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Adiabatic Quantum Computation, Decoherence-Free Subspaces & Noiseless Subsystems, and Dynamical Decoupling Lidar, Daniel 2010

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Hybrid quantum error prevention, reduction, and correction methods and correction methodsDaniel LidarUniversity of Southern California10th Canadian Summer School on Quantum InformationUBC, July 21, 2010DFS- encodedno encodingScience 291, 1013 (2001)probability of correct Grover answerPhys. Rev. Lett. 91, 217904 (2003) OutlineSymmetry and preserved quantum informationSystem-bath, decoherence and all thatUnified view of decoherence-free subspaces, noiseless subsystems, quantum error correcting codes, operator quantum error correctionDFS/NS examples: theory and experimentOutlineSymmetry and preserved quantum informationSystem-bath, decoherence and all thatUnified view of decoherence-free subspaces, noiseless subsystems, quantum error correcting codes, operator quantum error correctionDFS/NS examples: theory and experimentFeel free to interrupt and ask lots of questions!Symmetry Protects Q. InformationSymmetry �conserved quantity = quantum info.Symmetry Protects Q. InformationDFS/NS: Error Preventionuse an existingexisting exactexact symmetry to perfectlyperfectly hide q. info. from bathSymmetry �conserved quantity = quantum info.DFSH1)Symmetry Protects Q. InformationDFS/NS: Error Preventionuse an existingexisting exactexact symmetry to perfectlyperfectly hide q. info. from bathSymmetry �conserved quantity = quantum info.DFSHBANG + free evolutiontimeDynamical Decoupling: Open-loop controldynamically generategenerate a symmetrybath correlation timestrongSBH1)2)Whence the Errors?Whence the Errors? Decoherence from System-Bath Interactionsystem bathSBSBHα αα⊗=∑Every real quantum system is coupled to an environment (“bath”).Full Hamiltonian:B SBSHH HH + +=Whence the Errors?Whence the Errors? Decoherence from System-Bath Interactionsystem bathSBSBHα αα⊗=∑Every real quantum system is coupled to an environment (“bath”).Full Hamiltonian:B SBSHH HH + +=DecoherenceNon-unitary evolution of system†        System dynamics alone:() Tr (0)() (0) ()iHt iHtB SBkk kkte ecE t E tρρρ−⎡⎤⎢⎥⎣⎦==∑Whence the Errors?Whence the Errors? Decoherence from System-Bath Interactionsystem bathSBSBHα αα⊗=∑Every real quantum system is coupled to an environment (“bath”).Full Hamiltonian:B SBSHH HH + +=( )†† †,' ' ' ',' , 2]12[StaSSSSSSiHααααα α α αααρ ρρρρ +∂−−∂=−∑    Markovian master (Lindblad) equation:DecoherenceNon-unitary evolution of system†        System dynamics alone:() Tr (0)() (0) ()iHt iHtB SBkk kkte ecE t E tρρρ−⎡⎤⎢⎥⎣⎦==∑Whence the Errors?Whence the Errors? Decoherence from System-Bath InteractionWhat is there is a symmetry? Symmetric coin flipping noiseWhat is there is a symmetry? Symmetric coin flipping noiseHow to reliably store a single bit?What is there is a symmetry? Symmetric coin flipping noiselogical 0logical 1How to reliably store a single bit?A noiseless subspace.Unified view of Quantum Information Protection: Fixed CodesError Model:Trace-preserving completely-positive (CP) maps:ρ0=PkEkρE†k≡ E(ρ)PkE†kEk= IUnified view of Quantum Information Protection: Fixed CodesError Model:Trace-preserving completely-positive (CP) maps:ρ0=PkEkρE†k≡ E(ρ)PkE†kEk= IDecoherence-free subspace (DFS): H = A⊕BC = {all states ρ : A → A} such that E(ρ) = ρUnified view of Quantum Information Protection: Fixed CodesError Model:Trace-preserving completely-positive (CP) maps:passiveρ0=PkEkρE†k≡ E(ρ)PkE†kEk= IDecoherence-free subspace (DFS): H = A⊕BC = {all states ρ : A → A} such that E(ρ) = ρNoiseless subsystem (NS): H = A⊕B, A = N ⊗GC = {all states ρ : A → A} such that TrGE(ρ) = TrGρUnified view of Quantum Information Protection: Fixed CodesError Model:Trace-preserving completely-positive (CP) maps:passiveρ0=PkEkρE†k≡ E(ρ)PkE†kEk= IDecoherence-free subspace (DFS): H = A⊕BC = {all states ρ : A → A} such that E(ρ) = ρNoiseless subsystem (NS): H = A⊕B, A = N ⊗GC = {all states ρ : A → A} such that TrGE(ρ) = TrGρQuantum error correcting code (QEC): H = A⊕BC = {all states ρ : A → A} such that ∃CP map Rfor which R◦E(ρ) = ρUnified view of Quantum Information Protection: Fixed CodesError Model:Trace-preserving completely-positive (CP) maps:passiveactiveρ0=PkEkρE†k≡ E(ρ)PkE†kEk= IDecoherence-free subspace (DFS): H = A⊕BC = {all states ρ : A → A} such that E(ρ) = ρNoiseless subsystem (NS): H = A⊕B, A = N ⊗GC = {all states ρ : A → A} such that TrGE(ρ) = TrGρQuantum error correcting code (QEC): H = A⊕BC = {all states ρ : A → A} such that ∃CP map Rfor which R◦E(ρ) = ρOperator QEC (OQEC): H = A⊕B, A = N ⊗GC = {all states ρ : A → A} such that ∃CP map Rfor which TrGR◦E(ρ) = TrGρDFS as a QEC, QEC as a DFSDecoherence-free subspace (DFS): H = A⊕BC = {all states ρ : A → A} such that E(ρ) = ρQuantum error correcting code (QEC): H = A⊕BC = {all states ρ : A → A} such that ∃CP map Rfor which R◦E(ρ) = ρA DFS is a QEC with trivial recovery operation: R = IA QEC is a DFS with respect to the map R◦ENS as an OQEC, OQEC as an NSNoiseless subsystem (NS): H = A⊕B, A = N ⊗GC = {all states ρ : A → A} such that TrGE(ρ) = TrGρOperator QEC (OQEC): H = A⊕B, A = N ⊗GC = {all states ρ : A → A} such that ∃CP map Rfor which TrGR◦E(ρ) = TrGρAn NS is an OQEC with trivial recovery operation: R = IAn OQEC is an NS with respect to the map R◦EUnitarily Invariant DFSUnitarily Invariant DFS:=Subspace of full system Hilbert space in which evolution is purely unitaryUnitarily Invariant DFSUnitarily Invariant DFS:=Subspace of full system Hilbert space in which evolution is purely unitaryMore precisely:Let the system Hilbert space H decompose into a direct sum as H = HD⊕H⊥D,and partition the system state ρSaccordingly into blocks: ρS=µρDρ2ρ†2ρ3¶.Assume ρD(0) 6= 0.Then HDis called decoherence-free iff the initial and final DFS-blocks of ρSare unitarily related:ρD(t) = UDρD(0)U†D,where UDis a unitary matrix acting on HD.Note imperfect initialization!U.I. DFS Conditions for CP Mapsρ0=PkEkρE†k≡ E(ρ)PkE†kEk= IGiven a CP map:TheoremA necessary and sufficient condition for the existence of a DFS HDwithrespect to the CP map E is that all Kraus operators have a matrix representationof the formEk=µckUD00 Bk¶,where UDis unitary, ckare scalars satisfyingPk|ck|2= 1, and Bkare arbitraryoperators on H⊥DsatisfyingPkB†kBk= I.Meaning: Ekact unitarily on the DFSU.I. DFS Conditions for Master EquationsGiven a Markovian master equation:TheoremA necessary and sufficient condition for the existence of a DFS HDwithrespect to the Markovian master equation above is that the Lindblad operatorsFαand the system Hamiltonian HShave the block-diagonal formHS=µHD00 H⊥D¶, Fα=µcαI 00 Bα¶,where HDand H⊥Dare Hermitian, cαare scalars, and Bαare arbitrary operatorson H⊥D.dρdt= −i[HS,ρ] +12Pα2FαρF†α−ρF†αFα−F†αFαρMeaning: Fαact as identity on the DFS, while HSpreserves the DFSExercise1. Prove sufficiency (easy) and necessity (not so easy) of the U.I. DFS conditions for CP maps and Markovian master equations2. Generalize to NS, QEC, OQECWhere is the promised symmetry?How do we find and construct a DFS? Under Hamiltonian dynamics system and bath evolve jointly subject to the Schrodinger equation with thsubspace Hamiltonian .Find a  where acts trivially, i.e.:                e  SBBSSBH HH HSBHααα==+ +⊗∑                           makeSB S BHIO∝⊗U.I. DFS Conditions for Hamiltonian DynamicsAlso, remember that HSmust preserve the DFS. Under Hamiltonian dynamics system and bath evolve jointly subject to the Schrodinger equation with thsubspace Hamiltonian .Find a  where acts trivially, i.e.:                e  SBBSSBH HH HSBHααα==+ +⊗∑                           makeSB S BHIO∝⊗Theoremdegeneracy symmetryU.I. DFS Conditions for Hamiltonian DynamicsAlso, remember that HSmust preserve the DFS.Let A = alg{I,Sα,S†α}.Assume [HS,A] = 0.The dimension of the DFS HDequals the degeneracy of the 1-dimensionalirreducible representation (irrep) of A.Simplest DFS Example: Collective DephasingEffect: Random " ":01 0 1jj j j jj jj jiab aebθψ =+ +Collective Dephasing6random but j-independent phase  Find a  where acts trivially, i.e.:                                         su   makebspaceSBS BB SHHSBOIααα= ⊗∝⊗∑DFS idea:Permutation symmetry in zz direction:Long-wavelength magnetic field B (environment) couples to spins1ψ2ψ()BtzSimplest DFS Example: Collective DephasingEffect: Random " ":01 0 1jj j j jj jj jiab aebθψ =+ +Collective Dephasing6random but j-independent phase1212001110=⊗=⊗LLDFS encoding  Find a  where acts trivially, i.e.:                                         su   makebspaceSBS BB SHHSBOIααα= ⊗∝⊗∑DFS idea:11int()zzHg Bσ σ=+⊗=0220gBgB−⎛⎞⎜⎟⎜⎟⎝⎠12121212↓↑↑↓↓↓↑↑Permutation symmetry in zz direction:Long-wavelength magnetic field B (environment) couples to spins1ψ2ψ()BtzWhy it Works()()()()θθθθθθθ⊗⊗≡⊗⊗≡⊗⊗≡66662Case of two qubits:00 00001 0 1 0110 1 0 1011 1 1 1iiiiiiieeeeeeeθψ =+ +6Collective dephasing:01 0 1jj j jj jj j jiab aebWhy it Works()()()()θθθθθθθ⊗⊗≡⊗⊗≡⊗⊗≡66662Case of two qubits:00 00001 0 1 0110 1 0 1011 1 1 1iiiiiiieeeeeeeψ =+Global phase physically irrelevant:is decoherence-free:A 2-di pmensional rote01  cted subspace .LLLabθψ =+ +6Collective dephasing:01 0 1jj j jj jj j jiab aeb≡ 0L≡ 1LWhy it Works()()()()θθθθθθθ⊗⊗≡⊗⊗≡⊗⊗≡66662Case of two qubits:00 00001 0 1 0110 1 0 1011 1 1 1iiiiiiieeeeeeeψ =+Global phase physically irrelevant:is decoherence-free:A 2-di pmensional rote01  cted subspace .LLLabθψ =+ +6Collective dephasing:01 0 1jj j jj jj j jiab aeb≡ 0L≡ 1Lpop quiz:Are the states|00i and |11ialso in a DFS?1D irreps condition not needed…The dimension of the DFS HDequals the degeneracy of the 1-dimensionalirreducible representation (irrep) of A.Generalization: Noiseless Subsystemssystems [E. Knill, R. Laflamme and L. Viola, PRL 84, 2525 (2000)]α αα⊗=∑SBSBHModel of decoherence:α α=≅⊕ ⊗≅⊕ ⊗^^^^^†22Associative algebra polynomials{ , , }Matrix representation over :()Hilbert space decomposition:NJJNJJndJndJAISAIMmultiplicitydimensionirreducible representationsA theorem from C* algebras:1D irreps condition not needed…The dimension of the DFS HDequals the degeneracy of the 1-dimensionalirreducible representation (irrep) of A.Generalization: Noiseless Subsystemssystems [E. Knill, R. Laflamme and L. Viola, PRL 84, 2525 (2000)]α αα⊗=∑SBSBHModel of decoherence:α α=≅⊕ ⊗≅⊕ ⊗^^^^^†22Associative algebra polynomials{ , , }Matrix representation over :()Hilbert space decomposition:NJJNJJndJndJAISAIMmultiplicitydimensioncode subsystemirreducible representations1 iff   in system-env. interactionJn >∃symmetryA theorem from C* algebras:1D irreps condition not needed…The dimension of the DFS HDequals the degeneracy of the 1-dimensionalirreducible representation (irrep) of A.Generalization: Noiseless Subsystemssystems [E. Knill, R. Laflamme and L. Viola, PRL 84, 2525 (2000)]=1Each  labels an  NS codeDFS is the case -dimensionalJJJndIsotropic Quantum Errors: Collective Decoherence Modelˆ()YBty1ψˆ()ZBtz2ϕˆ()XBtxDescribes, e.g., low- T decoherence due to phonons in various solid state QC proposalsIsotropic Quantum Errors: Collective Decoherence Modelˆ()YBty1ψˆ()ZBtz2ϕˆ()XBtx()ααααασσ===++⊗∑	 "1,,total spin operatorError model,  qubits:"Collective Decoherence"NSBxyzSNHBDescribes, e.g., low- T decoherence due to phonons in various solid state QC proposalsIsotropic Quantum Errors: Collective Decoherence Modelˆ()YBty1ψˆ()ZBtz2ϕˆ()XBtxψψ⎧⎪⊗⊗⎪⎨⊗⊗⎪⎪⊗⊗⎩"6""0123Error model,  qubits:                          prob. (1) ( )   prob. (1) ( )    prob. (1) ( )    prob. XXYYZZNpUUNpUUNDescribes, e.g., low- T decoherence due to phonons in various solid state QC proposalsDo irreps analysis of n copies of su(2)…J01/213/2212513114914multiplicity nJ , counts paths;dJ =2J+11125 142 3 4 5 6 7 8All Decoherence-Free Subspaces/Subsystems for Collective Decoherencen≅⊕ ⊗^^^2Hilbert space decomposition:NJJndJ0L≡()101 102=−J01/213/2212513114914multiplicity nJ , counts paths;dJ =2J+11125 142 3 4 5 6 7 8All Decoherence-Free Subspaces/Subsystems for Collective Decoherence0L=1 2 3n01/21()101 10 12=−1L=1 2 3n01/21()21110 011 10133=− −n≅⊕ ⊗^^^2Hilbert space decomposition:NJJndJ0L≡()101 102=−J01/213/2212513114914multiplicity nJ , counts paths;dJ =2J+11125 142 3 4 5 6 7 8All Decoherence-Free Subspaces/Subsystems for Collective Decoherence0L=1 2 3n01/21()101 10 12=−1L=1 2 3n01/21()21110 011 10133=− −n0L≡()()101 10 01 102=− −1L≡()()12 0011 2 1100 0110 1001 1010 010123=+−+++≅⊕ ⊗^^^2Hilbert space decomposition:NJJndJWhat is the “Volume”” of a DFS/NS?()()!(2 1)Degeneracy for given ,  = dimension of DFS/NS ( )  /2 1 ! /2 !JnJJM D nnJ nJ+≡=++ −( )020 2physicallog ( ) logno. of  qubits 3  code rate     1no. of  encodedqubits 2=JnDn nnn=→∞⇒≡ ⎯⎯→−What is the “Volume”” of a DFS/NS?()()!(2 1)Degeneracy for given ,  = dimension of DFS/NS ( )  /2 1 ! /2 !JnJJM D nnJ nJ+≡=++ −DFS’s for collective decoherenceasymptotically fill the Hilbert space!( )020 2physicallog ( ) logno. of  qubits 3  code rate     1no. of  encodedqubits 2=JnDn nnn=→∞⇒≡ ⎯⎯→−ComputationComputation Inside a U.I. DFS/NSSo far have storage. What about computation?To prevent decoherence, computation should never leave DFS/NS.Which logic operations are compatible?ComputationComputation Inside a U.I. DFS/NSSo far have storage. What about computation?To prevent decoherence, computation should never leave DFS/NS.Which logic operations are compatible?≅⊕⊗≅⊕⊗⊗≅⊕^^^^^2Commutant = operators commuting with 'Error algebra:()'()The allowed logic operationCode subsy msst :!eJJJJNJJndJndJndJAMAM IAIComputationComputation Inside a U.I. DFS/NSSo far have storage. What about computation?To prevent decoherence, computation should never leave DFS/NS.Which logic operations are compatible?≅⊕⊗≅⊕⊗⊗≅⊕^^^^^2Commutant = operators commuting with 'Error algebra:()'()The allowed logic operationCode subsy msst :!eJJJJNJJndJndJndJAMAM IAIUniversal quantum computation over DFS/NS is possible using “exchange Hamiltonians”, e.g., Heisenberg interaction:Heis2ijyyx xzzij ij ijijHJσ σσσσσ⎛⎞⎜⎟⎝⎠= ++∑Heisenberg Computation over DFS/NS is Universal• Heisenberg exchange interaction:• Universal over collective-decoherence DFS [J. Kempe, D. Bacon, D.A.L., B. Whaley, Phys. Rev. A 63, 042307 (2001)]• Over 4-qubit DFS:CNOT involves 14 elementary steps (D. Bacon, Ph.D. thesis)• Implications for simplifying operation of spin-based quantum dot QCsHHeis=Pi,jJij(XiXj+YiYj+ ZiZj) ≡Pi,jJijEij≅⊕ ⊗^''()The allowed logic operationsJJndJAM IHeisenberg Computation over DFS/NS is Universal• Heisenberg exchange interaction:• Universal over collective-decoherence DFS [J. Kempe, D. Bacon, D.A.L., B. Whaley, Phys. Rev. A 63, 042307 (2001)]• Over 4-qubit DFS:CNOT involves 42 elementary steps (D. Bacon, Ph.D. thesis)• Implications for simplifying operation of spin-based quantum dot QCsHHeis=Pi,jJij(XiXj+YiYj+ ZiZj) ≡Pi,jJijEij¯X = −2√3(E13+12E12)¯Z = −E12eiθ¯Xand eiθ¯Zgenerate arbitrary single encoded qubit gates()()10 0110 01102L=− −()11 2 0011 2 1100 0110 1001 1010 010123L=+−+++≅⊕ ⊗^''()The allowed logic operationsJJndJAM IIn the beginning …In the beginning …13C-labeled alanine0Φ =1Φ =What about symmetry breaking? D.L., I.L. Chuang, K.B. Whaley, PRL 81, 2594 (1998); D. Bacon, D.L., K.B. Whaley, PRA 60, 1944 (1999)Symmetry breaking: unequal coupling constants, lowering of symmetry by a perturbation, etc.Introduce a perturbation via HSB7→ HSB+²∆H, k∆Hk = 1Theory shows that fidelity depends on ² only to second order.Robustness of DFS to symmetry breaking perturbations perturbationsangle strengthRobustness of DFS to symmetry breaking perturbations perturbationsDFS-encodedBare qubits12 1 2001 1 0    LL=⊗ =⊗Bare qubit: two hyperfine states of trapped 9Be+ ionChief decoherence sources:(i) fluctuating long-wavelength ambient magnetic fields;(ii) heating of ion CM motion during computation: a symmetry-breaking processDFS encoding: into pair of ionsStrong Symmetry BreakingNeed a way to deal with symmetry breaking…Intermission & Bathroom Break Hybrid quantum error prevention, reduction, and correction methods and correction methodsDaniel LidarUniversity of Southern California10th Canadian Summer School on Quantum InformationUBC, July 21, 2010DFS- encodedno encodingScience 291, 1013 (2001)probability of correct Grover answerPhys. Rev. Lett. 91, 217904 (2003) OutlineSymmetry and preserved quantum informationSystem-bath, decoherence and all thatUnified view of decoherence-free subspaces, noiseless subsystems, quantum error correcting codes, operator quantum error correctionDFS/NS examples: theory and experimentOutlineSymmetry and preserved quantum informationSystem-bath, decoherence and all thatUnified view of decoherence-free subspaces, noiseless subsystems, quantum error correcting codes, operator quantum error correctionDFS/NS examples: theory and experimentFeel free to interrupt and ask lots of questions!Symmetry Protects Q. InformationSymmetry �conserved quantity = quantum info.Symmetry Protects Q. InformationDFS/NS: Error Preventionuse an existingexisting exactexact symmetry to perfectlyperfectly hide q. info. from bathSymmetry �conserved quantity = quantum info.DFSH1)Symmetry Protects Q. InformationDFS/NS: Error Preventionuse an existingexisting exactexact symmetry to perfectlyperfectly hide q. info. from bathSymmetry �conserved quantity = quantum info.DFSHBANG + free evolutiontimeDynamical Decoupling: Open-loop controldynamically generategenerate a symmetrybath correlation timestrongSBH1)2)Whence the Errors?Whence the Errors? Decoherence from System-Bath Interactionsystem bathSBSBHα αα⊗=∑Every real quantum system is coupled to an environment (“bath”).Full Hamiltonian:B SBSHH HH + +=Whence the Errors?Whence the Errors? Decoherence from System-Bath Interactionsystem bathSBSBHα αα⊗=∑Every real quantum system is coupled to an environment (“bath”).Full Hamiltonian:B SBSHH HH + +=DecoherenceNon-unitary evolution of system†        System dynamics alone:() Tr (0)() (0) ()iHt iHtB SBkk kkte ecE t E tρρρ−⎡⎤⎢⎥⎣⎦==∑Whence the Errors?Whence the Errors? Decoherence from System-Bath Interactionsystem bathSBSBHα αα⊗=∑Every real quantum system is coupled to an environment (“bath”).Full Hamiltonian:B SBSHH HH + +=( )†† †,' ' ' ',' , 2]12[StaSSSSSSiHααααα α α αααρ ρρρρ +∂−−∂=−∑    Markovian master (Lindblad) equation:DecoherenceNon-unitary evolution of system†        System dynamics alone:() Tr (0)() (0) ()iHt iHtB SBkk kkte ecE t E tρρρ−⎡⎤⎢⎥⎣⎦==∑Whence the Errors?Whence the Errors? Decoherence from System-Bath InteractionWhat is there is a symmetry? Symmetric coin flipping noiseWhat is there is a symmetry? Symmetric coin flipping noiseHow to reliably store a single bit?What is there is a symmetry? Symmetric coin flipping noiselogical 0logical 1How to reliably store a single bit?A noiseless subspace.Unified view of Quantum Information Protection: Fixed CodesError Model:Trace-preserving completely-positive (CP) maps:ρ0=PkEkρE†k≡ E(ρ)PkE†kEk= IUnified view of Quantum Information Protection: Fixed CodesError Model:Trace-preserving completely-positive (CP) maps:ρ0=PkEkρE†k≡ E(ρ)PkE†kEk= IDecoherence-free subspace (DFS): H = A⊕BC = {all states ρ : A → A} such that E(ρ) = ρUnified view of Quantum Information Protection: Fixed CodesError Model:Trace-preserving completely-positive (CP) maps:passiveρ0=PkEkρE†k≡ E(ρ)PkE†kEk= IDecoherence-free subspace (DFS): H = A⊕BC = {all states ρ : A → A} such that E(ρ) = ρNoiseless subsystem (NS): H = A⊕B, A = N ⊗GC = {all states ρ : A → A} such that TrGE(ρ) = TrGρUnified view of Quantum Information Protection: Fixed CodesError Model:Trace-preserving completely-positive (CP) maps:passiveρ0=PkEkρE†k≡ E(ρ)PkE†kEk= IDecoherence-free subspace (DFS): H = A⊕BC = {all states ρ : A → A} such that E(ρ) = ρNoiseless subsystem (NS): H = A⊕B, A = N ⊗GC = {all states ρ : A → A} such that TrGE(ρ) = TrGρQuantum error correcting code (QEC): H = A⊕BC = {all states ρ : A → A} such that ∃CP map Rfor which R◦E(ρ) = ρUnified view of Quantum Information Protection: Fixed CodesError Model:Trace-preserving completely-positive (CP) maps:passiveactiveρ0=PkEkρE†k≡ E(ρ)PkE†kEk= IDecoherence-free subspace (DFS): H = A⊕BC = {all states ρ : A → A} such that E(ρ) = ρNoiseless subsystem (NS): H = A⊕B, A = N ⊗GC = {all states ρ : A → A} such that TrGE(ρ) = TrGρQuantum error correcting code (QEC): H = A⊕BC = {all states ρ : A → A} such that ∃CP map Rfor which R◦E(ρ) = ρOperator QEC (OQEC): H = A⊕B, A = N ⊗GC = {all states ρ : A → A} such that ∃CP map Rfor which TrGR◦E(ρ) = TrGρDFS as a QEC, QEC as a DFSDecoherence-free subspace (DFS): H = A⊕BC = {all states ρ : A → A} such that E(ρ) = ρQuantum error correcting code (QEC): H = A⊕BC = {all states ρ : A → A} such that ∃CP map Rfor which R◦E(ρ) = ρA DFS is a QEC with trivial recovery operation: R = IA QEC is a DFS with respect to the map R◦ENS as an OQEC, OQEC as an NSNoiseless subsystem (NS): H = A⊕B, A = N ⊗GC = {all states ρ : A → A} such that TrGE(ρ) = TrGρOperator QEC (OQEC): H = A⊕B, A = N ⊗GC = {all states ρ : A → A} such that ∃CP map Rfor which TrGR◦E(ρ) = TrGρAn NS is an OQEC with trivial recovery operation: R = IAn OQEC is an NS with respect to the map R◦EUnitarily Invariant DFSUnitarily Invariant DFS:=Subspace of full system Hilbert space in which evolution is purely unitaryUnitarily Invariant DFSUnitarily Invariant DFS:=Subspace of full system Hilbert space in which evolution is purely unitaryMore precisely:Let the system Hilbert space H decompose into a direct sum as H = HD⊕H⊥D,and partition the system state ρSaccordingly into blocks: ρS=µρDρ2ρ†2ρ3¶.Assume ρD(0) 6= 0.Then HDis called decoherence-free iff the initial and final DFS-blocks of ρSare unitarily related:ρD(t) = UDρD(0)U†D,where UDis a unitary matrix acting on HD.Note imperfect initialization!U.I. DFS Conditions for CP Mapsρ0=PkEkρE†k≡ E(ρ)PkE†kEk= IGiven a CP map:TheoremA necessary and sufficient condition for the existence of a DFS HDwithrespect to the CP map E is that all Kraus operators have a matrix representationof the formEk=µckUD00 Bk¶,where UDis unitary, ckare scalars satisfyingPk|ck|2= 1, and Bkare arbitraryoperators on H⊥DsatisfyingPkB†kBk= I.Meaning: Ekact unitarily on the DFSU.I. DFS Conditions for Master EquationsGiven a Markovian master equation:TheoremA necessary and sufficient condition for the existence of a DFS HDwithrespect to the Markovian master equation above is that the Lindblad operatorsFαand the system Hamiltonian HShave the block-diagonal formHS=µHD00 H⊥D¶, Fα=µcαI 00 Bα¶,where HDand H⊥Dare Hermitian, cαare scalars, and Bαare arbitrary operatorson H⊥D.dρdt= −i[HS,ρ] +12Pα2FαρF†α−ρF†αFα−F†αFαρMeaning: Fαact as identity on the DFS, while HSpreserves the DFSExercise1. Prove sufficiency (easy) and necessity (not so easy) of the U.I. DFS conditions for CP maps and Markovian master equations2. Generalize to NS, QEC, OQECWhere is the promised symmetry?How do we find and construct a DFS? Under Hamiltonian dynamics system and bath evolve jointly subject to the Schrodinger equation with thsubspace Hamiltonian .Find a  where acts trivially, i.e.:                e  SBBSSBH HH HSBHααα==+ +⊗∑                           makeSB S BHIO∝⊗U.I. DFS Conditions for Hamiltonian DynamicsAlso, remember that HSmust preserve the DFS. Under Hamiltonian dynamics system and bath evolve jointly subject to the Schrodinger equation with thsubspace Hamiltonian .Find a  where acts trivially, i.e.:                e  SBBSSBH HH HSBHααα==+ +⊗∑                           makeSB S BHIO∝⊗Theoremdegeneracy symmetryU.I. DFS Conditions for Hamiltonian DynamicsAlso, remember that HSmust preserve the DFS.Let A = alg{I,Sα,S†α}.Assume [HS,A] = 0.The dimension of the DFS HDequals the degeneracy of the 1-dimensionalirreducible representation (irrep) of A.Simplest DFS Example: Collective DephasingEffect: Random " ":01 0 1jj j j jj jj jiab aebθψ =+ +Collective Dephasing6random but j-independent phase  Find a  where acts trivially, i.e.:                                         su   makebspaceSBS BB SHHSBOIααα= ⊗∝⊗∑DFS idea:Permutation symmetry in zz direction:Long-wavelength magnetic field B (environment) couples to spins1ψ2ψ()BtzSimplest DFS Example: Collective DephasingEffect: Random " ":01 0 1jj j j jj jj jiab aebθψ =+ +Collective Dephasing6random but j-independent phase1212001110=⊗=⊗LLDFS encoding  Find a  where acts trivially, i.e.:                                         su   makebspaceSBS BB SHHSBOIααα= ⊗∝⊗∑DFS idea:11int()zzHg Bσ σ=+⊗=0220gBgB−⎛⎞⎜⎟⎜⎟⎝⎠12121212↓↑↑↓↓↓↑↑Permutation symmetry in zz direction:Long-wavelength magnetic field B (environment) couples to spins1ψ2ψ()BtzWhy it Works()()()()θθθθθθθ⊗⊗≡⊗⊗≡⊗⊗≡66662Case of two qubits:00 00001 0 1 0110 1 0 1011 1 1 1iiiiiiieeeeeeeθψ =+ +6Collective dephasing:01 0 1jj j jj jj j jiab aebWhy it Works()()()()θθθθθθθ⊗⊗≡⊗⊗≡⊗⊗≡66662Case of two qubits:00 00001 0 1 0110 1 0 1011 1 1 1iiiiiiieeeeeeeψ =+Global phase physically irrelevant:is decoherence-free:A 2-di pmensional rote01  cted subspace .LLLabθψ =+ +6Collective dephasing:01 0 1jj j jj jj j jiab aeb≡ 0L≡ 1LWhy it Works()()()()θθθθθθθ⊗⊗≡⊗⊗≡⊗⊗≡66662Case of two qubits:00 00001 0 1 0110 1 0 1011 1 1 1iiiiiiieeeeeeeψ =+Global phase physically irrelevant:is decoherence-free:A 2-di pmensional rote01  cted subspace .LLLabθψ =+ +6Collective dephasing:01 0 1jj j jj jj j jiab aeb≡ 0L≡ 1Lpop quiz:Are the states|00i and |11ialso in a DFS?1D irreps condition not needed…The dimension of the DFS HDequals the degeneracy of the 1-dimensionalirreducible representation (irrep) of A.Generalization: Noiseless Subsystemssystems [E. Knill, R. Laflamme and L. Viola, PRL 84, 2525 (2000)]α αα⊗=∑SBSBHModel of decoherence:α α=≅⊕ ⊗≅⊕ ⊗^^^^^†22Associative algebra polynomials{ , , }Matrix representation over :()Hilbert space decomposition:NJJNJJndJndJAISAIMmultiplicitydimensionirreducible representationsA theorem from C* algebras:1D irreps condition not needed…The dimension of the DFS HDequals the degeneracy of the 1-dimensionalirreducible representation (irrep) of A.Generalization: Noiseless Subsystemssystems [E. Knill, R. Laflamme and L. Viola, PRL 84, 2525 (2000)]α αα⊗=∑SBSBHModel of decoherence:α α=≅⊕ ⊗≅⊕ ⊗^^^^^†22Associative algebra polynomials{ , , }Matrix representation over :()Hilbert space decomposition:NJJNJJndJndJAISAIMmultiplicitydimensioncode subsystemirreducible representations1 iff   in system-env. interactionJn >∃symmetryA theorem from C* algebras:1D irreps condition not needed…The dimension of the DFS HDequals the degeneracy of the 1-dimensionalirreducible representation (irrep) of A.Generalization: Noiseless Subsystemssystems [E. Knill, R. Laflamme and L. Viola, PRL 84, 2525 (2000)]=1Each  labels an  NS codeDFS is the case -dimensionalJJJndIsotropic Quantum Errors: Collective Decoherence Modelˆ()YBty1ψˆ()ZBtz2ϕˆ()XBtxDescribes, e.g., low- T decoherence due to phonons in various solid state QC proposalsIsotropic Quantum Errors: Collective Decoherence Modelˆ()YBty1ψˆ()ZBtz2ϕˆ()XBtx()ααααασσ===++⊗∑	 "1,,total spin operatorError model,  qubits:"Collective Decoherence"NSBxyzSNHBDescribes, e.g., low- T decoherence due to phonons in various solid state QC proposalsIsotropic Quantum Errors: Collective Decoherence Modelˆ()YBty1ψˆ()ZBtz2ϕˆ()XBtxψψ⎧⎪⊗⊗⎪⎨⊗⊗⎪⎪⊗⊗⎩"6""0123Error model,  qubits:                          prob. (1) ( )   prob. (1) ( )    prob. (1) ( )    prob. XXYYZZNpUUNpUUNDescribes, e.g., low- T decoherence due to phonons in various solid state QC proposalsDo irreps analysis of n copies of su(2)…J01/213/2212513114914multiplicity nJ , counts paths;dJ =2J+11125 142 3 4 5 6 7 8All Decoherence-Free Subspaces/Subsystems for Collective Decoherencen≅⊕ ⊗^^^2Hilbert space decomposition:NJJndJ0L≡()101 102=−J01/213/2212513114914multiplicity nJ , counts paths;dJ =2J+11125 142 3 4 5 6 7 8All Decoherence-Free Subspaces/Subsystems for Collective Decoherence0L=1 2 3n01/21()101 10 12=−1L=1 2 3n01/21()21110 011 10133=− −n≅⊕ ⊗^^^2Hilbert space decomposition:NJJndJ0L≡()101 102=−J01/213/2212513114914multiplicity nJ , counts paths;dJ =2J+11125 142 3 4 5 6 7 8All Decoherence-Free Subspaces/Subsystems for Collective Decoherence0L=1 2 3n01/21()101 10 12=−1L=1 2 3n01/21()21110 011 10133=− −n0L≡()()101 10 01 102=− −1L≡()()12 0011 2 1100 0110 1001 1010 010123=+−+++≅⊕ ⊗^^^2Hilbert space decomposition:NJJndJWhat is the “Volume”” of a DFS/NS?()()!(2 1)Degeneracy for given ,  = dimension of DFS/NS ( )  /2 1 ! /2 !JnJJM D nnJ nJ+≡=++ −( )020 2physicallog ( ) logno. of  qubits 3  code rate     1no. of  encodedqubits 2=JnDn nnn=→∞⇒≡ ⎯⎯→−What is the “Volume”” of a DFS/NS?()()!(2 1)Degeneracy for given ,  = dimension of DFS/NS ( )  /2 1 ! /2 !JnJJM D nnJ nJ+≡=++ −DFS’s for collective decoherenceasymptotically fill the Hilbert space!( )020 2physicallog ( ) logno. of  qubits 3  code rate     1no. of  encodedqubits 2=JnDn nnn=→∞⇒≡ ⎯⎯→−ComputationComputation Inside a U.I. DFS/NSSo far have storage. What about computation?To prevent decoherence, computation should never leave DFS/NS.Which logic operations are compatible?ComputationComputation Inside a U.I. DFS/NSSo far have storage. What about computation?To prevent decoherence, computation should never leave DFS/NS.Which logic operations are compatible?≅⊕⊗≅⊕⊗⊗≅⊕^^^^^2Commutant = operators commuting with 'Error algebra:()'()The allowed logic operationCode subsy msst :!eJJJJNJJndJndJndJAMAM IAIComputationComputation Inside a U.I. DFS/NSSo far have storage. What about computation?To prevent decoherence, computation should never leave DFS/NS.Which logic operations are compatible?≅⊕⊗≅⊕⊗⊗≅⊕^^^^^2Commutant = operators commuting with 'Error algebra:()'()The allowed logic operationCode subsy msst :!eJJJJNJJndJndJndJAMAM IAIUniversal quantum computation over DFS/NS is possible using “exchange Hamiltonians”, e.g., Heisenberg interaction:Heis2ijyyx xzzij ij ijijHJσ σσσσσ⎛⎞⎜⎟⎝⎠= ++∑Heisenberg Computation over DFS/NS is Universal• Heisenberg exchange interaction:• Universal over collective-decoherence DFS [J. Kempe, D. Bacon, D.A.L., B. Whaley, Phys. Rev. A 63, 042307 (2001)]• Over 4-qubit DFS:CNOT involves 14 elementary steps (D. Bacon, Ph.D. thesis)• Implications for simplifying operation of spin-based quantum dot QCsHHeis=Pi,jJij(XiXj+YiYj+ ZiZj) ≡Pi,jJijEij≅⊕ ⊗^''()The allowed logic operationsJJndJAM IHeisenberg Computation over DFS/NS is Universal• Heisenberg exchange interaction:• Universal over collective-decoherence DFS [J. Kempe, D. Bacon, D.A.L., B. Whaley, Phys. Rev. A 63, 042307 (2001)]• Over 4-qubit DFS:CNOT involves 42 elementary steps (D. Bacon, Ph.D. thesis)• Implications for simplifying operation of spin-based quantum dot QCsHHeis=Pi,jJij(XiXj+YiYj+ ZiZj) ≡Pi,jJijEij¯X = −2√3(E13+12E12)¯Z = −E12eiθ¯Xand eiθ¯Zgenerate arbitrary single encoded qubit gates()()10 0110 01102L=− −()11 2 0011 2 1100 0110 1001 1010 010123L=+−+++≅⊕ ⊗^''()The allowed logic operationsJJndJAM IIn the beginning …In the beginning …13C-labeled alanine0Φ =1Φ =What about symmetry breaking? D.L., I.L. Chuang, K.B. Whaley, PRL 81, 2594 (1998); D. Bacon, D.L., K.B. Whaley, PRA 60, 1944 (1999)Symmetry breaking: unequal coupling constants, lowering of symmetry by a perturbation, etc.Introduce a perturbation via HSB7→ HSB+²∆H, k∆Hk = 1Theory shows that fidelity depends on ² only to second order.Robustness of DFS to symmetry breaking perturbations perturbationsangle strengthRobustness of DFS to symmetry breaking perturbations perturbationsDFS-encodedBare qubits12 1 2001 1 0    LL=⊗ =⊗Bare qubit: two hyperfine states of trapped 9Be+ ionChief decoherence sources:(i) fluctuating long-wavelength ambient magnetic fields;(ii) heating of ion CM motion during computation: a symmetry-breaking processDFS encoding: into pair of ionsStrong Symmetry BreakingNeed a way to deal with symmetry breaking…Intermission & Bathroom Break Need a way to deal with symmetry breaking…Part 2: Mostly Dynamical Decoupling NMR to the Rescue: Removal of Decoherence via Spin Echo=Time ReversalHahn spin echo ideaDynamical Decoupling BasicsA pulse producing a unitary evolution P, such that(CPMG, Hahn spin-echo)                                                           Ideal (zero-width) pulses, and ignoring      :†SB SB SB     i.e., { , } 0PH P H P H=−=††SB SB SB SBSB SBexp( ) exp( ) exp( )exp( )exp( ) exp( )P iH P iH iPH P iHiH iH Iτττ τ−−=− −=−τ τ τ††BHA pulse producing a unitary evolution P, such that(CPMG, spin-echo)                                                           Ideal (zero-width) pulses, and ignoring      :†SB SB SB     i.e., { , } 0PH P H P H=−=††SB SB SB SBSB SBexp( ) exp( ) exp( )exp( )exp( ) exp( )P iH P iH iPH P iHiH iH Iτττ τ−−=− −=−τ τ τ††stSB order Magnus ex n. pa )"time reversal",averaged (in to zero 1   XZX ZH=− ⇒SBH ZBλ= ⊗SBHSBHX XτBHDynamical Decoupling BasicsDynamical Decoupling = SymmetrizationDynamical Decoupling = Symmetrization Viola & Lloyd Phys. Rev. A 58, 2733 (1998); Byrd & Lidar, Q. Inf. Proc. 1, 19 (2002)system bathSystem-bath Hamiltonian:  SBSBHα αα⊗=∑zσxσyσSBHDynamical Decoupling = SymmetrizationDynamical Decoupling = Symmetrization Viola & Lloyd Phys. Rev. A 58, 2733 (1998); Byrd & Lidar, Q. Inf. Proc. 1, 19 (2002)system bathSystem-bath Hamiltonian:  SBSBHα αα⊗=∑zσxσyσSBHzσxσyσSBH−Apply rapid pulsesflipping sign of SαDynamical Decoupling = SymmetrizationDynamical Decoupling = Symmetrization Viola & Lloyd Phys. Rev. A 58, 2733 (1998); Byrd & Lidar, Q. Inf. Proc. 1, 19 (2002)system bathSystem-bath Hamiltonian:  SBSBHα αα⊗=∑zσxσyσSBHzσxσyσSBH−SBHApply rapid pulsesflipping sign of SαDynamical Decoupling = SymmetrizationDynamical Decoupling = Symmetrization Viola & Lloyd Phys. Rev. A 58, 2733 (1998); Byrd & Lidar, Q. Inf. Proc. 1, 19 (2002)system bathSystem-bath Hamiltonian:  SBSBHα αα⊗=∑zσxσyσSBHzσxσyσSBH−SBHApply rapid pulsesflipping sign of SαxσyσzσSBHMore general symmetrization:Dynamical Decoupling = SymmetrizationDynamical Decoupling = Symmetrization Viola & Lloyd Phys. Rev. A 58, 2733 (1998); Byrd & Lidar, Q. Inf. Proc. 1, 19 (2002)system bathSystem-bath Hamiltonian:  SBSBHα αα⊗=∑zσxσyσSBHzσxσyσSBH−SBHApply rapid pulsesflipping sign of SαxσyσzσSBHMore general symmetrization:Dynamical Decoupling = SymmetrizationDynamical Decoupling = Symmetrization Viola & Lloyd Phys. Rev. A 58, 2733 (1998); Byrd & Lidar, Q. Inf. Proc. 1, 19 (2002)system bathSystem-bath Hamiltonian:  SBSBHα αα⊗=∑zσxσyσSBHzσxσyσSBH−SBHApply rapid pulsesflipping sign of SαxσyσzσSBHMore general symmetrization:Dynamical Decoupling = SymmetrizationDynamical Decoupling = Symmetrization Viola & Lloyd Phys. Rev. A 58, 2733 (1998); Byrd & Lidar, Q. Inf. Proc. 1, 19 (2002)system bathSystem-bath Hamiltonian:  SBSBHα αα⊗=∑zσxσyσSBHzσxσyσSBH−SBHApply rapid pulsesflipping sign of SαxσyσzσSBHMore general symmetrization: averaged to zero.SBHDealing with Symmetry Breaking: Creating Collective Dephasing Conditions L.-A. Wu, D.A.L., Phys. Rev. Lett. 88, 207902 (2002)1122General two-qubit dephasing:   zzSBHBBσσ= ⊗+⊗()()()()12 1 2 12121212zzzzZBB BBσσσσ=⊗−++⊗+−	Dealing with Symmetry Breaking: Creating Collective Dephasing Conditions L.-A. Wu, D.A.L., Phys. Rev. Lett. 88, 207902 (2002)( )12 12 12Assume: controllable exchange1.2yyxx z zX σ σσσ ασσ= ++1122General two-qubit dephasing:   zzSBHBBσσ= ⊗+⊗()()()()12 1 2 12121212zzzzZBB BBσσσσ=⊗−++⊗+−	{,}0     - X ZXZXZ= ⇒=Dealing with Symmetry Breaking: Creating Collective Dephasing Conditions L.-A. Wu, D.A.L., Phys. Rev. Lett. 88, 207902 (2002)( )12 12 12Assume: controllable exchange1.2yyxx z zX σ σσσ ασσ= ++1122General two-qubit dephasing:   zzSBHBBσσ= ⊗+⊗()()()()12 1 2 12121212zzzzZBB BBσσσσ=⊗−++⊗+−	()()12 12Collective Dephasingexp( ) exp( )exp( )exp( ) exp( )22SB SBzzBBiH t i X iH t i X itπ πσσ−−− +− += ⊗⎡⎤⎢⎥⎣⎦	X XSBHtSBHt= 2t Coll.Deph.“Time reversal” Dynamical Decoupling pulse sequence:{,}0     - X ZXZXZ= ⇒=Heisenberg is “Super-Universal”Same method works, e.g., for spin-coupled quantum dotsquantum dots QC: ( )12 12 12HeisBy BB pulsing of col decoherenclective  conditions can be c2reat :e edyyx xzzJH σ σσσσσ= ++Details: L.-A. Wu, D.A.L.,  , 207902 Requires sequence of 6 /2 pulses to create collective decoherenceconditions over blocks of 4 qubits. Leakage elimination requires 7 more pulses.Phys. Rev. Lett.π88 (2002); L.A. Wu, M.S. Byrd, D.A.L.,  , 127901 (2002).Phys. Rev. Lett. 89Earlier DFS work showed universal QC with Heisenberg interaction alone possible [Bacon, Kempe, D.A.L., Whaley, Phys. Rev. Lett. 85, 1758 (2000)] :SB1    zyyx xzzii ii iiynizxxyyB BBHg g gSBSBSBσσσ== ++⊗⊗⊗⊗+⊗+⊗∑→All ingredients available for Heisenberg-only QCAnalysis of Dynamical DecouplingWe’ll need a formal detour…Decoherence: Isolated vs Open System EvolutionIsolated system: H = HS|ψ(t)i = US(t)|ψ(0)i˙US= −iHSUSUS(0) = Iequivalently: |ψ(t)ihψ(t)| = US(t)|ψ(0)ihψ(0)|U†S(t)Decoherence: Isolated vs Open System EvolutionIsolated system: H = HS|ψ(t)i = US(t)|ψ(0)i˙US= −iHSUSUS(0) = IOpen system: H = HS+HB+HSBρSB(t) = U(t)ρSB(0)U†(t)˙U = −iHU U(0) = IρS(t) = TrBρSB(t)6= unitary tranformation of ρS(0)(except when there is a decoherence-free subspace)equivalently: |ψ(t)ihψ(t)| = US(t)|ψ(0)ihψ(0)|U†S(t)Decoherence: Isolated vs Open System EvolutionIsolated system: H = HS|ψ(t)i = US(t)|ψ(0)i˙US= −iHSUSUS(0) = IOpen system: H = HS+HB+HSBρSB(t) = U(t)ρSB(0)U†(t)˙U = −iHU U(0) = IρS(t) = TrBρSB(t)6= unitary tranformation of ρS(0)(except when there is a decoherence-free subspace)kρS(t)−|ψ(t)ihψ(t)|k > 0decoherence:which norm?equivalently: |ψ(t)ihψ(t)| = US(t)|ψ(0)ihψ(0)|U†S(t)Kolmogorov Distance and Quantum Measurements (I)Given two classical probability distributions {p(1)i} and {p(2)i},their deviation is measured by the Kolmogorov distanceD(p(1),p(2)) ≡12Pi|p(1)i−p(2)i|Kolmogorov Distance and Quantum Measurements (I)A quantum measurement can always be described in terms of a POVM(positive operator valued measure), i.e., a set of positive operators EisatisfyingPiEi= I, where i enumerates the possible measurementoutcomes.For a system initially in the state ρ, outcome i occurs with probabilitypi= Pr(i|ρ) = Tr(ρEi)Given two classical probability distributions {p(1)i} and {p(2)i},their deviation is measured by the Kolmogorov distanceD(p(1),p(2)) ≡12Pi|p(1)i−p(2)i|Kolmogorov Distance and Quantum Measurements (I)A quantum measurement can always be described in terms of a POVM(positive operator valued measure), i.e., a set of positive operators EisatisfyingPiEi= I, where i enumerates the possible measurementoutcomes.For a system initially in the state ρ, outcome i occurs with probabilityThus quantum measurements produce classical probability distributions.pi= Pr(i|ρ) = Tr(ρEi)Given two classical probability distributions {p(1)i} and {p(2)i},their deviation is measured by the Kolmogorov distanceConsider two quantum states ρ(1)[= |ψ(t)ihψ(t)|] and ρ(2)[= ρS(t)]D(p(1),p(2)) ≡12Pi|p(1)i−p(2)i|Kolmogorov Distance and Quantum Measurements (II)Compare measurement outcomes of same POVM onρ(1)[= |ψ(t)ihψ(t)|] and ρ(2)[= ρS(t)]:Lemma: δ ≡ D(pρ(1),pρ(2)) ≤ kρ(1)−ρ(2)kTrThe bound is tight in the sense that it is saturated for the optimal measurement designed to distinguish the two states.kAkTr≡ Tr√A†A =P(singular values(A))Partial trace decreases trace distanceLemma: kρ(1)S−ρ(2)SkTr≤ kρ(1)SB−ρ(2)SBkTrPartial trace decreases trace distanceLemma: kρ(1)S−ρ(2)SkTr≤ kρ(1)SB−ρ(2)SBkTrPartial trace decreases trace distanceConclusion: we can compare dynamics of ideal and actual systems over thejoint system-bath space.Lemma: kρ(1)S−ρ(2)SkTr≤ kρ(1)SB−ρ(2)SBkTrIdeal vs Actual System EvolutionIdeal system: H = HS+HBρidealSB(t) = [US(t)⊗UB(t)]ρSB(0)[U†S(t)⊗U†B(t)]˙US/B= −iHS/BUS/BUS/B(0) = IIdeal vs Actual System EvolutionρSB(t) = U(t)ρSB(0)U†(t)˙U = −iHU U(0) = IIdeal system: H = HS+HBρidealSB(t) = [US(t)⊗UB(t)]ρSB(0)[U†S(t)⊗U†B(t)]˙US/B= −iHS/BUS/BUS/B(0) = IActual system: H = HS+ HB+HSBIdeal vs Actual System EvolutionρSB(t) = U(t)ρSB(0)U†(t)˙U = −iHU U(0) = IIdeal system: H = HS+HBρidealSB(t) = [US(t)⊗UB(t)]ρSB(0)[U†S(t)⊗U†B(t)]˙US/B= −iHS/BUS/BUS/B(0) = IActual system: H = HS+ HB+HSBDistance:kρSB(t)−ρidealSB(t)kTr= kV(t)ρSB(0)V†(t)−ρSB(0)kTrV (t) ≡ U†S(t)⊗U†B(t)U(t) ≡ exp[−itHeff(t)]Ideal vs Actual System EvolutionρSB(t) = U(t)ρSB(0)U†(t)˙U = −iHU U(0) = IIdeal system: H = HS+HBρidealSB(t) = [US(t)⊗UB(t)]ρSB(0)[U†S(t)⊗U†B(t)]˙US/B= −iHS/BUS/BUS/B(0) = IActual system: H = HS+ HB+HSBDistance:kρSB(t)−ρidealSB(t)kTr= kV(t)ρSB(0)V†(t)−ρSB(0)kTrV (t) ≡ U†S(t)⊗U†B(t)U(t) ≡ exp[−itHeff(t)]Lemma: kρSB(t)−ρidealSB(t)kTr≤ tkHeff(t)k∞follows from keiA−eiBk∞≤kA−Bk∞; kAk∞≡sup|vi,hv|vi=1qhv|A†A|vi = maxsing.val.(A)Kolmogorov Distance Bound from Effective HamiltonianV (t) ≡ U†S(t)⊗U†B(t)U(t)Lemma: δ ≡ D(pρ(1),pρ(2)) ≤ kρ(1)−ρ(2)kTrLemma: kρ(1)S−ρ(2)SkTr≤ kρ(1)SB−ρ(2)SBkTr(trace distance bounds Kolmogorov distance)(partial trace decreasesdistinguishability)≡ exp[−itHeff(t)]Lemma: kρSB(t)−ρidealSB(t)kTr≤ tkHeff(t)k∞δactual,ideal≤ tkHeff(t)k∞≡ η(t) ≡ noise strengthGoal: reduce effective Hamiltonian. Method: dynamical decoupling.δactual,ideal≤ tkHeff(t)k∞≡ η(t) ≡ noise strengthGoal: reduce effective Hamiltonian. Method: dynamical decoupling.How do we compute Heff(t)? The Magnus expansion˙U = −iH(t)UU(t) = e−itHeff(t)Heff(t) =1t∞Xj=1Ωj(t)Ω1(t) =Zt0dt1H(t1) Ω2(t) = −i2Zt0dt1Zt10dt2[H(t1),H(t2)]δactual,ideal≤ tkHeff(t)k∞≡ η(t) ≡ noise strengthAnalysis of Dynamical Decoupling“Symmetrizing group” of pulses { gi } and their inverses are applied in series:Periodic DD: periodic repetition of the universal DD pulse sequenceexp( )SBiH τ≡ −f†††22†11()()()exp( )iSBiNiNg g gg gg gHi gτ≈−∑"ffffirst order Magnus expansionDynamical Decoupling Theory“Symmetrizing group” of pulses { gi } and their inverses are applied in series:Choose the pulses so that: Periodic DD: periodic repetition of the universal DD pulse sequence†0SB i SB iiHH gHg≡=∑(1)eff6Dynamical Decoupling Conditionexp( )SBiH τ≡ −f†††22†11()()()exp( )iSBiNiNg g gg gg gHi gτ≈−∑"ffffirst order Magnus expansionDynamical Decoupling Theory“Symmetrizing group” of pulses { gi } and their inverses are applied in series:Choose the pulses so that: For a qubit the Pauli group G={X,Y,Z,I } (π pulses around all three axes) removes an arbitrary HSB :Periodic DD: periodic repetition of the universal DD pulse sequence(XfX)(YfY)(ZfZ)(IfI) = XfZfXfZf†0SB i SB iiHH gHg≡=∑(1)eff6Dynamical Decoupling Conditionexp( )SBiH τ≡ −f†††22†11()()()exp( )iSBiNiNg g gg gg gHi gτ≈−∑"ffffirst order Magnus expansionDynamical Decoupling TheoryAnother view of the universal decoupling sequence:The Effective Hamiltonianτexp[ ]SBiHτ≡ −fδX Z ZXf f f f'=fexp[ ( )]                    ( )=0, ideallyeffeffiTH THT≡ −f'Another view of the universal decoupling sequence:The Effective Hamiltonianτexp[ ]SBiHτ≡ −fδX Z ZX X"f f f f f'=fexp[ ( )]                    ( )=0, ideallyeffeffiTH THT≡ −f'But, errors accumulate…: () 0effHT≠Periodic Dynamical DecouplingPDD Strategy: repeat the basic XfZfZfXfZ cycle with total of N pulses. The total duration is fixed at T. N can be changed. Pulse interval:Recall noise strengthnorm of final effective system-bath Hamiltonian times the total duration.τ = T/Nη ≡ ||Heff(T)||TPeriodic Dynamical DecouplingPDD Strategy: repeat the basic XfZfZfXfZ cycle with total of N pulses. The total duration is fixed at T. N can be changed. Pulse interval:Recall noise strengthnorm of final effective system-bath Hamiltonian times the total duration.PDD leading order result for error:Can we do better?τ = T/Nη ∝N−1η ≡ ||Heff(T)||TDD as a Rescaling Transformation• Interaction terms are rescaled after the DD cycle • We need a mechanism to continue thisJ = J(0)7→ J(1)∝ max[τ(J(0))2,τβJ(0)]β7→ β +O((J(0))3τ2)J= kHSBk∞β= kHBk∞Concatenated Universal Dynamical DecouplingNest the universal DD pulse sequence into its own free evolution periods f :p(1)= X f   Z f   X f   Z fConcatenated Universal Dynamical DecouplingNest the universal DD pulse sequence into its own free evolution periods f :p(1)= X f   Z f   X f   Z fp(2)= X p(1)Z p(1)X p(1)Z p(1)p(n+1)= X p(n)Z p(n)X p(n)Z p(n)Concatenated Universal Dynamical DecouplingNest the universal DD pulse sequence into its own free evolution periods f :p(1)= X f   Z f   X f   Z fp(2)= X p(1)Z p(1)X p(1)Z p(1)p(n+1)= X p(n)Z p(n)X p(n)Z p(n)Level Concatenated DD Series after multiplying Pauli matrices1XfZfXfZf2fZfXfZfYfZfXfZffZfXfZfYfZfXfZf3XfZfXfZfYfZfXfZffZfXfZfYfZfXfZfZfZfXfZfYfZfXfZffZfXfZfYfZfXfZfXfZf XfZfYfZfXfZffZfXfZfYfZfXfZfZfZfXfZfYfZfXfZffZfXfZfYfZfXfZfLength grows exponentially; how about error reduction?Performance of Concatenated Sequences2 22 222 2error  (error)   ((error) )   (((error) ) )    (error)kk"66 6 66For fixed total time T=Nτ and N zero-width (ideal) pulses:Compare to  periodic DD: Performance of Concatenated Sequences[Khodjasteh & Lidar, PRA 75, 062310 (2007) ]2 22 222 2error  (error)   ((error) )   (((error) ) )    (error)kk"66666η ∝ NbN−clogNη ∝N−1Experimentsadamantene;qubit = 13CPDD=CDD1CDD2CDD3Concatenated DD on Adamantene Powder Dieter Suter, TU DortmundCDD Results25 sτ μ=50 sτ μ=85 sτ μ=10.5 sδ μ=Echo intensityEcho intensityEcho intensityConcatenated DD for electron spin of 31P donors in Si Steve Lyon, Princeton31P donor: Electron spin (S) = ½, Nuclear spin (I) = ½↑e,↓n↑e,↑n↓e,↑n↓e,↓nνrf1νμw1 νμw2νrf2|1〉|0〉29bath is Si~1% natural abundancePeriodic DD vs Concatenated DD012340.00.51.0 FidelityConcatenation LevelState +XState +Y1. Periodic(XfYfXfYf)2. Concatenated0 5 10 15 20 25 300.00.51.0 Number of Repeats FidelityState +XState +Y[log(time)]160nsδ =relative echo intensityrelative echo intensity~100ms~50msBetter than Concatenated DD?Does there exist an optimal pulse sequence?Optimal = removes maximum decoherence with least possible number of pulsesBetter than Concatenated DD?Better than Concatenated DD?ORBetter than Concatenated DD?ORBetter than Concatenated DD?Better than Concatenated DD?“Quadratic DD” eliminates the first n orders in the Dyson series of thejoint system-bath propagator using n2 pulsesConcatenated DD requires 4n pulses to do the same, approximatelyInner workings of Quadratic DDj,k ∈ {1,n}Inner workings of Quadratic DDj,k ∈ {1,n}For every value of n, the first√n terms in the Dyson series are removedComparison of DD SequencesSummary• Symmetry as a unifying principle for both passive and active error prevention/correction strategies• A comprehensive strategy can take advantage of a layered approach:Hybrid Q. Error Correction: The Big PictureDFS encodingQECCHybrid Q. Error Correction: The Big PictureDFS encoding- symmetry not for free…QECCHybrid Q. Error Correction: The Big PictureDFS encoding- symmetry not for free…DDQECCHybrid Q. Error Correction: The Big PictureDFS encoding- symmetry not for free…DDQECC-pulse errors,Markovian effectsHybrid Q. Error Correction: The Big PictureDFS encoding- symmetry not for free…DDFT-QEC-pulse errors,Markovian effectsOpen Questions• What is the optimal hybrid strategy?• Is the fault tolerance threshold better for a  hybrid strategy?see: H.‐K. Ng, D.A.L., and J. Preskill, “Combining dynamical decoupling with  fault‐tolerant quantum computation”, arXiv:0911:3202 

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