Hybrid quantum error prevention, reduction, and correction methods 10th Canadian Summer School on Quantum Information UBC, July 21, 2010 Phys. Rev. Lett. 91, 217904 (2003) Daniel Lidar DFSencoded no encoding Science 291, 291, 1013 (2001) University of Southern California probability of correct Grover answer Outline Symmetry and preserved quantum information System-bath, decoherence and all that Unified view of decoherence-free subspaces, noiseless subsystems, quantum error correcting codes, operator quantum error correction DFS/NS examples: theory and experiment Outline Symmetry and preserved quantum information System-bath, decoherence and all that Unified view of decoherence-free subspaces, noiseless subsystems, quantum error correcting codes, operator quantum error correction DFS/NS examples: theory and experiment Feel free to interrupt and ask lots of questions! Symmetry Protects Q. Information Symmetry Î conserved quantity = quantum info. Symmetry Protects Q. Information Symmetry Î conserved quantity = quantum info. 1) DFS/NS: Error Prevention use an existing exact symmetry to perfectly hide q. info. from bath H DFS Symmetry Protects Q. Information Symmetry Î conserved quantity = quantum info. 1) DFS/NS: Error Prevention use an existing exact symmetry to perfectly hide q. info. from bath H DFS 2) Dynamical Decoupling: Open-loop control dynamically generate a symmetry strong H BANG + free evolution bath correlation time SB time Whence the Errors? Decoherence from System-Bath Interaction Whence the Errors? Decoherence from System-Bath Interaction Every real quantum system is coupled to an environment (“bath”). Full Hamiltonian: H = H S + H B + H SB H SB = ∑ Sα ⊗ Bα α system bath Whence the Errors? Decoherence from System-Bath Interaction Every real quantum system is coupled to an environment (“bath”). H = H S + H B + H SB Full Hamiltonian: H SB = ∑ Sα ⊗ Bα α system System dynamics alone: ρ (t ) = TrB ⎡⎢e−iHt ρSB (0)eiHt ⎤⎥ ⎣ bath Decoherence ⎦ = ∑ k ck Ek (t ) ρ (0) Ek (t ) † Non-unitary evolution of system Whence the Errors? Decoherence from System-Bath Interaction Every real quantum system is coupled to an environment (“bath”). H = H S + H B + H SB Full Hamiltonian: H SB = ∑ Sα ⊗ Bα α system System dynamics alone: bath ρ (t ) = TrB ⎡⎢e−iHt ρSB (0)eiHt ⎤⎥ ⎣ Decoherence ⎦ = ∑ k ck Ek (t ) ρ (0) Ek (t ) † Markovian master (Lindblad) equation: ∂ρ ∂t Non-unitary evolution of system = −i[ H , ρ ] + 1 ∑ aα ,α ' ( 2Sα ρ Sα† ' − ρ Sα† 'Sα − Sα† 'Sα ρ ) S 2 α ,α ' What is there is a symmetry? Symmetric coin flipping noise What is there is a symmetry? Symmetric coin flipping noise How to reliably store a single bit? What is there is a symmetry? Symmetric coin flipping noise How to reliably store a single bit? logical 1 logical 0 A noiseless subspace. Unified view of Quantum Information Protection: Fixed Codes Error Model: Trace-preserving completely-positive (CP) maps: ρ0 = P † E ρE k k k ≡ E(ρ) P † E k k Ek = I Unified view of Quantum Information Protection: Fixed Codes Error Model: Trace-preserving completely-positive (CP) maps: ρ0 = P † E ρE k k k ≡ E(ρ) P † E k k Ek = I Decoherence-free subspace (DFS): H = A ⊕ B C = {all states ρ : A → A} such that E(ρ) = ρ Unified view of Quantum Information Protection: Fixed Codes Error Model: Trace-preserving completely-positive (CP) maps: ρ0 = P † E ρE k k k ≡ E(ρ) P † E k k Ek = I Decoherence-free subspace (DFS): H = A ⊕ B C = {all states ρ : A → A} such that E(ρ) = ρ Noiseless subsystem (NS): H = A ⊕ B, A = N ⊗ G C = {all states ρ : A → A} such that TrG E(ρ) = TrG ρ passive Unified view of Quantum Information Protection: Fixed Codes Error Model: Trace-preserving completely-positive (CP) maps: ρ0 = P † E ρE k k k ≡ E(ρ) P † E k k Ek = I Decoherence-free subspace (DFS): H = A ⊕ B C = {all states ρ : A → A} such that E(ρ) = ρ Noiseless subsystem (NS): H = A ⊕ B, A = N ⊗ G C = {all states ρ : A → A} such that TrG E(ρ) = TrG ρ Quantum error correcting code (QEC): H = A ⊕ B C = {all states ρ : A → A} such that ∃ CP map R for which R ◦ E(ρ) = ρ passive Unified view of Quantum Information Protection: Fixed Codes Error Model: Trace-preserving completely-positive (CP) maps: ρ0 = P † E ρE k k k ≡ E(ρ) P † E k k Ek = I Decoherence-free subspace (DFS): H = A ⊕ B C = {all states ρ : A → A} such that E(ρ) = ρ Noiseless subsystem (NS): H = A ⊕ B, A = N ⊗ G C = {all states ρ : A → A} such that TrG E(ρ) = TrG ρ Quantum error correcting code (QEC): H = A ⊕ B C = {all states ρ : A → A} such that ∃ CP map R for which R ◦ E(ρ) = ρ Operator QEC (OQEC): H = A ⊕ B, A = N ⊗ G C = {all states ρ : A → A} such that ∃CP map R for which TrG R ◦ E(ρ) = TrG ρ passive active DFS as a QEC, QEC as a DFS Decoherence-free subspace (DFS): H = A ⊕ B C = {all states ρ : A → A} such that E(ρ) = ρ Quantum error correcting code (QEC): H = A ⊕ B C = {all states ρ : A → A} such that ∃ CP map R for which R ◦ E(ρ) = ρ A DFS is a QEC with trivial recovery operation: R = I A QEC is a DFS with respect to the map R ◦ E NS as an OQEC, OQEC as an NS Noiseless subsystem (NS): H = A ⊕ B, A = N ⊗ G C = {all states ρ : A → A} such that TrG E(ρ) = TrG ρ Operator QEC (OQEC): H = A ⊕ B, A = N ⊗ G C = {all states ρ : A → A} such that ∃CP map R for which TrG R ◦ E(ρ) = TrG ρ An NS is an OQEC with trivial recovery operation: R = I An OQEC is an NS with respect to the map R ◦ E Unitarily Invariant DFS Unitarily Invariant DFS:=Subspace of full system Hilbert space in which evolution is purely unitary Unitarily Invariant DFS Unitarily Invariant DFS:=Subspace of full system Hilbert space in which evolution is purely unitary More precisely: ⊥ Let the system Hilbert space H decompose into a direct sum as Hµ= HD ⊕H¶ D, ρD ρ2 . and partition the system state ρS accordingly into blocks: ρS = ρ†2 ρ3 Assume ρD (0) 6= 0. Note imperfect initialization! Then HD is called decoherence-free iﬀ the initial and final DFS-blocks of ρS are unitarily related: ρD (t) = UD ρD (0)UD† , where UD is a unitary matrix acting on HD . U.I. DFS Conditions for CP Maps Given a CP map: ρ0 = P † E ρE k k k ≡ E(ρ) P † E k k Ek = I Theorem A necessary and suﬃcient condition for the existence of a DFS HD with respect to the CP map E is that all Kraus operators have a matrix representation of the form ¶ µ ck UD 0 , Ek = 0 Bk where UD is unitary, ck are scalars satisfying P † ⊥ operators on HD satisfying k Bk Bk = I. P k |ck |2 = 1, and Bk are arbitrary Meaning: Ek act unitarily on the DFS U.I. DFS Conditions for Master Equations Given a Markovian master equation: dρ dt = −i[HS , ρ] + 1 2 P † † † 2F ρF − ρF F − F Fα ρ α α α α α α Theorem A necessary and suﬃcient condition for the existence of a DFS HD with respect to the Markovian master equation above is that the Lindblad operators Fα and the system Hamiltonian HS have the block-diagonal form ¶ ¶ µ µ 0 HD cα I 0 , F , = HS = α ⊥ 0 Bα 0 HD ⊥ where HD and HD are Hermitian, cα are scalars, and Bα are arbitrary operators ⊥ on HD . Meaning: Fα act as identity on the DFS, while HS preserves the DFS Exercise 1. Prove sufficiency (easy) and necessity (not so easy) of the U.I. DFS conditions for CP maps and Markovian master equations 2. Generalize to NS, QEC, OQEC Where is the promised symmetry? How do we find and construct a DFS? U.I. DFS Conditions for Hamiltonian Dynamics Under Hamiltonian dynamics system and bath evolve jointly subject to the Schrodinger equation with the Hamiltonian H = H S + H SB + H B. Find a subspace where H SB = ∑α Sα ⊗ Bα acts trivially, i.e.: make H SB ∝ I S ⊗ OB Also, remember that HS must preserve the DFS. U.I. DFS Conditions for Hamiltonian Dynamics Under Hamiltonian dynamics system and bath evolve jointly subject to the Schrodinger equation with the Hamiltonian H = H S + H SB + H B. Find a subspace where H SB = ∑α Sα ⊗ Bα acts trivially, i.e.: make H SB ∝ I S ⊗ OB Also, remember that HS must preserve the DFS. Theorem Let A = alg{I, Sα , Sα† }. Assume [HS , A] = 0. The dimension of the DFS HD equals the degeneracy of the 1-dimensional irreducible representation (irrep) of A. degeneracy symmetry Simplest DFS Example: Collective Dephasing DFS idea: Find a subspace where H SB = ∑α Sα ⊗ Bα acts trivially, make H SB ∝ I S ⊗ OB i.e.: Permutation symmetry in z direction: B(t ) z ψ1 ψ2 Long-wavelength magnetic field B (environment) couples to spins Effect: Random "Collective Dephasing": ψ j = a j 0 j + b j 1 j 6 a j 0 j + eiθ b j 1 j random but j-independent phase Simplest DFS Example: Collective Dephasing DFS idea: Find a subspace where H SB = ∑α Sα ⊗ Bα acts trivially, make H SB ∝ I S ⊗ OB i.e.: Permutation symmetry in z direction: B(t ) z H int = g (σ1z + σ1z ) ⊗ B = ⎛ −2 gB ⎞ ⎜ ⎟ 0 ⎜ ⎟ ⎜ ⎟ 0 ⎜ ⎟ 2 gB ⎠ ⎝ ↓ ↓ ↑ ↑ 1 1 1 1 ↓ ↑ ↓ ↑ 2 2 ψ1 2 ψ2 2 Long-wavelength magnetic field B (environment) couples to spins Effect: Random "Collective Dephasing": iθ ψ j = a j 0 j + bj 1 j 6 a j 0 j + e bj 1 DFS encoding j 0 random but j-independent phase L = 0 ⊗1 1 1 =1 ⊗0 L 1 2 2 Why it Works Collective dephasing: ψ j = a j 0 j + b j 1 j 6 a j 0 j + e iθ b j 1 Case of two qubits: 0 ⊗ 0 6 0 ⊗ 0 ≡ 00 ( ) 0 ⊗ 1 6 0 ⊗ e iθ 1 ≡ e iθ 01 ( 6 (e ) 1 ) ⊗ (e 1 ⊗ 0 6 e iθ 1 ⊗ 0 ≡ e iθ 10 1 ⊗ 1 iθ iθ ) 1 ≡ e 2iθ 11 j Why it Works Collective dephasing: ψ j = a j 0 j + b j 1 j 6 a j 0 j + e iθ b j 1 j Case of two qubits: 0 ⊗ 0 6 0 ⊗ 0 ≡ 00 ( ) 0 ⊗ 1 6 0 ⊗ e iθ 1 ≡ e iθ 01 ( 6 (e ) 1 ) ⊗ (e 1 ⊗ 0 6 e iθ 1 ⊗ 0 ≡ e iθ 10 1 ⊗ 1 iθ iθ ) ≡ 0 L ≡ 1L 1 ≡ e 2iθ 11 Global phase physically irrelevant: ψ L =a 0 L +b 1 L is decoherence-free: A 2-dimensional protected subspace. Why it Works Collective dephasing: ψ j = a j 0 j + b j 1 j 6 a j 0 j + e iθ b j 1 j pop quiz: Are the states |00i and |11i also in a DFS? Case of two qubits: 0 ⊗ 0 6 0 ⊗ 0 ≡ 00 ( ) 0 ⊗ 1 6 0 ⊗ e iθ 1 ≡ e iθ 01 ( 6 (e ) 1 ) ⊗ (e 1 ⊗ 0 6 e iθ 1 ⊗ 0 ≡ e iθ 10 1 ⊗ 1 iθ iθ ) ≡ 0 L ≡ 1L 1 ≡ e 2iθ 11 Global phase physically irrelevant: ψ L =a 0 L +b 1 L is decoherence-free: A 2-dimensional protected subspace. Generalization: Noiseless Subsystems [E. Knill, R. Laflamme and L. Viola, PRL 84, 2525 (2000)] The dimension of the DFS HD equals the degeneracy of the 1-dimensional irreducible representation (irrep) of A. 1D irreps condition not needed… Generalization: Noiseless Subsystems [E. Knill, R. Laflamme and L. Viola, PRL 84, 2525 (2000)] The dimension of the DFS HD equals the degeneracy of the 1-dimensional irreducible representation (irrep) of A. 1D irreps condition not needed… A theorem from C* algebras: Associative algebra A = polynomials{I, Sα , Sα† } Model of decoherence: Matrix representation over ^2 : HSB = ∑ Sα ⊗ Bα A ≅ ⊕ InJ ⊗ MdJ (^) N J α irreducible representations dimension multiplicity Hilbert space decomposition: N ^2 ≅ ⊕ ^nJ ⊗ ^dJ J Generalization: Noiseless Subsystems [E. Knill, R. Laflamme and L. Viola, PRL 84, 2525 (2000)] The dimension of the DFS HD equals the degeneracy of the 1-dimensional irreducible representation (irrep) of A. 1D irreps condition not needed… A theorem from C* algebras: Associative algebra A = polynomials{I, Sα , Sα† } Model of decoherence: Matrix representation over ^2 : HSB = ∑ Sα ⊗ Bα A ≅ ⊕ InJ ⊗ MdJ (^) N J α irreducible representations dimension multiplicity Hilbert space decomposition: Each J labels an nJ -dimensional NS code DFS is the case dJ = 1 N ^2 ≅ ⊕ ^nJ ⊗ ^dJ J code subsystem nJ > 1 iff ∃ symmetry in system-env. interaction Isotropic Quantum Errors: Collective Decoherence Model Describes, e.g., lowT decoherence due to phonons in various solid state QC proposals ψ ϕ 1 B (t )zˆ Z B (t )yˆ Y 2 B (t )xˆ X Isotropic Quantum Errors: Collective Decoherence Model Describes, e.g., lowT decoherence due to phonons in various solid state QC proposals ψ Error model, N qubits: "Collective Decoherence" HSB = ∑ (σ 1α + " + σ Nα ) ⊗ Bα α = x,y ,z Sα =total spin operator ϕ 1 B (t )zˆ Z B (t )yˆ Y 2 B (t )xˆ X Isotropic Quantum Errors: Collective Decoherence Model Describes, e.g., lowT decoherence due to phonons in various solid state QC proposals ψ ϕ 1 B (t )yˆ Y 2 Error model, N qubits: ⎧ψ ⎪ ⎪U X (1) ⊗ " ⊗ U X (N ) ψ 6⎨ ⎪UY (1) ⊗ " ⊗ UY (N ) ⎪U (1) ⊗ " ⊗ U (N ) Z ⎩ Z prob. p0 prob. p1 B (t )zˆ Z B (t )xˆ X prob. p2 prob. p3 Do irreps analysis of n copies of su(2)… All Decoherence-Free Subspaces/Subsystems for Collective Decoherence Hilbert space decomposition: J 2 N ^2 ≅ ⊕ ^ J ⊗ ^ n J 1 2 = 14 5 5 2 3 multiplicity nJ, counts paths; dJ=2J+1 9 2 1 1 0L ≡ 4 3 1 1 0 1 1 3/2 1/2 dJ 4 1 ( 01 − 10 2 5 ) 6 14 7 8 n All Decoherence-Free Subspaces/Subsystems for Collective Decoherence Hilbert space decomposition: J 2 N ^2 ≅ ⊕ ^ J ⊗ ^ n J 1 2 = 5 ) 6 = 14 7 8 = 1 ( 01 − 10 ) 1 2 = 0L 1/2 0 5 4 1 ( 01 − 10 2 1 14 5 2 3 multiplicity nJ, counts paths; dJ=2J+1 9 2 1 1 0L ≡ 4 3 1 1 0 1 1 3/2 1/2 dJ 1 2 n 3 2 1 110 − ( 011 − 101 3 3 1 n = 1L 1/2 0 1 2 3 n ) All Decoherence-Free Subspaces/Subsystems for Collective Decoherence Hilbert space decomposition: J N ^2 ≅ ⊕ ^ J ⊗ ^ 2 n dJ J 1 4 1 3/2 3 1 1 1 0 1 2 4 0 5 5 6 = 14 7 8 1 ( 01 − 10 ) 1 2 = 0L 1 2 n 3 2 1 110 − ( 011 − 101 3 3 1 n = 1L 1/2 0 1 2 n 3 1L ≡ 0L ≡ = = 1/2 14 2 3 1 9 5 2 1 1/2 multiplicity nJ, counts paths; dJ=2J+1 1 ( 01 − 10 2 )( 01 − 10 ) = 1 2 2 0011 + 2 1100 − ( 0110 ( 3 + 1001 + 1010 + 0101 )) ) What is the “Volume” of a DFS/NS? Degeneracy for given J , M = dimension of DFS/NS ≡ DJ (n) = no. of encoded qubits ⇒ code rate ≡ no. of physical qubits ( J =0 ) = n!(2 J + 1) ( n / 2 + J + 1)!( n / 2 − J )! log 2 D0 (n) n→∞ 3 log 2 n ⎯⎯⎯ →1 − 2 n n What is the “Volume” of a DFS/NS? Degeneracy for given J , M = dimension of DFS/NS ≡ DJ (n) = no. of encoded qubits ⇒ code rate ≡ no. of physical qubits ( J =0 ) = n!(2 J + 1) ( n / 2 + J + 1)!( n / 2 − J )! log 2 D0 (n) n→∞ 3 log 2 n ⎯⎯⎯ →1 − 2 n n DFS’s for collective decoherence asymptotically fill the Hilbert space! Computation Inside a U.I. DFS/NS So far have storage. What about computation? To prevent decoherence, computation should never leave DFS/NS. Which logic operations are compatible? Computation Inside a U.I. DFS/NS So far have storage. What about computation? To prevent decoherence, computation should never leave DFS/NS. Which logic operations are compatible? Error algebra: A ≅ ⊕ InJ ⊗ MdJ (^) J Code subsystem: N ^2 ≅ ⊕ ^nJ ⊗ ^dJ J Commutant = operators commuting with A A ' ≅ ⊕ M 'nJ (^ ) ⊗ IdJ J The allowed logic operations! Computation Inside a U.I. DFS/NS So far have storage. What about computation? To prevent decoherence, computation should never leave DFS/NS. Which logic operations are compatible? Error algebra: A ≅ ⊕ InJ ⊗ MdJ (^) J Code subsystem: N ^2 ≅ ⊕ ^nJ ⊗ ^dJ J Commutant = operators commuting with A A ' ≅ ⊕ M 'nJ (^ ) ⊗ IdJ J The allowed logic operations! Universal quantum computation over DFS/NS is possible using “exchange Hamiltonians”, e.g., Heisenberg interaction: Jij ⎛ x x H Heis = ∑ ⎜σ i σ j + σ iyσ jy + σ izσ zj ⎞⎟ ⎠ ij 2 ⎝ Heisenberg Computation over DFS/NS is Universal • Heisenberg exchange interaction: HHeis = P i,j Jij (Xi Xj + Yi Yj + Zi Zj ) ≡ P i,j Jij Eij • Universal over collective-decoherence DFS [J. Kempe, D. Bacon, D.A.L., B. Whaley, Phys. Rev. A 63, 042307 (2001)] • Over 4-qubit DFS: A ' ≅ ⊕ M 'nJ (^) ⊗ IdJ J The allowed logic operations CNOT involves 14 elementary steps (D. Bacon, Ph.D. thesis) • Implications for simplifying operation of spin-based quantum dot QCs Heisenberg Computation over DFS/NS is Universal • Heisenberg exchange interaction: HHeis = P i,j Jij (Xi Xj + Yi Yj + Zi Zj ) ≡ P i,j Jij Eij • Universal over collective-decoherence DFS [J. Kempe, D. Bacon, D.A.L., B. Whaley, Phys. Rev. A 63, 042307 (2001)] A ' ≅ ⊕ M 'nJ (^) ⊗ IdJ • Over 4-qubit DFS: J 0L = 1L = X̄ = − √23 (E13 + 12 E12 ) 1 ( 01 − 10 2 1 2 )( 01 − 10 ) The allowed logic operations 2 0011 + 2 1100 − ( 0110 ( 3 + 1001 + 1010 + 0101 )) Z̄ = −E12 eiθX̄ and eiθZ̄ generate arbitrary single encoded qubit gates CNOT involves 42 elementary steps (D. Bacon, Ph.D. thesis) • Implications for simplifying operation of spin-based quantum dot QCs In the beginning … In the beginning … 13C-labeled alanine Φ0 = Φ1 = 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 HSB 7→ HSB + ²∆H, k∆Hk = 1 Theory shows that fidelity depends on ² only to second order. Robustness of DFS to symmetry breaking perturbations Robustness of DFS to symmetry breaking perturbations angle strength Strong Symmetry Breaking Bare qubit: two hyperfine states of trapped 9Be+ ion Chief decoherence sources: (i) fluctuating long-wavelength ambient magnetic fields; (ii) heating of ion CM motion during computation: a symmetry-breaking process DFS encoding: into pair of ions 0 L = 0 1 ⊗ 1 2 1 L = 11⊗ 0 2 DFS-encoded Bare qubits Need a way to deal with symmetry breaking… Intermission & Bathroom Break Part 2: Mostly Dynamical Decoupling Need a way to deal with symmetry breaking… NMR to the Rescue: Removal of Decoherence via Spin Echo=Time Reversal Hahn spin echo idea Dynamical Decoupling Basics A pulse producing a unitary evolution P, such that PH SB P † = − H SB i.e., {P, H SB } = 0 (CPMG, Hahn spin-echo) † † τ τ τ Ideal (zero-width) pulses, and ignoring H B: P exp(−iτ H SB ) P † exp(−iτ H SB ) = exp(−iτ PH SB P † ) exp(−iτ H SB ) = exp(iτ H SB ) exp(−iτ H SB ) = I Dynamical Decoupling Basics A pulse producing a unitary evolution P, such that PH SB P † = − H SB i.e., {P, H SB } = 0 (CPMG, spin-echo) X X HSB † † HSB τ τ HSB = λ Z ⊗ B τ XZX = −Z ⇒ "time reversal", Hτ averaged to zero SB (in 1st order Magnus expan.) Ideal (zero-width) pulses, and ignoring H B: P exp(−iτ H SB ) P † exp(−iτ H SB ) = exp(−iτ PH SB P † ) exp(−iτ H SB ) = exp(iτ H SB ) exp(−iτ H SB ) = I Dynamical Decoupling = Symmetrization Viola & Lloyd Phys. Rev. A 58, 2733 (1998); Byrd & Lidar, Q. Inf. Proc. 1, 19 (2002) System-bath Hamiltonian: H SB = ∑ Sα ⊗ Bα α σz system σy σx H SB bath Dynamical Decoupling = Symmetrization Viola & Lloyd Phys. Rev. A 58, 2733 (1998); Byrd & Lidar, Q. Inf. Proc. 1, 19 (2002) System-bath Hamiltonian: H SB = ∑ Sα ⊗ Bα α σz system σy σx H SB bath Apply rapid pulses flipping sign of Sα σz − H SB σy σx Dynamical Decoupling = Symmetrization Viola & Lloyd Phys. Rev. A 58, 2733 (1998); Byrd & Lidar, Q. Inf. Proc. 1, 19 (2002) System-bath Hamiltonian: H SB = ∑ Sα ⊗ Bα α σz system σy σx H SB bath Apply rapid pulses flipping sign of Sα σz − H SB σy σx H SB Dynamical Decoupling = Symmetrization Viola & Lloyd Phys. Rev. A 58, 2733 (1998); Byrd & Lidar, Q. Inf. Proc. 1, 19 (2002) System-bath Hamiltonian: H SB = ∑ Sα ⊗ Bα α σz system σy σx H SB σz More general symmetrization: σy σx σz − H SB Apply rapid pulses flipping sign of Sα σy σx bath H SB H SB Dynamical Decoupling = Symmetrization Viola & Lloyd Phys. Rev. A 58, 2733 (1998); Byrd & Lidar, Q. Inf. Proc. 1, 19 (2002) System-bath Hamiltonian: H SB = ∑ Sα ⊗ Bα α σz system σy σx H SB σz More general symmetrization: σy σx σz − H SB Apply rapid pulses flipping sign of Sα σy σx bath H SB H SB Dynamical Decoupling = Symmetrization Viola & Lloyd Phys. Rev. A 58, 2733 (1998); Byrd & Lidar, Q. Inf. Proc. 1, 19 (2002) System-bath Hamiltonian: H SB = ∑ Sα ⊗ Bα α σz system σy σx H SB σz More general symmetrization: σy σx σz − H SB Apply rapid pulses flipping sign of Sα σy σx bath H SB H SB Dynamical Decoupling = Symmetrization Viola & Lloyd Phys. Rev. A 58, 2733 (1998); Byrd & Lidar, Q. Inf. Proc. 1, 19 (2002) System-bath Hamiltonian: H SB = ∑ Sα ⊗ Bα α σz system σy σx H SB σz More general symmetrization: σy σx σz − H SB Apply rapid pulses flipping sign of Sα σy σx bath H SB H SB H SB averaged to zero. Dealing with Symmetry Breaking: Creating Collective Dephasing Conditions L.L.-A. Wu, D.A.L., Phys. Rev. Lett. Lett. 88, 88, 207902 (2002) General two-qubit dephasing: H SB = σ 1z ⊗ B1 + σ 2z ⊗ B2 1 1 = σ 1z − σ 2z ⊗ ( B1 − B2 ) + σ 1z + σ 2z ⊗ ( B1 + B2 ) 2 2 Z ( ) ( ) Dealing with Symmetry Breaking: Creating Collective Dephasing Conditions L.L.-A. Wu, D.A.L., Phys. Rev. Lett. Lett. 88, 88, 207902 (2002) General two-qubit dephasing: H SB = σ 1z ⊗ B1 + σ 2z ⊗ B2 1 1 = σ 1z − σ 2z ⊗ ( B1 − B2 ) + σ 1z + σ 2z ⊗ ( B1 + B2 ) 2 2 Z 1 Assume: controllable exchange X = (σ1xσ 2x + σ1yσ 2y ) + ασ1zσ 2z . ( ) ( ) 2 {X , Z} = 0 ⇒ XZX = -Z Dealing with Symmetry Breaking: Creating Collective Dephasing Conditions L.L.-A. Wu, D.A.L., Phys. Rev. Lett. Lett. 88, 88, 207902 (2002) General two-qubit dephasing: H SB = σ 1z ⊗ B1 + σ 2z ⊗ B2 1 1 = σ 1z − σ 2z ⊗ ( B1 − B2 ) + σ 1z + σ 2z ⊗ ( B1 + B2 ) 2 2 Z 1 Assume: controllable exchange X = (σ1xσ 2x + σ1yσ 2y ) + ασ1zσ 2z . ( ) ( ) 2 {X , Z} = 0 ⇒ XZX = -Z “Time reversal” Dynamical Decoupling pulse sequence: π π ⎡ ⎤ exp(−iH SB t ) ⎢ exp(−i X ) exp(−iH SB t ) exp(i X ) ⎥ = exp(−it (σ 1z + σ 2z ) ⊗ ( B1 + B2 )) 2 2 ⎣ ⎦ Collective Dephasing X H SB t X H SB t = 2t Coll.Deph. Heisenberg is “Super-Universal” Same method works, e.g., for spin-coupled quantum dots QC: J By BB pulsing of H = σ xσ x + σ yσ y + σ zσ z ( 1 2 1 2 2 1 2 collective decoherence conditions can be created: Heis ) HSB = ∑i=1 gixσ ix ⊗ Bix + giyσ iy ⊗ Biy + gizσ iz ⊗ Biz n → Sx ⊗ Bx + S y ⊗ By + Sz ⊗ B z Requires sequence of 6 π /2 pulses to create collective decoherence conditions over blocks of 4 qubits. Leakage elimination requires 7 more pulses. Details: L.-A. Wu, D.A.L., Phys. Rev. Lett. 88, 207902 (2002); L.A. Wu, M.S. Byrd, D.A.L., Phys. Rev. Lett. 89, 127901 (2002). Earlier DFS work showed universal QC with Heisenberg interaction alone possible [Bacon, Kempe, D.A.L., Whaley, Phys. Rev. Lett. 85, 1758 (2000)]: All ingredients available for Heisenberg-only QC Analysis of Dynamical Decoupling We’ll need a formal detour… Decoherence: Isolated vs Open System Evolution Isolated system: H = HS |ψ(t)i = US (t)|ψ(0)i U̇S = −iHS US equivalently: |ψ(t)ihψ(t)| = US (t)|ψ(0)ihψ(0)|US† (t) US (0) = I Decoherence: Isolated vs Open System Evolution Isolated system: H = HS |ψ(t)i = US (t)|ψ(0)i U̇S = −iHS US US (0) = I equivalently: |ψ(t)ihψ(t)| = US (t)|ψ(0)ihψ(0)|US† (t) Open system: H = HS + HB + HSB ρSB (t) = U (t)ρSB (0)U † (t) U̇ = −iHU U (0) = I ρS (t) = TrB ρSB (t) 6 unitary tranformation of ρS (0) = (except when there is a decoherence-free subspace) Decoherence: Isolated vs Open System Evolution Isolated system: H = HS |ψ(t)i = US (t)|ψ(0)i U̇S = −iHS US US (0) = I equivalently: |ψ(t)ihψ(t)| = US (t)|ψ(0)ihψ(0)|US† (t) Open system: H = HS + HB + HSB ρSB (t) = U (t)ρSB (0)U † (t) U̇ = −iHU U (0) = I ρS (t) = TrB ρSB (t) 6 unitary tranformation of ρS (0) = (except when there is a decoherence-free subspace) decoherence: kρS (t) − |ψ(t)ihψ(t)|k > 0 which norm? Kolmogorov Distance and Quantum Measurements (I) (1) (2) Given two classical probability distributions {pi } and {pi }, their deviation is measured by the Kolmogorov distance P (1) (2) D(p(1) , p(2) ) ≡ 12 i |pi − pi | Kolmogorov Distance and Quantum Measurements (I) (1) (2) Given two classical probability distributions {pi } and {pi }, their deviation is measured by the Kolmogorov distance P (1) (2) D(p(1) , p(2) ) ≡ 12 i |pi − pi | A quantum measurement can always be described in terms of a POVM (positive operator valued measure), i.e., a set of positive operators Ei P satisfying i Ei = I, where i enumerates the possible measurement outcomes. For a system initially in the state ρ, outcome i occurs with probability pi = Pr(i|ρ) = Tr(ρEi ) Kolmogorov Distance and Quantum Measurements (I) (1) (2) Given two classical probability distributions {pi } and {pi }, their deviation is measured by the Kolmogorov distance P (1) (2) D(p(1) , p(2) ) ≡ 12 i |pi − pi | A quantum measurement can always be described in terms of a POVM (positive operator valued measure), i.e., a set of positive operators Ei P satisfying i Ei = I, where i enumerates the possible measurement outcomes. For a system initially in the state ρ, outcome i occurs with probability pi = Pr(i|ρ) = Tr(ρEi ) Thus quantum measurements produce classical probability distributions. Consider two quantum states ρ(1) [= |ψ(t)ihψ(t)|] and ρ(2) [= ρS (t)] Kolmogorov Distance and Quantum Measurements (II) Compare measurement outcomes of same POVM on ρ(1) [= |ψ(t)ihψ(t)|] and ρ(2) [= ρS (t)]: Lemma: δ ≡ D(pρ(1) , pρ(2) ) ≤ kρ(1) − ρ(2) kTr kAkTr √ P † ≡ Tr A A = (singular values(A)) The bound is tight in the sense that it is saturated for the optimal measurement designed to distinguish the two states. Partial trace decreases trace distance (1) (2) (1) (2) Lemma: kρS − ρS kTr ≤ kρSB − ρSB kTr Partial trace decreases trace distance (1) (2) (1) (2) Lemma: kρS − ρS kTr ≤ kρSB − ρSB kTr Partial trace decreases trace distance (1) (2) (1) (2) Lemma: kρS − ρS kTr ≤ kρSB − ρSB kTr Conclusion: we can compare dynamics of ideal and actual systems over the joint system-bath space. Ideal vs Actual System Evolution Ideal system: H = HS + HB † † ρideal (t) = [U (t) ⊗ U (t)]ρ (0)[U (t) ⊗ U S B SB SB S B (t)] U̇S/B = −iHS/B US/B US/B (0) = I Ideal vs Actual System Evolution Ideal system: H = HS + HB † † ρideal (t) = [U (t) ⊗ U (t)]ρ (0)[U (t) ⊗ U S B SB SB S B (t)] U̇S/B = −iHS/B US/B US/B (0) = I Actual system: H = HS + HB + HSB ρSB (t) = U (t)ρSB (0)U † (t) U̇ = −iHU U (0) = I Ideal vs Actual System Evolution Ideal system: H = HS + HB † † ρideal (t) = [U (t) ⊗ U (t)]ρ (0)[U (t) ⊗ U S B SB SB S B (t)] U̇S/B = −iHS/B US/B US/B (0) = I Actual system: H = HS + HB + HSB ρSB (t) = U (t)ρSB (0)U † (t) U̇ = −iHU U (0) = I Distance: † kρSB (t) − ρideal SB (t)kTr = kV (t)ρSB (0)V (t) − ρSB (0)kTr V (t) ≡ US† (t) ⊗ UB† (t)U (t) ≡ exp[−itHeﬀ (t)] Ideal vs Actual System Evolution Ideal system: H = HS + HB † † ρideal (t) = [U (t) ⊗ U (t)]ρ (0)[U (t) ⊗ U S B SB SB S B (t)] U̇S/B = −iHS/B US/B US/B (0) = I Actual system: H = HS + HB + HSB ρSB (t) = U (t)ρSB (0)U † (t) U̇ = −iHU U (0) = I Distance: † kρSB (t) − ρideal SB (t)kTr = kV (t)ρSB (0)V (t) − ρSB (0)kTr V (t) ≡ US† (t) ⊗ UB† (t)U (t) ≡ exp[−itHeﬀ (t)] Lemma: kρSB (t) − ρideal SB (t)kTr ≤ tkHeﬀ (t)k∞ follows from keiA − eiB k∞ ≤ kA − Bk∞ ; kAk∞ ≡ sup|vi,hv|vi=1 q hv|A† A|vi = max sing.val.(A) Kolmogorov Distance Bound from Effective Hamiltonian Lemma: δ ≡ D(pρ(1) , pρ(2) ) ≤ kρ(1) − ρ(2) kTr (1) (2) (1) (2) Lemma: kρS − ρS kTr ≤ kρSB − ρSB kTr (trace distance bounds Kolmogorov distance) (partial trace decreases distinguishability) V (t) ≡ US† (t) ⊗ UB† (t)U (t) ≡ exp[−itHeﬀ (t)] Lemma: kρSB (t) − ρideal SB (t)kTr ≤ tkHeﬀ (t)k∞ δactual,ideal ≤ tkHeﬀ (t)k∞ ≡ η(t) ≡ noise strength δactual,ideal ≤ tkHeﬀ (t)k∞ ≡ η(t) ≡ noise strength Goal: reduce effective Hamiltonian. Method: dynamical decoupling. δactual,ideal ≤ tkHeﬀ (t)k∞ ≡ η(t) ≡ noise strength Goal: reduce effective Hamiltonian. Method: dynamical decoupling. How do we compute Heﬀ (t)? U̇ = −iH(t)U U (t) = e−itHeff (t) ∞ 1X Heﬀ (t) = Ωj (t) t j=1 Z t Ω1 (t) = dt1 H(t1 ) 0 The Magnus expansion i Ω2 (t) = − 2 Z 0 t dt1 Z 0 t1 dt2 [H(t1 ), H(t2 )] Analysis of Dynamical Decoupling Dynamical Decoupling Theory “Symmetrizing group” of pulses { gi } and their inverses are applied in series: ( g N† fg N )" ( g 2†fg 2 )( g1†fg1 ) ≈ exp(−iτ ∑ i gi † H SB gi ) f ≡ exp(−iH SBτ ) first order Magnus expansion Periodic DD: periodic repetition of the universal DD pulse sequence Dynamical Decoupling Theory “Symmetrizing group” of pulses { gi } and their inverses are applied in series: ( g N† fg N )" ( g 2†fg 2 )( g1†fg1 ) ≈ exp(−iτ ∑ i gi † H SB gi ) f ≡ exp(−iH SBτ ) Choose the pulses so that: (1) H SB 6 H eff ≡ ∑ i gi † H SB gi = 0 first order Magnus expansion Dynamical Decoupling Condition Periodic DD: periodic repetition of the universal DD pulse sequence Dynamical Decoupling Theory “Symmetrizing group” of pulses { gi } and their inverses are applied in series: ( g N† fg N )" ( g 2†fg 2 )( g1†fg1 ) ≈ exp(−iτ ∑ i gi † H SB gi ) f ≡ exp(−iH SBτ ) Choose the pulses so that: (1) H SB 6 H eff ≡ ∑ i gi † H SB gi = 0 first order Magnus expansion Dynamical Decoupling Condition For a qubit the Pauli group G={X,Y,Z,I } (π pulses around all three axes) removes an arbitrary HSB : (XfX)(YfY)(ZfZ)(IfI) = XfZfXfZf Periodic DD: periodic repetition of the universal DD pulse sequence The Effective Hamiltonian Another view of the universal decoupling sequence: X f f δ X Z f Z f τ f ≡ exp[−iτ H SB ] = f' f' ≡ exp[−iTH eff (T )] H eff (T )=0, ideally The Effective Hamiltonian Another view of the universal decoupling sequence: X f f δ X Z f Z f X f " τ f ≡ exp[−iτ H SB ] But, errors accumulate…: Heff (T ) ≠ 0 = f' f' ≡ exp[−iTH eff (T )] H eff (T )=0, ideally Periodic Dynamical Decoupling PDD Strategy: repeat the basic XfZfZfXfZ cycle with total of N pulses. The total duration is fixed at T. N can be changed. Pulse interval: τ = T /N η ≡ ||H (T )||T Recall noise strength eﬀ norm of final effective system-bath Hamiltonian times the total duration. Periodic Dynamical Decoupling PDD Strategy: repeat the basic XfZfZfXfZ cycle with total of N pulses. The total duration is fixed at T. N can be changed. Pulse interval: τ = T /N η ≡ ||H (T )||T Recall noise strength eﬀ norm of final effective system-bath Hamiltonian times the total duration. PDD leading order result for error: η∝N Can we do better? −1 DD as a Rescaling Transformation J = kHSB k∞ β= kHB k∞ • Interaction terms are rescaled after the DD cycle J = J (0) 7→ J (1) ∝ max[τ (J (0) )2 , τ βJ (0) ] β7→ β + O((J (0) )3 τ 2 ) • We need a mechanism to continue this Concatenated Universal Dynamical Decoupling Nest the universal DD pulse sequence into its own free evolution periods f : p(1)= X f Z f X f Z f Concatenated Universal Dynamical Decoupling Nest the universal DD pulse sequence into its own free evolution periods f : p(1)= X f Z f X f Z f p(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 Decoupling Nest the universal DD pulse sequence into its own free evolution periods f : p(1)= X f Z f X f Z f p(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 matrices 1 XfZfXfZf 2 fZfXfZfYfZfXfZffZfXfZfYfZfXfZf 3 XfZfXfZfYfZfXfZffZfXfZfYfZfXfZfZfZfXfZfYfZfXfZffZfXfZfYfZfXfZfXfZf XfZfYfZfXfZffZfXfZfYfZfXfZfZfZfXfZfYfZfXfZffZfXfZfYfZfXfZf Length grows exponentially; how about error reduction? Performance of Concatenated Sequences error 6 (error)2 6 ((error)2)2 6 (((error)2)2)2 6 " 6 (error)2k k Performance of Concatenated Sequences error 6 (error)2 6 ((error)2)2 6 (((error)2)2)2 6 " 6 (error)2k k For fixed total time T=Nτ and N zero-width (ideal) pulses: b η∝N N Compare to periodic DD: −c log N η∝N −1 [Khodjasteh & Lidar, PRA 75, 062310 (2007) ] Experiments Concatenated DD on Adamantene Powder Dieter Suter, TU Dortmund adamantene; qubit = 13C PDD=CDD1 CDD2 CDD3 Echo intensity τ = 25μ s δ = 10.5μ s Echo intensity Echo intensity CDD Results τ = 85μ s τ = 50μ s Concatenated DD for electron spin of 31P donors in Si Steve Lyon, Princeton 31P donor: Electron spin (S) = ½, Nuclear spin (I) = ½ ↓e,↓n ↓e,↑n νμw1 νrf1 νμw2 ↑e,↑n bath is 29Si ~1% natural abundance ↑e,↓n |1〉 |0〉 νrf2 1. Periodic (XfYfXfYf) relativeFidelity echo intensity Periodic DD vs Concatenated DD 1.0 0.5 State +X 0.0 0 δ = 160ns 5 10 15 20 25 ~50ms 30 Number of Repeats Fidelity relative echo intensity 2. Concatenated State +Y 1.0 State +Y State +X 0.5 0.0 0 1 2 3 4 ~100ms Concatenation Level [log(time)] Better than Concatenated DD? Does there exist an optimal pulse sequence? Optimal = removes maximum decoherence with least possible number of pulses Better than Concatenated DD? Better than Concatenated DD? OR Better than Concatenated DD? OR Better than Concatenated DD? Better than Concatenated DD? “Quadratic DD” eliminates the first n orders in the Dyson series of the joint system-bath propagator using n2 pulses Concatenated DD requires 4n pulses to do the same, approximately Inner workings of Quadratic DD j, k ∈ {1, n} Inner workings of Quadratic DD j, k ∈ {1, n} For every value of n, the first √ n terms in the Dyson series are removed Comparison of DD Sequences Summary • 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 Picture DFS encoding QECC Hybrid Q. Error Correction: The Big Picture - symmetry not for free… DFS encoding QECC Hybrid Q. Error Correction: The Big Picture - symmetry not for free… DFS encoding DD QECC Hybrid Q. Error Correction: The Big Picture - symmetry not for free… DFS encoding DD QECC -pulse errors, Markovian effects Hybrid Q. Error Correction: The Big Picture - symmetry not for free… DFS encoding DD FT-QEC -pulse errors, Markovian effects Open 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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Canadian Summer School on Quantum Information (CSSQI) (10th : 2010)
Adiabatic Quantum Computation, Decoherence-Free Subspaces & Noiseless Subsystems, and Dynamical Decoupling Lidar, Daniel 2010-07-21
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Title | Adiabatic Quantum Computation, Decoherence-Free Subspaces & Noiseless Subsystems, and Dynamical Decoupling |
Alternate Title | Quantum Error Prevention, Reduction and Correction 1;Hybrid quantum error prevention, reduction and correction methods |
Creator |
Lidar, Daniel |
Contributor |
Summer School on Quantum Information (10th : 2010 : Vancouver, B.C.) University of British Columbia. Department of Physics and Astronomy Pacific Institute for the Mathematical Sciences |
Date Issued | 2010-07-21 |
Description | The second part of the day will start by covering several advanced topics in AQC, including a sketch of the proof of the equivalence between AQC and the circuit model, rigorous formulations of the adiabatic theorem, the geometry of AQC, and a time-optimized ("brachistochrone") approach to AQC. We'll then switch gears and provide an introduction to decoherence-free subspaces, noiseless subsystems, dynamical decoupling, and hybrid methods in which they are combined. The emphasis will be on the underlying unifying symmetry principles which enable quantum errors to be avoided by encoding. Time permitting, we'll return to AQC and discuss how it can be made resilient to decoherence. |
Subject |
adibatic quantum computation quantum error correction dynamical decoupling |
Type |
Moving Image |
Language | eng |
Date Available | 2016-11-22 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0040935 |
URI | http://hdl.handle.net/2429/30252 |
Affiliation |
Non UBC |
Peer Review Status | Unreviewed |
Scholarly Level | Faculty |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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