetry. We s udy th realization of such defects in the oric c de model, showing t t a process where defects are braided and fused has the ame outc me as if they were Ising anyons. These ideas can also be applied in th context of pological c s. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 emRme = e!R!e = Rm!R!m = −1 (1) {1, e,m, $} m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g = − 1 #= q #= q′ #= 1 e↔ m X Z A := X Zk+iXk+i+jZk+j. Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rm = 1 R!! = −1 Rem e = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e× = $ e× $ = m m× $ = e e e m×m = $× $ = 1 (2) e,m→ bos s $→ fermion (a) (b) (c) e m (d) a c r g b g b r g (c) (d (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g = − 1 #= q #= q′ #= 1 e↔ m X Z Ak := XkZk+iXk+i jZk+j. Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon mod ls ca be symmetric under some permutati ns of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code m del, showing that a process where defe ts are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 Re Rme = e! !e = Rm! !m −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e × e = m×m = × $ = 1 (2) e,m→ bo ons $→ fermion (a) (b) (c) e m (d) a c r g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b r g gr b b r r g b r g b b g r r g g b g = − 1 #= q #= q′ #= 1 e↔ m X Z Ak := XkZk+iXk+i+j j. Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter I stitute for Theoretical Physics, 31 Carolin St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon odels ca be symm tr c under so e permutations f their topological charges. One can then conc ive topological defect that, under monodro y, transform an o s cc rding t a symmetry. We study the realiza ion of such defects in the toric code model, showing that a pr cess where defects are braided and fused has the same outcome a if they were Ising anyons. These ideas can also be appli d in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 Re Rme = e!R! = Rm! !m = −1 (1) {1, e,m, $} e×m = $ $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bo ons $ fermion (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g = − 1 #= q #= q′ #= 1 e↔ m X Z Ak := XkZk+iX +i+jZ +j. Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theore ical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon mod ls can be sy m tric under s me pe mutati ns of their topological charges. One can then conceive topological d fects that, under monodromy, transform anyons according t a symmetry. We tudy the realiza ion of such defects in the oric o m del, showing that a pr ess where defe ts are brai d and fused has the same outcome as if they were Ising any ns. These ideas can also be applied in the contex of topological c d s. PACS numbers: 05.30.Pr, 03.67.Lx, 73 43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 !! = −1 RemRme = Re! !e = Rm = −1 (1) {1, e,m, $} e×m = $ $ = m × $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g r r g g b g = − 1 #= q #= q′ #= 1 e↔ m X Z Ak := XkZk+iX +i+j j. Topological Order with a Twist: Ising Anyons fr m an belian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroli e S . N., Wat rloo, Ontario N2L 2Y5, Canada Anyo models can be sy etric under some per utations of their topological ch rges. One can then conceive topol gical defe ts that, under mono ro y, transform anyons according to a symmetry. We study the realization of such efects in the t ric c de model, showing hat a process where defects are b aided an fus d has t e same outcome as if they were Ising anyons. Thes id as can also be applied in the context of opol gical codes. PACS numbers: 05.3 .Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g = − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := kZk+iX jZk+j. Generalized charges EVEN number of twists → 4 possible charges ODD number of twists → 2 possible charges Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c = − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk+i+jZk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological Topological Order with a Twist: Ising Anyons f om an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c = − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk+i+jZk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ Particle tati tics are particularly rich in two spat al di- mensions, where beyond the usual fermio s and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological Topological Orde wi h a Twist: Ising Anyons from an Abeli n Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Carolin St. N., Waterloo, O tari N2L 2Y5, Canada Anyon models can be symmetric un r some permutations of their topological c arges. One can then conceive topological defects that, under monodromy, tra sform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, ,m, $} e×m = $ e× $ = m m× $ = e e× e = m $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c = − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk+i+jZk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological Top logical Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perim ter Insti ute for Theoretical Physics, 31 Caroline S . N., Wat loo, Ontario N2L 2Y5, C n da Anyon models can be sy metric un er some permu ations of their top logical charges. O e can then conceive top logical defects that, under m nodromy, transform anyons according to a sy metry. We study the realization of such defects in the toric code model, showing that a process wh re defects are brai ed and fused has the same outcome as if they w re Ising anyons. Th se ideas can also be applied i the context of top logical codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a b, . . } Q′ Q×Q′ = q1 + q2 + . . |ψ〉 −→ ? 1 2q3 . . q −→ pi(q) Ree = R m = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1 e, , $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ b sons $→ fermion (a) (b) (c) e m (d) a c = − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk i+jZk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ Particle s atistics are particularly rich in two spatial di- mensions, wh re beyond the sual fermions and b sons th r exist more generally anyons (see [? ] for a com- pilation of the basic r f rences). Anyon c s atistics are complex enough to give rise to the notion of top logical T pological Order wi h a wis : Ising Anyons from an Abelian Model H. Bombi Perim ter Insti ute for Theoretical Physics, 31 Caroline St N., Waterloo, Ontario N2L 2Y5, C ada A yon models can be sy metric under some permu ations of their t p logi al charges. One can then conceive t p logical d fects that, under monodro y, transform a yons according to a sy metry. We study the realization of such d fects in he tori code model, showing that a process wh re d fects a e braide and fused has the same outcome as if they w re Isi g a yons. Th s ideas can also be applied in the context of t p logi al codes. PACS numbers: 5.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1 2q3 . . . q −→ pi(q) Ree = R m = 1 R!! = −1 Re Rme = e!R!e = m!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e × e = m×m = × $ = 1 (2) e,m→ b sons $→ fermion (a) (b) (c) e m (d) a c = − 1 # q #= q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk i+jZk+j. H := − ∑ k Ck, Se Sm − 1 1 em $ Particle stati tics are particularly rich in two spatial di- me sions, wh re beyond the usual fermions and b sons th r exist more generally a yons (see [? ] for a com- pilati n of the basic f r nces). A yonic stati tics are complex enough to give rise to the notion f top logical T pological Order with a Twi t: Ising Anyons from an Ab lian Model H. Bombin Perim ter Insti u e for Theoretical Physics, 31 Caroline S N., Waterlo , Ontario N L 2Y5, Can da A yon models can be sy metric under some permu ations of heir top ogi al charges. One ca then conceive top ogical d fects that, under m nodromy, t ansform a yons according to a sy metry. We study the real zation of such d fects in he tori c model, showing that a process wh re d fects e br ided an fused has th same outcome as if they w re Isi g a yons. Thes ideas can also be appl ed in th context f top ogi al codes. PACS numbers: 5.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 1 ⊗ σ z 2 y 2 ⊗ σ x 3 #= Q ∈ {a b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1 2q3 . . . q −→ pi(q) Ree = R m = 1 R!! = −1 Re Rme = e!R!e = R !R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m × $ = e × e = ×m = × $ = 1 (2) e,m→ b ons $→ fermion (a) (b) (c) e m (d) a c = − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := XkZk iXk i+jZk+j. H := − ∑ k Ck, Se Sm − 1 em $ Par icle s ati tics are particularly r ch in two spatial di- mensions, wh r beyond the sual fermions and b ons th r exist mor generally a yons (see [? ] for a com- p lati n of the basic f r nces). A yonic s ati tics are complex enough to give rise to the oti n f top logical +1 +1 +1 -1 -1 +1 -1 -1 Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroli e St. N., Waterloo, Ont rio N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodro y, transform anyons according to a symmetry. We study the realization of such defects in the toric code m del, showing that a pr cess where def t ar braid d and fuse ha the same utco e as if hey were Ising anyons. These ideas can also b applied in the ont x of topological codes. PACS num ers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1 2q3 . . . q −→ pi(q) Ree = Rmm 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} m = $ e× m m× $ = e e× e = m×m = $ $ 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c = − 1 # q # q′ #= 1 e↔ m X Z Y Ck := XkZk+i k+i+jZk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ Particle statistics are particularly rich i two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the otion of topological Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric u d r s me ermutations of their topological charges. One can then conceive opol gical defects that, under monodro y, transform anyons ccording to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fu ed has the same outcome as if they were Ising any ns. These ideas can also be applied in the context of topological codes. PACS numb rs: 05.30.Pr, 03.67.Lx, 73.43.C Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } ′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) ee = Rmm = 1 R!! = −1 Re me = e! !e = R !R!m = −1 (1) {1 e m, $} e×m = $ e× $ = m× $ = e e e = m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c = − 1 #= q #= q′ # 1 e↔ m X Z Y Ck := XkZk iX i jZk+j. H := − ∑ k Ck, e Sm − 1 em $ Par icle s ti tics are particula ly rich in two sp tial di- me ions, wh re beyond the usual fermions and bosons th r exist more generally a yons (s e [? ] for a com- pilation of the basic efer nces). A yonic statistics are complex enough to give rise to the notion of topological Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric cod odel, showing that a process where defects are braided and fused has the same outcome as if th y w r Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c = − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk+i+jZk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological +i -i T p logic l Or er with a Twist: Isi g Anyons from an Abelia Model H. B mbin Peri eter Insti te for Theoretical Physics, 31 Caroline St. N., Wat rloo, O tario N2L 2Y5, Canada Anyon models can be symmetric under s me per utations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= ∈ {a, b, . . . } Q′ Q×Q′ = q1 + + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk+i+jZk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ S σ+ σ− Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute f r Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defec s that, under monodromy, transform a yons according to a symmetry. We study the realization f su h defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e Rm!R!m = −1 (1) {1, e,m, $} e× = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c = − 1 # q #= q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk+i+jZk+j. H := − ∑ k Ck, S Sm − 1 1 1 em $ S σ+ σ− Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are Topological Order with a Twist: Ising Anyons fro an Abelian Model H. B mbin Perimeter Institute for Theoretical Physics, 31 Caroline S . N., Waterl , Ontar o N2L 2Y5, Canada Anyon mo ls c b symmetric nder so e permutations of their topological charges. One can th co c ive opological d fe ts that, under monodromy, tra sform anyon ccording to a symmetry. W tudy the r aliza i n of such defects in the toric code model, showing that a process where defects are braided and fus d has the sam outcome as if they were Ising anyons. These ideas can also be applie in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.L , 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rm = R!! = −1 RemRme = Re! !e = R !R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c = − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk+i+jZk+j. H := − ∑ k Ck, S Sm − 1 1 1 em $ S σ+ σ− Particle statistics are particularly rich in tw spatial di- mensions, where beyond the usual fermions and bosons there exist more g nerally anyon (see [? ] for a com- pila ion of the basic ref ences). Anyonic statistics are Top logical Order with a Twist: Ising Any ns from an Abelian Model H. Bombi P rimeter Insti u for T or ti al Physics, 31 Ca line St. N., Waterl o, Ont rio N2L 2Y5, Canada Anyon mod l can b symme ric under some permutations f th i topol gic l harg . One can th n co ceive topol gical d fects at, under monodromy, transform any ns ac or ing to a symmetr . We stu y the realization of s ch defects in the oric code model, sh wing that a process where defects are b aided an fuse has the same outc e as if they were Ising a yons. These ideas can also b applied in the context of opol gical codes. PACS umbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2 3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 Rem e = e! !e = R ! !m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× = m×m = $× = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c = − 1 #= q # q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk+i+jZk+j. H := − ∑ k Ck, S Sm − 1 1 1 em $ S σ+ σ− Partic statistics are pa ticularly rich in two s atial di- mensio , where beyond the usual fermions a d b sons there xist more gen rally a yo s (see [? ] for a com- pilation of th basic fe nces). Anyo ic statistics are Fusion rules Twists are sinks for fermions: Non-abelian fusion rules! We recover Ising rules: Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c = − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk+i+jZk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ Particl statistics ar particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon odels can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under onodro y, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c = − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk+i+jZk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ Particle statistics are particularly rich in two spatial di- mensions, where b yond th usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological T pological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter I stitute for Theoretical Physics, 31 Carolin St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under som permu ations of their topological charges. On can th n conceive topological defects that, under mon dromy, ransform anyons according to a symmetry. We study the realiza ion such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 Re Rme = e!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c = − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk+i+jZk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological Topological Order with a Twist: Ising Anyons from n Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion (a) (b) (c) e m (d) a c = − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk+i+jZk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ S σ+ σ− Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73 43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion = − 1 #= q # q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk+i+jZk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ S σ+ σ− σ± × σ± = 1 + $ σ± × σ∓ = e+m σ± × $ =σ± σ± × e = σ± ×m = σ∓ (3) P rticle statistics are pa ticularly rich in two spatial di- mensions, where beyond th usual fermions and b sons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- formation on fusion channels and the topological nature of braiding makes TQC naturally robust against local perturbations, providing a complement to fault-tolerant quantum computation [? ? ]. In condensed matter, anyons emerge as excitations in systems that exhibit topological order [? ]. A possible way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not computationally powerful. In this paper we address an strategy to recover compu- tationally interesting anyon-like behavior from systems with very simple anyonic statistics. Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bo bin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 # Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = R !R!e = m!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion = − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk+i+jZk+j. H := − ∑ k Ck, Se Sm − 1 1 em $ S σ+ σ− σ± × σ± = 1 + $ σ± × σ∓ = e+m σ± × $ =σ± σ± × e = σ± ×m = σ∓ (3) σ × σ = 1+ ψ σ × ψ = σ ψ × ψ = 1 (4) Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- formation on fusion channels and the topological nature of brai i g makes TQC naturally robust against local perturbations, providing a complem nt to fault-tolerant quantum computation [? ? ]. In condensed matter, anyons emerg as excitations in systems that exhibit t pol gical order [? ]. A possible way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not computationally powerful. Majorana operators All closed string ops can be expressed in terms of a set of open string ops → Majorana operators Braiding is also Ising-like → 1-qubit Clifford gates 2 (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g In this paper we address an strategy to recover compu- tationally interesting anyon-like behavior from systems with very simple anyonic statistics. Our starting point are the symmetries that anyons may exhibit. Anyon models have three main ingredients: (i) a set of labels that identify the superselection sectors or topological charges, (ii) fusion/splitting rules that dictate the charges of composite systems, and (iii) braiding rules that dictate the effect of particle exchanges. A symme- try is a label permutation that leaves braiding and fusion rules unchanged —for a recent survey, see [? ]—. Given a symmetry s, we can imagine cutting the system along an open curve, as in Fig. ??(d), and then gluing it again “up to s”. Ideally the location of the cut itself is unphysical, only its endpoints have a measurable effect. In partic- ular, transporting an anyon around one end of the line changes the charge of the anyon according to the action of s. Our aim is to explore to which extent these topolog- ical defects, that we call twists for short, can be “treated as anyons” and used in TQC. Twists are being indepen- dently studied by Kong and Kitaev [? ]. An interesting precedent are the Alice strings appearing in some gauge (a) (b) (b) (c) (a) s sm (e) models [? ], which can cause charge conjugation under monodromy, whereas the twists that we will discuss here exchange electric and magnetic charges. Rather than trying a general, abstract approach, we will focus on a well-known spin model, the toric code model, and address twists constructively. In this model anyons have no computational power, but we will show that twists behave as Ising anyons [? ], which are com- putationally interesting. In fact, they do not directly allow universal computation, but there exist strategies to overcome this difficulty [? ? ? ]. In [? ], Wootton et al. also try to mimic the non-abelian behavior in an abelian system, using an entirely different approach and philosophy. We remark that, although the discussion will mainly be in terms of topological order, it has direct application in the closely related context of topological codes [? ? ? ? ]. Anyon models— Anyon models are mathematically characterized by modular tensor categories, but we will not need such generalities (for an introduction, see for example [? ]). Instead, we will illustrate the content of anyon models with an example: Ising anyons. The first element of an anyon model is a set of la- bels that identify the superselection sectors or topological charges of the model. For Ising anyons there are three: 1, σ and ψ. Any given anyon carries such a charge, which cannot be changed locally. We can also attach a charge to a set of anyons or a to a given region. A region without Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion = − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk+i+jZk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ S σ+ σ− cjck + ckcj = 2δjk c1c2c3 σ± × σ± = 1 + $ σ± × σ∓ = e+m σ± × $ =σ± σ± × e = σ± ×m = σ∓ (3) σ × σ = 1+ ψ σ × ψ = σ ψ × ψ = 1 (4) Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- formation on fusion channels and the topological nature of braiding makes TQC naturally robust against local perturbations, providing a complement to fault-tolerant quantum computation [? ? ]. In condensed matter, anyons emerge as excitations in systems that exhibit topological order [? ]. A possible way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not computationally powerful. Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a pr cess where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion = − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk+i+jZk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ S σ+ σ− cjck + ckcj = 2δjk c1c2c3 σ± × σ± = 1 + $ σ± × σ∓ = e+m σ± × $ =σ± σ± × e = σ± ×m = σ∓ (3) σ × σ = 1+ ψ σ × ψ = σ ψ × ψ = 1 (4) Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic atistics are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding d fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- formation on fusion channels and the topological natur of braiding makes TQC naturally robust against l cal perturbations, providing a complement to fault-tol rant quantum computation [? ? ]. In condensed matter, anyons emerge as excitations in systems that exhibit t pological order [? ]. A possible way to obtain these exotic phases i by engineering su - able Hamiltonians on attice spin systems [? ? ? ? ? ]. In ed, implement tions on optical lattices have bee proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not computationally powerful. Top logical Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Phy ics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, C n da Anyon models can be sy metric under some permutations of their t pological charges. One can then conceive t pological defects that, under m odromy, transform anyons according t a sy metry. We study th realization of such defects in he toric code model, showi g th t a process where defects are brai ed and fused has the same outcome as if they were I ing anyons. Thes ideas can also be applied in the context of t pological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 7 .43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . } Q′ Q×Q′ = q1 + q2 + . . |ψ〉 −→ ? 1 2q3 . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1 e, , $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ b sons $→ fermion = − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk i+jZk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ S σ+ σ− jck + kcj = 2δjk 1 2c3 σ± × σ± = 1 + $ σ± × σ∓ = e+m σ± × $ =σ± σ± × e = σ± ×m = σ∓ (3) σ × σ = 1+ ψ σ × ψ = σ ψ × ψ = 1 (4) Particle s a stics are particularly rich in two spatial di- mensions, wh re beyond the sual fe mion nd b sons th r exist more g nerally anyons (see [? ] for a com- pilation of the basic ref rences). Anyon c s a stics a e complex enough to give rise to the notion of t pological quantum compu ation (TQC) [? ? ? ], wh re compu- ations are carried out by braiding and fusing any s, see Fig. ??(a-c). The onlocal encoding of quantum in- formation on fusion channels and the t pologi al nature of braiding makes TQC naturally robust gainst local perturbations, providing a complement to fault-tolerant quantum compu ation [ ? ]. In condensed matter, anyons emerge as exci a ions in yst ms that exhibit t pological order [? ]. A possible way t obtain th se exotic phase i by engine ring suit- able Hamilton a s on lattice spin stems [? ? ? ? ? ]. Indeed, impl men ations on optical l ttices h ve been pr posed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not compu ationally powe f l. Top logical O der with a Twist: Ising A yons from an Abelian Model H. Bombin P rimeter Institute for Theoretical Phy ics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, C n da Anyon models can be sy metric under some permutations of their t pologic l charges. One can the conceive t pological defects that, under m nodro y, transform anyons according to a sy metry. We study th realization of such def cts in the toric cod model, showing th t a process wh r defects are brai ed and fused as the same outcome as if they were Ising anyons. These ideas can also be applied in the context of t pological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 7 .43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . } Q′ ×Q′ = q1 q2 + . . |ψ〉 −→ ? q1 2q3 . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 emRme = R !R!e = !R!m = −1 (1) {1 e,m, $} e×m = $ e× $ = m m× $ = e e× e = ×m = $× $ = 1 (2) e,m→ b sons $→ fermion = − 1 # # q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk i jZk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ S σ+ σ− jck + kcj = 2δjk c1 2c3 σ± × σ± = 1 + $ σ± × σ∓ = e+m σ± × $ =σ± σ± × e = σ± ×m = σ∓ (3) × σ = 1+ ψ σ × ψ = σ × ψ = 1 (4) Particle a stics are particularly rich in two spatial di- mensions, wh re beyond the sual fermio s and b sons ther exist more g ner ll anyons (see [? ] for a com- pilation of the basic ref rences). Anyonic a stics re complex enough to g v rise to the notion of t pological q antu compu ation (TQC) [? ? ? ], wh re compu- ations are carried out by braidi g and fusi g anyons, see Fig. ??(a-c). The onlocal encoding of q antum in- formation on fusion channels and the t pological nature of braiding makes TQC n turally robust ga nst local perturbations, providing a co plement to fau t-tolerant q antu compu ation [ ? ]. I condensed matter, anyons emerg as exci s in y tems that exhibit t pological order [? ]. A possible way t ob ain th se exotic phase is by engi eering suit- able Ha ltonia s on lattice spin y tems [? ? ? ? ? ]. Indeed, i plemen ati s on optical lattices have been pr p sed [? ]. Unfor unately, the anyon models that ap- pear in simple models are not compu ationally powerful. Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can the co ceive topologic l defects that, under monodromy, transform anyons according to a symmetry. We study the realizatio of such defects in th toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied i the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Re = m = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = ×m = $× $ = 1 (2) e,m→ bosons $→ fermion = − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk+i+jZk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ S σ+ σ− cjck + ckcj = 2δjk c1c2c3 σ± × σ± = 1 + $ σ± × σ∓ = e+m σ± × $ =σ± σ± × e = σ± ×m = σ∓ (3) σ × σ = 1+ ψ σ × ψ = σ ψ × ψ = 1 (4) Particle statistics are particularly rich in two spatial di- ensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- formation on fusion channels and the topological nature of braiding makes TQC naturally robust against local perturbations, providing a complement to fault-tolerant quantum computation [? ? ]. In condensed matter, anyons emerge as excitations in systems that exhibit topological order [? ]. A possible way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not computationally powerful. 2 (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g In this paper we address an strategy to recover compu- tationally in eresting nyon-like b havior from syste s with very simple anyonic statistics. Our starting point are the symmetries that anyons may exhibit. Anyon models have three ain ingredients: (i) a set of labels that identify the sup rsel ction sectors or topological charges, (ii) fusion/splitting rules that dictate the charges of composite systems, and (iii) braiding rules that dictate the effect of particle exchanges. A symme- try is a label permutation that leaves braiding and fusion rules unchanged —for a recent survey, see [? ]—. Given a symmetry s, we can imagine cutting the system along an open curve, as in Fig. ??(d), and then gluing it again “up to s”. Ideally the location of the cut itself is unphysical, only its endpoints have a measurable effect. In partic- ular, transporting an nyon around ne e d of the line ch nges the charge of the anyon according to the action of s. Our aim is to explore to which extent these topolo - ical defects, tha we call twists for short, can be “treat d as anyons” and used in TQC. Twists are being indepen- dently studied by Kong and Kitaev [? ]. An interesting precedent are the Alice strings appearing in some gauge (a) (b) (b) (c) (a) s sm (e) models [? ], which can cause charge conjugation under monodromy, whereas the twists that we will discuss here exchange electric and magnetic charges. Rather than trying a general, abstract approach, we will focus on a well-known spin model, the toric code model, and address twists constructively. In this model anyons have no computational power, but we will show that twists behave as Isi g anyons [? ], which are com- putati nally interesting. In fact, they do n t directly allow universal co putation, but ther ex st str tegies to overcome this difficulty [? ? ? ]. In [? ], Wootton et al. also try to mimic the non-abelian behavior in an abelian system, using an entirely different approach and philosophy. We remark that, although the discussion will mainly be in terms of topological order, it has direct application in the closely related context of topological codes [? ? ? ? ]. Anyon models— Anyon models are mathematically characterized by modular tensor categories, but we will not need such generalities (for an introduction, see for example [? ]). Instead, we will illustrate the content of anyon models with an example: Ising anyons. The first elem t of an anyon model is a set of la- bels that ide tify the superselection sec ors or topological charges of the model. For Ising anyons there are hree: 1, σ and ψ. Any given anyon carries such a charge, which cannot be changed locally. We can also attach a charge to a set of anyons or a to a given region. A region without 2 (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g r g gr b b r r g b r g b b g br r g g b g I this paper we address strategy to recover compu- tationally interesting anyon-like behavior from systems with very simple anyonic statistics. Our starting point are the symmetries that anyons may exhibit. Anyon models have three main ingredients: (i) a set of labels that identify the superselection sectors or topological charges, (ii) fusion/splitting rules that dictate the charges of composite systems, and ( ii) braiding rules that dictate the effect of particle exchanges. A symme- try is a label permutation that leaves braiding and fusion rules unchanged —for a recent survey, see [? ]—. Given a symmetry s, we can imagine cu ting the system along an open curve, as in Fig. ??(d), and then gluing it again “up to s”. Ideally the location of the cut itself is unphysical, only its endpoints have a measurable effect. In partic- ular, transporting an anyon around one end of the line changes the charge of the anyon according to the action of s. Our aim is to explore to which extent these topolog- ical defects, that we call t ists for short, can be “treated as anyons” and used in TQC. Twists are being indepen- dently studied by Kong and Kitaev [? ]. An interesting precedent are the Alice strings appearing in some gauge (a) (b) (b) (c) (a) s sm (e) models [? ], which can cause charge conjugation under monodromy, whereas the twists that we will discuss here exchange electric and magnetic charges. Rather than trying a general, abstract approach, we will focus on a well-known spin model, the toric code model, and address twists constructively. In this model anyons have no computational power, but we will show that twists behave as Ising anyons [? ], which are com- putationally interesting. In fact, they do not directly allow universal computation, but there exist strategies to overcome this difficulty [? ? ? ]. In [? ], Wootton et al. also try to mimic the non-abelian behavior in an abelian system, using an entirely different approach and philosophy. We r mark that, although the discussion will mainly be in terms of topological order, it has direct application in the closely related context of topological codes [? ? ? ? ]. Anyon models— Anyon models are mathematically characterized by modular tensor categories, but we will not need such generalities (for an introduction, see for example [? ]). Instead, we will illustrate the content of anyon models with an example: Ising anyons. The first element of an anyon model is a set of la- bels that identify the superselection sectors or topological charges of the model. For Ising anyons there are three: 1, σ and ψ. Any given anyon carries such a charge, which cannot be changed locally. We can also attach a charge to a set of anyons or a to a given region. A region without Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models c n be symmetric u der some permutations of th ir topol gical charges. One can then conceive top logical defects tha , under monodromy, tr nsform any s according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = R m = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion = − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk+i+jZk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ S σ+ σ− cjck + ckcj = 2δjk c1c2c3 ± × σ± = 1 + $ σ± × σ∓ = e+m σ± × $ =σ± σ± × e = σ± ×m = σ∓ (3) σ × σ = 1+ ψ σ × ψ = σ ψ × ψ = 1 (4) cj → cj+1 cj+1 → −cj ck → ck Particle statistics are particularly rich in two spatial di- m nsions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum omputation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusi g anyons, se Fig. ??(a-c). The nonlocal encodi g f quantum in- formation on fusion channels and the topological nature of braiding makes TQC naturally robust against local Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e × e = m×m = $× $ = 1 (2) e,m→ bosons $→ fer ion = − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk+i+jZk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ S σ+ σ− cjck + ckcj = 2δjk c1c2c3 σ± × σ± = 1 + $ σ± × σ∓ = e+m σ± × $ =σ± σ± × e = σ± ×m = σ∓ (3) σ × σ = 1+ ψ σ × ψ = σ ψ × ψ = 1 (4) cj → cj+1 cj+1 → −cj ck → ck Pa ti le statistic are particul rly rich in two spatial di- mensi ns, wher bey d the usual fermions and bosons here exist more genera ly a yons (see [? ] for a com- pilation of the bas c referen es). A ic statistics are omplex nough to g ve r se t the otion of topological quantum computation (TQC) [? ? ? ], where compu- tation are c ried out y braid g and fusing anyons, see F g. ??(a-c). The nonloca encodi g of qua tum in- formation on fusion channels and the topologi al nature of braiding makes TQC natur lly robust against local Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e, , $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion = − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk+i+jZk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ S σ+ σ− cjck + ckcj = 2δjk c1c2c3 σ± σ± = 1 + $ σ± × σ∓ = e+m σ± × $ =σ± σ± × e σ± ×m = σ∓ (3) σ × σ = 1+ ψ σ × ψ = σ ψ × ψ = 1 (4) cj → cj+1 cj+1 → −cj ck → ck Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions an bosons there exist mor g nerally anyons (see [? ] f r a com- pilation of the asic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum c mput tion (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- for ation o fusio channels and the t pological nature of braiding makes TQC naturally robust against local Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform any ns accordin to a symmetry. We study the realization of such defects in the toric code model, showing th t a process where defects are braided and fused has the same outcome as if they were Ising anyons. T ese ideas can also b applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 z 1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q× ′ = q1 + q2 . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm! !m −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion = − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk+i+jZk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ S σ+ σ− cjck + ckcj = 2δjk c1c2c3 σ± × σ± = 1 + $ σ± × σ∓ = e+m σ± × $ =σ± σ± × e = σ± ×m = σ∓ (3) σ × σ = 1+ ψ σ × ψ = σ ψ × ψ = 1 (4) cj → cj+1 cj+1 → −cj ck → ck Par icle s atistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exis more generally anyons (see [? ] for a com- pilation of the basic referenc s). Any ic statistics are complex enough to give rise to the notion f topological quantum computation (TQC) [? ? ? ], where compu- tation are carried out by braiding a d fusi g a y s, see Fig. ??(a-c). The nonlocal encoding of quantum in- formation on fusion channels and the topological nature of braiding makes TQC naturally robust against loc l TSC twists Charges: Like in toric code, but all are fermions Symmetries: any permutation of fermions Non-commutative fusion rules! (unlike anyons) Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion = − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk+i+jZk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ S σ+ σ− cjck + ckcj = 2δjk c1c2c3 σ± × σ± = 1 + $ σ± × σ∓ = e+m σ± × $ =σ± σ± × e = σ± ×m = σ∓ (3) σ × σ = 1+ ψ σ × ψ = σ ψ × ψ = 1 (4) cj → cj+1 cj+1 → −cj ck → ck {1, r, g, b} Γ = {e, ζ+, ζ−,σr,σg,σb} Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological Topological O der with a Twist: Ising A yons from an Abelian Model H. Bombin P rimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, C nada Anyon models can be symmetric under some permutations of their topologic l charges. One can the conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defec s in the toric code model, showing that a process wh r defects are braided and fused as the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.3 .Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ ×Q′ = 1 q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 emRme = R !R!e = !R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = ×m = $× $ = 1 (2) e,m→ bosons $→ fermion = − 1 # q # q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk+i Zk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ S σ+ σ− cjck + ckcj = 2δjk c1c2c3 σ± × σ± = 1 + $ σ± × σ∓ = e+m σ± × $ =σ± σ± × e = σ± ×m = σ∓ (3) × σ = 1+ ψ σ × ψ = σ × ψ = 1 (4) cj → cj+1 cj+1 → −cj ck → ck {1, r, g, b} Γ = {e, ζ+, ζ−,σr,σg,σb} Particle atistics re particularly rich in two spatial di- mensions, where beyond the usual fermio s and bosons there exist more gener ll anyons (see [? ] for a com- pilation of the basic references). Anyonic atistics are complex enough to g v rise to the notion of t pological Topological O der with a Twist: Ising A yons from an Abelian Model H. Bombin Perimet r Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, C nada Any n model can be symmetric under some permutations of the r topolo ic l charges. One a then conceive topological defects that, un er monodromy, transform anyo s according to a symmetry. W study the realization of such defec s in the t ric code model, showing that a process wh re defects are braid and fused a the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbe s: 05.3 .Pr, 03.6 .Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ ×Q′ = 1 q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 emRme = Re!R!e = !R!m = − (1) {1, e,m, $} e×m = e× $ = m m× $ = e e e = m = $× $ = 1 (2) e,m→ bosons $→ fermion = − 1 # q # q′ #= 1 e↔ m X Z Y C := XkZ iXk+i Zk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ S σ+ σ− + c cj = 2δjk c1c2c3 σ± × σ± = 1 + $ σ± × σ∓ = e+m × $ =σ± σ± e = σ± ×m = σ∓ (3) σ × σ = 1+ ψ σ × ψ = σ × ψ = 1 (4) cj → cj+1 cj+1 → −cj ck → ck {1, r, g, b} Γ = {e, ζ+, ζ−,σr,σg,σb} Particle atistics e particularly rich in two spatial di- mensions, where b yond the usual fermions and bosons there xist more ge er ll anyons (see [? ] for a com- pilation of the basic refere ces). Anyonic atistics are complex enough to give rise o the ti n of topological Topological O der with a Twist: Ising A yons from an Ab lian Model H. Bombin Perime r Institute for Theoretical Physics, 31 Caroline St. N., Wa erloo, Ontario N2L 2Y5, C nada Anyon model can be symmetric und s me permutations of the r topolo ic l charges. One a hen conceive topological def cts that, un er mon dromy, transform anyo s according to a symmetry. W study he realization of such defe s in th t ric code model, showing that a process wh fects are braid and fus d as th same outcom as if the were Ising anyons. These ideas can also b applied in he c ntext f topological codes. PACS numbe s: 05.3 .Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ ×Q′ = 1 q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = mm = R!! = −1 emRme = e!R!e = !R!m = − (1) {1, e,m, $} e×m e× $ = m m× $ = e e e m = $× $ = 1 (2) e,m→ bosons $→ fermion = − 1 # q # q′ #= 1 e↔ m X Z Y C := XkZk iXk+i Zk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ S σ+ σ− cj + j = 2δjk c1c2c3 σ± × σ± = 1 + $ σ± × σ∓ = e+m × $ =σ± e = σ± ×m = σ∓ (3) × σ = 1+ ψ σ × ψ = σ × ψ = 1 (4) cj → cj+1 j+1 → −cj ck → ck {1, r, g, b} Γ = {e, ζ+, ζ−,σr,σg,σb} Particle atistics e particularly rich in two spatial di- m nsions, w re beyond the usual fermio s and bosons there xist more gener ll anyons (see [? ] for a com- pilation o the basic refere ces). Anyonic atistics are complex nough to g ve rise o he n t n of t pological Topological Order wi h a Twist: Ising A yo s from an Ab ian Model H. Bombin Perimet r Institute for Theoret al Physics, 31 Carolin St. N., Waterlo , Ontario N2L 2Y5, Canada Anyon mod ls can be symm tric under s me permuta i ns f their t pologi al charges. One can the c nceive t pological defects that, under onodromy, tr sform anyons according to a symme ry. We study the r alization of such def c s n the t ric code model, showing that a process where defects are braid nd fus d has the sa utcome a if th y were Isi g anyons. These ideas can also be applied in the contex of t p logical codes. PACS numbers: 5. 0.Pr 0 67 Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? 1q2q3 . . . −→ pi(q) Ree Rmm = 1 R!! = −1 e e = e R!e = m!R!m = −1 (1) {1, e,m, $} e× $ e× $ = m m× $ = e e m m = × $ = 1 (2) e,m→ bos s $→ fermion = − # #= q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk+i+jZk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ S σ+ σ− cjck + ckcj = 2δjk 1c2c3 σ± × σ± = 1 + $ σ± × σ∓ = e+m σ × $ =σ± σ± × e = σ± ×m = σ∓ (3) × σ = 1+ ψ × ψ = σ × ψ = 1 (4) cj → cj+1 cj+1 → −cj ck → ck {1, r, g, b} Γ = {e, ζ+, ζ−,σr,σg,σb} P rticl sta stics are parti ul rly rich in two spatial di- m ions, where b yond th usual fermio s and bosons th re exist m re ge erally a yons (see [? ] for a com- p lation of th basi feren e ). A yonic statistics are complex noug to giv rise t the notion of t pological Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion = − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk+i+jZk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ S σ+ σ− cjck + ckcj = 2δjk c1c2c3 σ± × σ± = 1 + $ σ± × σ∓ = e+m σ± × $ =σ± σ± × e = σ± ×m = σ∓ (3) σ × σ = 1+ ψ σ × ψ = σ ψ × ψ = 1 (4) cj → cj+1 cj+1 → −cj ck → ck {1, r, g, b} Γ = {e, ζ+, ζ−,σr,σg,σb} Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological Topological Order with a Tw st: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion = − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk+i+jZk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ S σ+ σ− cjck + ckcj = 2δjk c1c2c3 σ± × σ± = 1 + $ σ± × σ∓ = e+m σ± × $ =σ± σ± × e = σ± ×m = σ∓ (3) σ × σ = 1+ ψ σ × ψ = σ ψ × ψ = 1 (4) cj → cj+1 cj+1 → −cj ck → ck {1, r, g, b} Γ = {e, ζ+, ζ−,σr,σg,σb} Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be sy metric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. W tudy the realization of such def cts in the t ric code model, showing t t a process where defects are braided and fused has the same outcome as if they were Ising a yons. These ideas can also be applie in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion = − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk+i+jZk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ S σ+ σ− cjck + ckcj = 2δjk c1c2c3 σ± × σ± = 1 + $ σ± × σ∓ = e+ σ± × $ =σ± σ± × e = σ± ×m = σ∓ (3) σ × σ = 1+ ψ σ × ψ = σ ψ × ψ = 1 (4) cj → cj+1 cj+1 → −cj ck → ck {1, r, g, b} Γ = {e, ζ+, ζ−,σr,σg,σb} Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological opological rder ith a ist: Ising nyons fro an belian odel H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., a erl o, Ontario N2L 2Y5, Ca ada Anyon models can b symmetric under some permutations f their topol gical charges. One can then onceive topological d fects that, under m nodromy, transform anyons a cording to a symmetry. e study the realization of such defe ts in the toric code model, showing that a proce s where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 − ? q1q2q3 . . . q − pi(q) R e = Rmm = 1 R ! = −1 RemRme = Re!R!e = Rm!R!m −1 (1) {1, e, , $} e× = $ e× $ = × $ = e e× e = × = $× $ = 1 (2) e, bosons $ fermion = − 1 #= q #= q′ #= 1 e X Z Y Ck := XkZk+iXk+i+jZk+j. H := − k Ck, Se Sm − 1 1 1 e $ S σ+ σ− cjck + ckcj = 2δjk c c2c3 σ± × σ± = 1 + $ σ± × σ∓ = e+ σ± × $ =σ± σ± × e = σ± × = σ∓ (3) σ × σ = 1+ ψ σ × ψ = σ ψ × ψ = 1 (4) cj cj+1 cj+1 −cj ck ck {1, , g, b} Γ = { , ζ+, ζ−,σr,σg,σb} Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more genera ly anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological 2 (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- formation on fusion channels and the topological nature of braiding makes TQC naturally robust against local perturbations, providing a complement to fault-tolerant quantum computation [? ? ]. In condensed matter, anyons emerge as excitations in systems that exhibit topological order [? ]. A possible way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not computationally powerful. In this paper we address an strategy to recover compu- tationally interesting anyon-like behavior from systems with very simple anyonic statistics. Our starting point are the symmetries that anyons may exhibit. Anyon models have three main ingredients: (i) a set of labels that identify the superselection sectors or topological charges, (ii) fusion/splitting rules that dictate the charges of composite systems, and (iii) braiding rules (a) (b) (b) (c) (a) s sm (e) that dictate the effect of particle exchanges. A symme- try is a label permutation that leaves braiding and fusion rules unchanged—for a recent survey, see [? ]—. Given a symmetry s, we can imagine cutting the system along an open curve, as in Fig. ??(d), and then gluing it again “up to s”. Ideally the location of the cut itself is unphysical, only its endpoints have a measurable effect. In partic- ular, transporting an anyon around one end of the line changes the charge of the anyon according to the action of s. Our aim is to explore to which extent these topolog- ical defects, that we call twists for short, can be “treated as anyons” and used in TQC. Twists are being indepen- dently studied by Kong and Kitaev [? ]. An interesting precedent are the Alice strings appearing in some gauge models [? ], which can cause charge conjugation under monodromy, whereas the twists that we will discuss here exchange electric and magnetic charges. Rather than trying a general, abstract approach, we will focus on a well-known spin model, the toric code model, and address twists constructively. In this model anyons have no computational power, but we will show that twists behave as Ising anyons [? ], which are com- putationally interesting. In fact, they do not directly allow universal computation, but there exist strategies to overcome this difficulty [? ? ? ]. In [? ], Wootton et al. also try to mimic the non-abelian behavior in an abelian system, using an entirely different approach and philosophy. We remark that, although the discussion will mainly Topological Order with a Tw st: Ising Any ns from an Abelian Model H. Bombin Peri et r Institut for Theoretical Physics, 31 Caroline St. N., Waterl o, Ontario N2L 2Y5, Canada Anyon models c n b symmet ic under some pe muta ions of their topological charges. One can th n conc ive topologi al defects that, u de onodromy, tr sf rm nyons acc rding to a s mmetry. We study he realization o such defects in the toric code m del, s owing that a process where d fects are braided and fused has the sa e outcom as if they were Isi g anyons. These ideas can lso be appli d i the context f topological codes. PACS numbers: 5.30.Pr, 03.67.Lx, 73.43.Cd C = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . −→ pi(q) Ree Rmm = 1 R!! = −1 Rem me = Re!R e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m $× $ = 1 (2) e,m→ bosons $→ fermion = − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := Zk iXk+i+jZk+j. H := − ∑ k Ck, e Sm − 1 1 em $ S + σ− cjck + ckcj = δjk c c2c3 σ± × σ± = 1 + $ σ± × σ∓ = e+m σ± × $ = σ± × e = σ± ×m = σ∓ (3) σ × σ = 1+ ψ σ × ψ = σ ψ × ψ = 1 (4) cj → cj+1 cj+1 → −cj ck → ck {1, r, g, b} Γ = {e, ζ+, ζ−,σr,σg,σb} Par icle t tistics are particularly rich in wo spatial di- mensions, where beyond the usual fermions a d bosons there exist mo e generally anyons (see [? ] for a com- pilation of the basic references). Anyonic t tistics are complex enough to give rise t the no ion f topological Topological Order with a Tw st: Ising Any ns from an Abelian Model H. Bombin Perimeter Institut for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmet ic under some permut ions of their topological ch rges. One can then conceive topological defects that, u de onodromy, tr sf rm anyons acc rding to a sy m try. W tudy the ealization such def c s i the r c c de mode , s owing t t a process where defects are br ided and fused has the sa e outcom s i they were Ising a yons. These ideas can al o be applie n he c ntext of topological c des. PACS numbers: 5.30.Pr, 03.67.Lx, 73.43.Cd C = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . −→ pi(q) Ree Rmm = 1 R!! = −1 RemRme = Re!R e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m $× $ = 1 (2) e,m→ bosons $→ fermion = − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := Zk+iXk+i+jZk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ S σ+ σ− cjck + ckcj = δjk c1c2c3 σ± σ± = 1 + $ ± × σ∓ = e+m σ± × $ =σ± σ± × e = σ± ×m = σ∓ (3) σ × σ = 1+ ψ σ × ψ = σ ψ × ψ = 1 (4) cj → cj+1 cj+1 → −cj ck → ck {1, r, g, b} Γ = {e, ζ+, ζ−,σr,σg,σb} Particle tatistics are particularly rich in two spatial di- mensions, where beyond the usual fermions a d bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic t tistics are complex enough to give rise to the notion f topological opological rd r ith t: Ising ny ns fro an belian odel H. Bombin Perimeter Institut for Theoreti al Physics, 31 Caroli e St. N., a erl , Ontario N2L 2Y5, Ca da Anyon mod ls can b symm t ic under som permuta ions f t eir topol gical charges. One can then nceive topologic l d fects that, u e nodromy, tr sf rm anyon a c rding to a sym e ry. e study the realizati n o such def ts n the toric code model, s owing that a proce s where defects are braided and fus d has the sa outcom as if they were Ising anyons. These ideas can also be applied in the context of top logical codes. PACS numbers: 5.30.Pr, 0 .67.Lx, 73.43.Cd C = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 − ? q1q2q3 . . . − pi(q) R e Rmm = 1 R ! = −1 RemRme Re!R e = Rm!R!m = −1 (1) {1, e, , $} e× = $ e× $ = × $ = e e e = $× $ = 1 (2) e, bosons $ fermion = − 1 # q #= q′ #= 1 e X Z Y Ck := Zk+iXk+i+jZk+j. H := − k Ck, Se Sm − 1 1 1 e $ S σ+ σ− jck + ckcj δjk c c2c3 σ± × σ± = 1 + $ σ± × σ∓ = e+ × $ =σ± σ± × e = σ± × = σ∓ (3) σ × σ = 1+ ψ × ψ = σ ψ × ψ = 1 (4) cj cj+1 cj+1 −cj ck ck {1, r, g, b} Γ = { , ζ+, ζ−,σr,σg,σb} P rticle tatistics are part ularly rich in two spatial di- mensions, where b yond th usual fermion a d bosons there exist more ge era ly anyons (see [? ] for a com- pilation of th basi refere ces). Anyon t tistics are complex enough to give rise to the notion f topological 2 (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g g r b b r r g b r g b b g b r r g g b g quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- formation on fusion channels and the topological nature of braiding makes TQC naturally robust against local perturbations, providing a complement to fault-tolerant quantum computation [? ? ]. In cond nsed matter, anyons emerge as excitations in systems that exhibit topological order [? ]. A possible way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not computationally powerful. In this paper we address an strategy to recover compu- tationally interesting anyon-like behavior from systems with very simple anyonic statistics. Our starting point are the symmetries that anyons may exhibit. Anyon models have three main ingredients: (i) a set of labels that identify the superselection sectors or topological charges, (ii) fusion/splitting rules that dictate the charges of composite systems, and (iii) braiding rules (a) (b) (b) (c) (a) s sm (e) that dictate the effect of particle exchanges. A symme- try is a label permutation that leaves braiding and fusion rules unchanged—for a recent survey, see [? ]—. Given a symmetry s, we can imagine cutting the system along an open curve, as in Fig. ??(d), and then gluing it again “up to s”. Ideally the location of the cut itself is unphysical, only its endpoints have a measurable effect. In partic- ular, transporting an anyon around one end of the line changes the charge of the anyon according to the action of s. Our aim is to explore to which extent these topolog- ical defects, that we call twists for short, can be “treated as anyons” and used in TQC. Twists are being indepen- dently studied by Kong and Kitaev [? ]. An interesting precedent are the Alice strings appearing in some gauge models [? ], which can cause charge conjugation under monodromy, whereas the twists that we will discuss here exchange electric and magnetic charges. Rather than trying a general, abstract approach, we will focus on a well-known spin model, the toric code model, and address twists constructively. In this model anyons have no computational power, but we will show that twists behave as Ising anyons [? ], which are com- putationally interesting. In fact, they do not directly allow universal computation, but there exist strategies to overcome this difficulty [? ? ? ]. In [? ], Wootton et al. also try to mimic the non-abelian behavior in an abelian system, using an entirely different approach and philosophy. We remark that, although the discussion will mainly Topo ogical Order with a Tw st: I ing An ns from an Abelian Model H. Bo bin P ri e r Ins tut for Theoretic l Physics, 31 Caroline St. N., Waterl o, Ont rio N2L 2Y5, Canada Anyon od ls n b sym et ic under some permu a i ns of t ei to ological charges. One can th n onc iv opologi al fects th t, u de nodromy, tr sf rm nyons acc rding to a s mmet y. W s udy he realiza ion o such defects in th toric code m del, s owing that a process where fects ar br ided and fused has the sa outcom as if they were Isi g anyons. These ideas can lso be appli d i the context f topological codes. PACS numbers: 5 0 Pr, 03.67.Lx, 73.43.Cd C = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . −→ pi(q) Ree Rmm = 1 R!! = −1 m me = Re!R e = Rm!R!m −1 (1) {1, e,m, $} e×m = $ e m m× $ = e e× e = m×m $× $ = 1 (2) e,m bosons $→ fermion = − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := Zk iXk+i+jZk+j. H := − ∑ k Ck, e Sm − 1 1 em $ S + σ− cjck + kcj = δjk c1c2c3 σ± × = 1 + $ σ± × σ∓ = e+m $ σ± × e = σ± ×m = σ∓ (3) σ × σ = 1+ ψ σ ψ σ ψ × ψ = 1 (4) cj → cj+1 cj+1 → −cj ck → ck {1, r, g, b} Γ = {e, ζ+, ζ−,σr,σg,σb} P cle tistics are par icularly rich in wo spatial di- mensi s, where beyond the usual fermions a d bosons there exist mo e gen rally any ns (see [? ] for a com- pila ion of th basic references). Anyonic t tistics are complex enough to give rise t the no ion f topological Topological Order with a Tw st: Ising Any ns from an Abelian Model H. Bombin Peri eter Institut for Theoretical Physics, 31 Caroline St. N., Waterloo, Ont rio N2L 2Y5, Canada Anyon mod ls an b symmet ic under some permut ions of t ei topologic l ch rges. One can then onceive topological defects that, u de nodromy, tr sf rm anyons according to a symm t . W tudy th realiza ion such def c s i the r c de mod , s owing t t a process where efe ts ar br ided and fused has the sa outc s i they were Ising a yons. These ideas can al o be applie n he c ntext of opol gical c des. PACS numbers: 5 30 Pr, 03.67.Lx, 73.43.Cd C = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . −→ pi(q) Ree Rmm = 1 R!! = −1 R mRme = Re!R e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m $× $ = 1 (2) e,m bosons $→ fermion = − 1 #= q #= q′ #= 1 e↔ m X Z Y C := kZk+iXk+i+jZk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ S + σ− cjck + kcj = δjk c1c2c3 σ± × σ± = 1 + $ σ± × σ∓ = e+m $ σ± σ± × e = σ± ×m = σ∓ (3) σ × σ = 1+ ψ σ × ψ = σ ψ × ψ = 1 (4) cj → cj+1 cj+1 → −cj ck → ck {1, r, g, b} Γ = {e, ζ+, ζ−,σr,σg,σb} P ticle tatistics are particularly rich in two spatial di- mensions, where beyond the usual fermions a d bosons there exist more gen rally anyons (see [? ] for a com- pilation of th basic references). Anyonic t tistics are complex enough to give rise to the notion f topological opol g cal rder ith a st: Ising ny ns fro a bel an odel H. Bombin Perimeter Institu for Theoreti al Physics, 31 C roli e St. N., a erl , Ont ri N2L 2Y5, Ca ada Anyon od ls a b y m t ic nder om p rmuta i ns f ei top l gical charges. One can th n nc ive topologic l d fe ts that, u e nodromy, tr sf rm anyon a cording to a symm y. e study the realiza i o such def ts n the toric code model, s owing that a proce s wher efects ar br ided and fus d has the sa outcom as if they were Ising nyons. These ideas ca also be applied in the context of top logical codes. PACS numbers: 5 30 Pr, 0 .67.Lx, 73.43.Cd C = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 − ? q1q2q3 . . . − pi(q) R e Rmm = 1 R ! = −1 R mRme = Re!R e = Rm!R!m = −1 (1) {1, e, , $} e× = $ e× $ = × $ = e e e = $× $ = 1 (2) e, → bosons $ fermion = − 1 # q #= q′ #= 1 e X Z Y Ck := kZk+iXk+i+jZk+j. H := − k Ck, Se Sm − 1 1 1 e $ S σ+ σ− j k + kcj = δjk c c2c3 σ± × σ± = 1 + $ σ± × σ∓ = e+ $ σ± × e = σ± × = σ∓ (3) σ × σ = 1+ ψ × ψ = σ ψ × ψ = 1 (4) cj cj+1 cj+1 −cj ck ck {1, r, g, b} Γ = { , ζ+, ζ−,σr,σg,σb} P ticle tatistics are part ularly rich in two spatial di- m nsions, where b yond th usual fermion a d bosons th r exist more ge ra ly anyons (see [? ] for a com- pilation of th basi refere ces). Anyon t tistics are c mpl x enough to give rise the notion f topological Topo ogical Or er with a Tw st: I i g Any s fr m an Abelian Model H. Bo bin P ri e er Ins tut for Th o tic l Physics, 31 Carol ne St. N., Water o, Ont rio N2L 2Y5, Canada Anyon mod ls n b sy e ic un er s e e mu ati ns of thei to ologic l charg s. One can th n onceiv opologi al f cts h t, u de odromy, tr nsf rm ny ns acc rding to a s m ry. We s u y he realiza i n of such def ct in th toric code m del, s owing that a process where fect ar br id d and fused has the sa outcom a if t y w re I i g anyons. These ideas can lso be a li d i the context f topol gical codes. PACS numbers: 05. 0.Pr, 03.67.Lx, 73.43.Cd C = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree Rmm = 1 R!! = −1 m me = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ m m× $ = e× e = m×m $× $ = 1 (2) , bosons $→ fer ion = − 1 #= q #= q′ #= 1 e↔ X Z Y Ck := Zk iXk+i+jZk+j. H := ∑ k Ck, e Sm − 1 1 em $ S + σ− c k + kcj = δjk c1c2c3 σ± × = 1 + $ σ± × σ∓ = e+m σ $ σ± × e = σ± ×m = σ∓ (3) σ × = + ψ σ σ ψ × ψ = 1 (4) cj → cj+1 cj+1 → −cj ck → ck {1, r, g, b} Γ = {e ζ+, ζ−,σr,σg,σb} P icle tist cs are r icularly rich in wo spatial di- mensi s, where beyo the usual fermions a d bosons there exist mo e gen rally any ns (see [? ] for a com- pila ion of th basic r ferences). Anyonic t tistics are complex enough to give rise t the no ion f topological Topo ogical Order with a Tw st: Isi g Any s fr m an Abelian Model H. Bombin Perime er Inst tut for Theor tical Physics, 31 Caroline St. N., Waterloo, Ont rio N2L 2Y5, Canada Anyon mod ls an b sy me ic und s me pe mut tions of thei topological ch rges. One can th n onceiv opological efects h t, u de odromy, tr nsf rm anyons acc rding to a . We tu y th ealiz i f suc def ct i the ric ode mod , s owing t t a process where fe t ar br i d and fused has the am outc s i they w e I ing a yons. These ideas c n ls be a plie i the c nt x of pol gical codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd C = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree Rmm = 1 R!! = −1 R m me = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m $× $ = 1 (2) , bosons $→ fer ion = − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := Zk+iXk+i+jZk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ S σ+ σ− cjck + kcj = δjk c1c2c3 σ± × σ± = 1 + $ σ± × σ∓ = e+m σ $ σ σ± × = σ± ×m = σ∓ (3) σ × σ = 1+ ψ σ × ψ = σ ψ × ψ = 1 (4) cj → cj+1 cj+1 → −cj ck → ck {1, r, g, b} Γ = {e, ζ+, ζ−,σr,σg,σb} P icle t tistics are rticularly rich in two spatial di- mensions, where beyo the usual fermions a d bosons there exist ore gen rally anyons (see [? ] for a com- pila ion of th basic references). Anyonic t tistics are complex enough to give rise to the notion f topological opo og cal rd r ith a st: Isi g ny s fr an belian odel H. Bombin P ri r In t tu f r Theor t l Physi s, 31 C roli e St. N., a erl , O t rio N2L 2Y5, Ca ada Anyon o ls n b sy m c n r som p rmuta i ns f i top l gical charges. One can th n nc iv opologic l fe ts h t, u de dromy, tr sf rm anyons a c rding to a sy m ry. stu y the r aliza i of such def n the oric code model, s owing that a proce s wher efect ar br ided and fus d has the sa outcom as if th y w re I ing nyons. These ideas a also be a plied i the context of top logical codes. PACS numbers: 5 30 Pr 0 67 Lx, 73.43.Cd C = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 − ? q1q2q3 . . . − pi(q) R e Rmm = 1 R ! = −1 R m me = Re!R!e = Rm!R!m = −1 (1) {1, e, , $} e× = e× $ = × $ = e e e = $× $ = 1 (2) , → bosons $ fer ion = − 1 # q #= q′ #= 1 e m X Z Y Ck := Zk+iXk+i+jZk+j. H := − k Ck, Se Sm − 1 1 1 e $ S σ+ σ− cj k kcj = δjk c c2c3 σ± × σ± = 1 + $ σ± × σ∓ = e+ $ σ σ± × e = σ± × = σ∓ (3) σ × σ = 1+ ψ × ψ = σ ψ × ψ = 1 (4) cj cj+1 cj+1 −cj ck ck {1, r, g, b} Γ = { , ζ+, ζ−,σr,σg,σb} P icle t tistics are rti ularly rich in two spatial di- m nsions, where beyo th u ual fermions a d bosons th r exi t more ge ra ly anyons (see [? ] for a com- pila ion of th basi references). Anyonic t tistics are c mpl x nough t give rise the notion f topological T ological Order with a Twist: I i g ns fr m an Abelian Model H. Bombi P eter In titut for Th retic l Phy ics, 31 Carolin St. N., Waterloo, Ont ri N2L 2Y5, Canada A yon m d ls can b sy tric under s e er u ti ns of their topol gic l charges. One c th n onc iv t p l gi l fects that, und m nod om , tra sf rm ny n according to a s mm ry. W s udy realizati n f su h defects in h tori code m del, sh wing that a process where d fects ar br id d and fused h s the me utcom a if t y were Isi g anyons. These ideas can ls be ap li i he c n ext of topol gical codes. PACS numbers: 05. 0.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) ee = Rmm = 1 R!! = −1 em e = e! !e = Rm!R!m = −1 (1) {1, e,m, $} m = $ m m× $ = e× e = m×m $× $ = 1 (2) e, bosons $→ fer i n = − 1 #= q #= q′ #= 1 e↔ m X Y C := Zk iX i+jZk+j. H := − ∑ k Ck, e Sm − 1 1 em $ S + σ− k + ckcj = 2δjk c1c2c3 σ± × = 1 + $ σ± × σ∓ = e+m σ± × $ = σ± × e = σ± ×m = σ∓ (3) σ × = 1+ ψ σ σ ψ × ψ = 1 (4) cj → cj+1 cj+1 → −cj ck → ck {1, r, g, b} Γ = {e, ζ+ ζ− r,σg,σb} Pa icle s ti t cs e p icularly rich in wo spatial di- mensi , wh e beyond the usual fermions and bosons ther xist mo generally any ns (see [? ] for a com- pil tion of the basi r f rences). Anyonic statistics are complex e ugh give ris t he no i n of topological Topological Order with a Twist: I ing ny ns fr m an Abelian Model H. Bombin P rimeter Institute for Theoretic l Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada A yon mod s can b sy r c under some mu i ns of their topological ch rges. One can th n onceive t p logi al fects that, unde monodromy, transf rm nyons according to a y. W udy the alization f suc def cts h t ric code m d l, showing t t a process where are br id and fus d has the me out me i t y were I i g a yo s. These ideas can l b ap li i the con x f topol gical codes. PACS numbers: 05. 0 Pr, 03.67.Lx, 73.43 Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q → pi(q) Ree = Rmm = 1 R!! = −1 em me = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ m m× $ = e× e = m×m $× $ = 1 (2) e, bosons $→ fer ion = − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := kZk iXk+i+jZk+j. H := − ∑ k Ck, e Sm − 1 1 em $ S + σ− c k ckcj = 2δjk c1c2c3 σ± × = 1 + $ σ± × σ∓ = e+m σ± × $ = σ± × e = σ± ×m = σ∓ (3) σ × = 1+ ψ σ σ ψ × ψ = 1 (4) cj → cj+1 cj+1 → −cj ck → ck {1, r, g, b} Γ = {e, ζ+, ζ−,σr,σg,σb} Par icle s ti t cs re par icularly rich in wo spatial di- mensi s, where beyond the usual fermions and bosons there xist mo e generally any ns (see [? ] for a com- pilation of the basic ref rences). Anyonic statistics are complex en ugh t give rise t the no ion of topological p logical rder ith a st: I i g A y s f an belian odel H. Bo bi P ri ter Inst tut for Th or tic l Physic , 31 C roline St. N., erl o, Ont io N2L 2Y5, Ca ada A yon mod l n b sy e ic under s m e mu a i ns f thei to ol gical charges. One can n o c iv op logi l f cts h t, u de odromy, tr sf rm nyons a c rding to a s mme ry. s u y he realiza i n suc def t in h t ric code m del, s owing that a proce s w ere fect ar br id d and fused has the s e outcome a if t y w re I i g anyons. These ideas can ls be li d i the context f topol gical codes. PACS numbers: 5. 0.Pr, 03.67.Lx, 73.43.Cd C = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 − ? q1q2q3 . . . − pi(q) R e Rmm = 1 R ! = −1 m me = Re!R e = Rm!R!m = −1 (1) {1, e, , $} e× = $ × $ = e× e = × $× $ = 1 (2) , bosons $ fer ion = − 1 #= q #= q′ #= 1 e X Z Y Ck := Zk iXk+i+jZk+j. H := − k Ck, e Sm − 1 1 $ S + σ− c k + kcj = δjk c c2c3 σ± × = 1 + $ σ± × σ∓ = e+ σ± $ = σ± × e = σ± × = σ∓ (3) σ × = 1+ ψ σ σ ψ × ψ = 1 (4) cj cj+1 cj+1 −cj ck ck {1, r, g, } Γ = { , ζ+, ζ−,σr,σg,σb} P icle ti t cs re r icularly rich in wo spatial di- mensi s, where ey the usual fermions a d bosons there xist mo e genera ly any ns (see [? ] for a com- pila ion of th basic ref rences). Anyonic t tistics are complex en ugh t ve rise t the no ion f topological TSC twists Transpositions → 2 possible charges As in toric codes, we fix one Colored Majorana operators: The i-th twist is and i<j Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion = − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk+i+jZk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ S σ+ σ− cjck + ckcj = 2δjk c1c2c3 σ± × σ± = 1 + $ σ± × σ∓ = e+m σ± × $ =σ± σ± × e = σ± ×m = σ∓ (3) σ × σ = 1+ ψ σ × ψ = σ ψ × ψ = 1 (4) cj → cj+1 cj+1 → −cj ck → ck {1, r, g, b} Γ = {e, ζ+, ζ−,σr,σg,σb} σci kikj = { kjki if ci = ζ+(cj), −kikj otherwise. (5) Topological Order with a Twist: Ising Anyons from an Abelian Model H. Bombin Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realiz tion of such defects in the t ric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes. PACS numbers: 05.30.Pr, 03.67.Lx, 73.43.Cd Ci|ψ〉 = ci|ψ〉 σz1 ⊗ σ z 2 σy2 ⊗ σ x 3 #= Q ∈ {a, b, . . . } Q′ Q×Q′ = q1 + q2 + . . . |ψ〉 −→ ? q1q2q3 . . . q −→ pi(q) Ree = Rmm = 1 R!! = −1 RemRme = Re!R!e = Rm!R!m = −1 (1) {1, e,m, $} e×m = $ e× $ = m m× $ = e e× e = m×m = $× $ = 1 (2) e,m→ bosons $→ fermion = − 1 #= q #= q′ #= 1 e↔ m X Z Y Ck := XkZk+iXk+i+jZk+j. H := − ∑ k Ck, Se Sm − 1 1 1 em $ S σ+ σ− cjck + ckcj = 2δjk c1c2c3 σ± × σ± = 1 + $ σ± × σ∓ = e+m σ± × $ =σ± σ± × e = σ± ×m = σ∓ (3) σ × σ = 1+ ψ σ × ψ = σ ψ × ψ = 1 (4) cj → cj+1 cj+1 → −cj ck → ck {1, r, g, b} Γ = {e, ζ+, ζ−,σr,σg,σb} σci kikj = { kjki if ci = ζ+(cj), −kikj otherwise. (5) TSC twists Braiding is now more interesting This gives the whole Clifford group!!! 2 (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g (a) (b) (b) (c) (a) s sm (e) kj → kj+1, kj+1 → −kj if cj = cj+1, ikjkj+1 if cj = ζ−(cj+1), −kjkj+1 otherwise. (6) Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the basic references). Anyonic statistics are complex enough to give rise to the notion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- formation on fusion channels and the topological nature of braiding makes TQC naturally robust against local perturbations, providing a complement to fault-tolerant quantum computation [? ? ]. In condensed matter, anyons emerge as excitations in systems that exhibit topological order [? ]. A possible way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not computationally powerful. In this paper we address an strategy to recover compu- tationally interesting anyon-like behavior from systems with very simple anyonic statistics. Our starting point are the symmetries that anyons may exhibit. Anyon models have three main ingredients: (i) a set of labels that identify the superselection sectors or topological charges, (ii) fusion/splitting rules that dictate the charges of composite systems, and (iii) braiding rules that dictate the effect of particle exchanges. A symme- try is a label permutation that leaves braiding and fusion rules unchanged—for a recent survey, see [? ]—. Given a symmetry s, we can imagine cutting the system along an open curve, as in Fig. ??(d), and then gluing it again “up to s”. Ideally the location of the cut itself is unphysical, only its endpoints have a measurable effect. In partic- ular, transporting an anyon around one end of the line changes the charge of the anyon according to the action of s. Our aim is to explore to which extent these topolog- ical defects, that we call twists for short, can be “treated as anyons” and used in TQC. Twists are being indepen- dently studied by Kong and Kitaev [? ]. An interesting precedent are the Alice strings appearing in some gauge models [? ], which can cause charge conjugation under monodromy, whereas the twists that we will discuss here exchange electric and magnetic charges. 2 (a) (b) (c) e m (d) a c r g b g b r g (c) (d) (e) (a) (b) 1 2 3 X Y ZI r g b g b r g gr b b r r g b r g b b g br r g g b g (a) (b) (b) (c) (a) s sm (e) kj → kj+1, kj+1 → −kj if cj = cj+1, ikjkj+1 if cj = ζ−(cj+1), −kjkj+1 otherwise. (6) Particle statistics are particularly rich in two spatial di- mensions, where beyond the usual fermions and bosons there exist more generally anyons (see [? ] for a com- pilation of the b sic references). Anyonic statistics are complex enough t give ise to the n tion of topological quantum computation (TQC) [? ? ? ], where compu- tations are carried out by braiding and fusing anyons, see Fig. ??(a-c). The nonlocal encoding of quantum in- formation on fusion channels and the topological nature of braiding makes TQC naturally robust against local perturbations, providing a complement to fault-tolerant quantum computation [? ? ]. In condensed matter, anyons emerge as excitations in systems that exhibit topological order [? ]. A possible way to obtain these exotic phases is by engineering suit- able Hamiltonians on lattice spin systems [? ? ? ? ? ]. Indeed, implementations on optical lattices have been proposed [? ]. Unfortunately, the anyon models that ap- pear in simple models are not computationally powerful. In this paper we address an strategy to recover compu- tationally interesting anyo -lik behavior from systems with very simple anyonic tatistics. Our starting point are the symmetries that anyons may exhibit. Anyon models have three main ingredients: (i) a set of labels that identify the superselection sectors or topological charges, (ii) fusion/splitting rules that dictate the charges of composite systems, and (iii) braiding rules that dictate the effect of particle exchanges. A symme- try is a label permutation that leaves braiding and fusion rules unchanged—for a recent survey, see [? ]—. Given a symmetry s, we can imagine cutting the system along an open curve, as in Fig. ??(d), and then gluing it again “up to s”. Ideally the location of the cut itself is unphysical, only its endpoints have a measurable effect. In partic- ular, transporting an anyon around one end of the line changes the charge of the anyon according to the action of s. Our aim is to explore to which extent these topolog- ical defects, that we call twists for sh rt, can be “treated as anyons” and used in TQC. Twists are being indepen- dently studied by Kong and Kitaev [? ]. An interesting precedent are the Alice strings appearing in some gauge models [? ], which can cause charge conjugation under monodromy, whereas the twists that we will discuss here exchange electric and magnetic charges. Conclusions Topological codes stand out for their locality TSCs only require 2-local measurements Twists reflect anyon symmetries Twist are a tool to improve topological codes With twists, via code deformation, we can implement all Clifford gates on TSCs PRL 105.030403 / arXiv:1004.1838 arXiv:1006.5260
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Canadian Summer School on Quantum Information (CSSQI) (10th : 2010)
Twists in Topological Codes Bombin, Hector 2010-07-25
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Title | Twists in Topological Codes |
Creator |
Bombin, Hector |
Contributor |
University of British Columbia. Department of Physics and Astronomy Workshop on Quantum Algorithms, Computational Models and Foundations of Quantum Mechanics Pacific Institute for the Mathematical Sciences Summer School on Quantum Information (10th : 2010 : Vancouver, B.C.) |
Date Issued | 2010-07-25 |
Description | There exists a close relationship between topological quantum error-correcting codes and topological order in condensed matter systems. Indeed, a topological stabilizer code can be regarded as the ground state of a suitable Hamiltonian model, so that "wrong" syndromes correspond to excitations. These excitations are anyons, quasiparticles that carry a topological charge and exhibit exotic statistics. Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We call these defects twists. Twists give rise to new topological degrees of freedom in the ground state, useful as a quantum memory. Moreover, twists can be braided to perform topologically protected gates on these topological qubits. Thus, twists provide a new way to encode and compute with topological codes through code deformations. Because the properties of twists depend on the anyon model, codes with different anyon content give rise to different computational capabilities. E.g., in the well-known toric code a process where suitable twists are braided and fused has the same outcome as if they were Ising anyons. These are non-abelian anyons: braiding produces non-trivial gates on encoded qubits. |
Subject |
Topological quantum error-correcting code anyons symmetry topological defect code deformation |
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Presentation |
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Text Moving Image |
Language | eng |
Date Available | 2016-11-22 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0103163 |
URI | http://hdl.handle.net/2429/30066 |
Affiliation |
Non UBC |
Peer Review Status | Reviewed |
Scholarly Level | Postdoctoral |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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