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Adiabatic Quantum Computation, Decoherence-Free Subspaces & Noiseless Subsystems, and Dynamical Decoupling 2010
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Title | Adiabatic Quantum Computation, Decoherence-Free Subspaces & Noiseless Subsystems, and Dynamical Decoupling |
Creator |
Lidar, Daniel |
Date Created | 2010-12-01T00:17:44Z |
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 |
Sound Moving Image |
Language | Eng |
Collection |
10th Canadian Summer School on Quantum Information |
Date Available | 2010-12-01T00:17:44Z |
DOI | 10.14288/1.0040935 |
Affiliation |
Non UBC |
Peer Review Status | Unreviewed |
Scholarly Level | Faculty |
URI | http://hdl.handle.net/2429/30252 |
Digital Resource Original Record | https://open.library.ubc.ca/collections/29055/items/1.0040935/source |
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Hybrid quantum error prevention, reduction, and correction methods Daniel Lidar University of Southern California 10th Canadian Summer School on Quantum Information UBC, July 21, 2010 DFS- encoded no encoding Science 291, 1013 (2001) probability of correct Grover answer Phys. Rev. Lett. 91, 217904 (2003) 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 DFS/NS: Error Prevention use an existing exact symmetry to perfectly hide q. info. from bath Symmetry Î conserved quantity = quantum info. DFS H 1) Symmetry Protects Q. Information DFS/NS: Error Prevention use an existing exact symmetry to perfectly hide q. info. from bath Symmetry Î conserved quantity = quantum info. DFS H BANG + free evolution time Dynamical Decoupling: Open-loop control dynamically generate a symmetry bath correlation time strong SBH 1) 2) Whence the Errors? Decoherence from System-Bath Interaction system bath SB S BH α αα ⊗=∑ Every real quantum system is coupled to an environment (“bath”). Full Hamiltonian: B SBS HH HH + += Whence the Errors? Decoherence from System-Bath Interaction system bath SB S BH α αα ⊗=∑ Every real quantum system is coupled to an environment (“bath”). Full Hamiltonian: B SBS HH HH + += Decoherence Non-unitary evolution of system† System dynamics alone: ( ) Tr (0) ( ) (0) ( ) iHt iHt B SB k k kk t e e c E t E t ρ ρ ρ −⎡ ⎤⎢ ⎥⎣ ⎦= =∑ Whence the Errors? Decoherence from System-Bath Interaction system bath SB S BH α αα ⊗=∑ Every real quantum system is coupled to an environment (“bath”). Full Hamiltonian: B SBS HH HH + += ( )† † †, ' ' ' ' , ' , 2] 12[ St a S S S S S Si H α α αα α α α αα α ρ ρ ρρ ρ +∂ − −∂ =− ∑ Markovian master (Lindblad) equation: Decoherence Non-unitary evolution of system† System dynamics alone: ( ) Tr (0) ( ) (0) ( ) iHt iHt B SB k k kk t e e c E t E t ρ ρ ρ −⎡ ⎤⎢ ⎥⎣ ⎦= =∑ Whence the Errors? Decoherence from System-Bath Interaction 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 logical 0 logical 1 How to reliably store a single bit? A noiseless subspace. Unified view of Quantum Information Protection: Fixed Codes Error Model: Trace-preserving completely-positive (CP) maps: ρ0 = P k EkρE † k ≡ E(ρ) P k E † kEk = I Unified view of Quantum Information Protection: Fixed Codes Error Model: Trace-preserving completely-positive (CP) maps: ρ0 = P k EkρE † k ≡ E(ρ) P k E † kEk = 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: passive ρ0 = P k EkρE † k ≡ E(ρ) P k E † kEk = 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 TrGE(ρ) = TrGρ Unified view of Quantum Information Protection: Fixed Codes Error Model: Trace-preserving completely-positive (CP) maps: passive ρ0 = P k EkρE † k ≡ E(ρ) P k E † kEk = 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 TrGE(ρ) = TrGρ 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 Unified view of Quantum Information Protection: Fixed Codes Error Model: Trace-preserving completely-positive (CP) maps: passive active ρ0 = P k EkρE † k ≡ E(ρ) P k E † kEk = 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 TrGE(ρ) = TrGρ 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 TrGR ◦ E(ρ) = TrGρ 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 TrGE(ρ) = TrGρ Operator QEC (OQEC): H = A⊕ B, A = N ⊗ G C = {all states ρ : A→ A} such that ∃CP map R for which TrGR ◦ E(ρ) = TrGρ 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 H = HD⊕H⊥D, and partition the system state ρS accordingly into blocks: ρS = µ ρD ρ2 ρ†2 ρ3 ¶ . Assume ρD(0) 6= 0. Then HD is called decoherence-free iff the initial and final DFS-blocks of ρS are unitarily related: ρD(t) = UDρD(0)U†D, where UD is a unitary matrix acting on HD. Note imperfect initialization! U.I. DFS Conditions for CP Maps ρ0 = P k EkρE † k ≡ E(ρ) P k E † kEk = I Given a CP map: 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 Ek = µ ckUD 0 0 Bk ¶ , where UD is unitary, ck are scalars satisfying P k |ck|2 = 1, and Bk are arbitrary operators on H⊥D satisfying P k B † kBk = I. Meaning: Ek act unitarily on the DFS U.I. DFS Conditions for Master Equations Given a Markovian master equation: 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 HS = µ HD 0 0 H⊥D ¶ , Fα = µ cαI 0 0 Bα ¶ , where HD and H ⊥ D are Hermitian, cα are scalars, and Bα are arbitrary operators on H⊥D. dρ dt = −i[HS , ρ] + 1 2 P α 2FαρF †α − ρF †αFα − F †αFαρ 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? Under Hamiltonian dynamics system and bath evolve jointly subject to the Schrodinger equation with th subspac e Hamiltonian . Find a where acts trivially, i.e.: e SB BS SBH H H H S BH α αα= = + + ⊗∑ make SB S BH I O∝ ⊗ U.I. DFS Conditions for Hamiltonian Dynamics Also, remember that HS must preserve the DFS. Under Hamiltonian dynamics system and bath evolve jointly subject to the Schrodinger equation with th subspac e Hamiltonian . Find a where acts trivially, i.e.: e SB BS SBH H H H S BH α αα= = + + ⊗∑ make SB S BH I O∝ ⊗ Theorem degeneracy symmetry U.I. DFS Conditions for Hamiltonian Dynamics Also, remember that HS must preserve the DFS. Let A = alg{I, Sα, S†α}. Assume [HS , A] = 0. The dimension of the DFS HD equals the degeneracy of the 1-dimensional irreducible representation (irrep) of A. Simplest DFS Example: Collective Dephasing Effect: Random " ": 0 1 0 1j j j j jj j j j ia b a e bθψ = + + Collective Dephasing 6 random but j-independent phase Find a where acts trivially, i.e.: su make bspace SB S BB S H H S B OI α αα= ⊗ ∝ ⊗ ∑DFS idea: Permutation symmetry in z direction: Long-wavelength magnetic field B (environment) couples to spins 1ψ 2ψ ( )B t z Simplest DFS Example: Collective Dephasing Effect: Random " ": 0 1 0 1j j j j jj j j j ia b a e bθψ = + + Collective Dephasing 6 random but j-independent phase 1 2 1 2 0 0 1 1 1 0 = ⊗ = ⊗ L L DFS encoding Find a where acts trivially, i.e.: su make bspace SB S BB S H H S B OI α αα= ⊗ ∝ ⊗ ∑DFS idea: 1 1int ( ) z zH g Bσ σ= + ⊗ = 0 2 2 0 gB gB −⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ 1 2 1 2 1 2 1 2 ↓ ↑ ↑ ↓ ↓ ↓ ↑ ↑ Permutation symmetry in z direction: Long-wavelength magnetic field B (environment) couples to spins 1ψ 2ψ ( )B t z Why it Works ( ) ( ) ( ) ( ) θ θ θ θ θ θ θ ⊗ ⊗ ≡ ⊗ ⊗ ≡ ⊗ ⊗ ≡ ⊗ ⊗ ≡ 6 6 6 6 2 Case of two qubits: 0 0 0 0 00 0 1 0 1 01 1 0 1 0 10 1 1 1 1 11 i i i i i i i e e e e e e e θψ = + +6 Collective dephasing: 0 1 0 1j j j jj j j j j ia b a e b Why it Works ( ) ( ) ( ) ( ) θ θ θ θ θ θ θ ⊗ ⊗ ≡ ⊗ ⊗ ≡ ⊗ ⊗ ≡ ⊗ ⊗ ≡ 6 6 6 6 2 Case of two qubits: 0 0 0 0 00 0 1 0 1 01 1 0 1 0 10 1 1 1 1 11 i i i i i i i e e e e e e e ψ = + Global phase physically irrelevant: is decoherence-free: A 2-di pmensional rote 0 1 cted subspace . L L La b θψ = + +6 Collective dephasing: 0 1 0 1j j j jj j j j j ia b a e b ≡ 0 L ≡ 1 L Why it Works ( ) ( ) ( ) ( ) θ θ θ θ θ θ θ ⊗ ⊗ ≡ ⊗ ⊗ ≡ ⊗ ⊗ ≡ ⊗ ⊗ ≡ 6 6 6 6 2 Case of two qubits: 0 0 0 0 00 0 1 0 1 01 1 0 1 0 10 1 1 1 1 11 i i i i i i i e e e e e e e ψ = + Global phase physically irrelevant: is decoherence-free: A 2-di pmensional rote 0 1 cted subspace . L L La b θψ = + +6 Collective dephasing: 0 1 0 1j j j jj j j j j ia b a e b ≡ 0 L ≡ 1 L pop quiz: Are the states |00i and |11i also in a DFS? 1D irreps condition not needed… The dimension of the DFS HD equals the degeneracy of the 1-dimensional irreducible representation (irrep) of A. Generalization: Noiseless Subsystems [E. Knill, R. Laflamme and L. Viola, PRL 84, 2525 (2000)] α αα ⊗=∑SB S BH Model of decoherence: α α= ≅ ⊕ ⊗ ≅ ⊕ ⊗ ^ ^ ^ ^ ^ † 2 2 Associative algebra polynomials{ , , } Matrix representation over : ( ) Hilbert space decomposition: N J J N J J n dJ n d J A I S S A I M multiplicity dimension irreducible representations A theorem from C* algebras: 1D irreps condition not needed… The dimension of the DFS HD equals the degeneracy of the 1-dimensional irreducible representation (irrep) of A. Generalization: Noiseless Subsystems [E. Knill, R. Laflamme and L. Viola, PRL 84, 2525 (2000)] α αα ⊗=∑SB S BH Model of decoherence: α α= ≅ ⊕ ⊗ ≅ ⊕ ⊗ ^ ^ ^ ^ ^ † 2 2 Associative algebra polynomials{ , , } Matrix representation over : ( ) Hilbert space decomposition: N J J N J J n dJ n d J A I S S A I M multiplicity dimension code subsystem irreducible representations 1 iff in system-env. interactionJn > ∃ symmetry A theorem from C* algebras: 1D irreps condition not needed… The dimension of the DFS HD equals the degeneracy of the 1-dimensional irreducible representation (irrep) of A. Generalization: Noiseless Subsystems [E. Knill, R. Laflamme and L. Viola, PRL 84, 2525 (2000)] =1 Each labels an NS code DFS is the case -dimensionalJ J J n d Isotropic Quantum Errors: Collective Decoherence Model ˆ( )YB t y 1ψ ˆ( )ZB t z 2ϕ ˆ( )XB t x Describes, e.g., low- T decoherence due to phonons in various solid state QC proposals Isotropic Quantum Errors: Collective Decoherence Model ˆ( )YB t y 1ψ ˆ( )ZB t z 2ϕ ˆ( )XB t x( ) α α α αα σ σ = = = + + ⊗∑ "1, , total spin operator Error model, qubits: "Collective Decoherence" NSB x y z S N H B Describes, e.g., low- T decoherence due to phonons in various solid state QC proposals Isotropic Quantum Errors: Collective Decoherence Model ˆ( )YB t y 1ψ ˆ( )ZB t z 2ϕ ˆ( )XB t x ψ ψ ⎧⎪ ⊗ ⊗⎪⎨ ⊗ ⊗⎪⎪ ⊗ ⊗⎩ "6 " " 0 1 2 3 Error model, qubits: prob. (1) ( ) prob. (1) ( ) prob. (1) ( ) prob. X X Y Y Z Z N p U U N p U U N p U U N p Describes, e.g., low- T decoherence due to phonons in various solid state QC proposals Do irreps analysis of n copies of su(2)… J0 1/2 1 3/2 2 1 2 5 1 3 1 1 4 9 14 multiplicity nJ , counts paths; dJ =2J+1 1 1 2 5 14 2 3 4 5 6 7 8 All Decoherence-Free Subspaces/Subsystems for Collective Decoherence n ≅ ⊕ ⊗^ ^ ^2 Hilbert space decomposition: N J Jn d J 0L ≡ ( )1 01 102= − J0 1/2 1 3/2 2 1 2 5 1 3 1 1 4 9 14 multiplicity nJ , counts paths; dJ =2J+1 1 1 2 5 14 2 3 4 5 6 7 8 All Decoherence-Free Subspaces/Subsystems for Collective Decoherence 0L= 1 2 3 n0 1/2 1 ( )1 01 10 1 2 = − 1L= 1 2 3 n0 1/2 1 ( )2 1110 011 101 3 3 = − − n ≅ ⊕ ⊗^ ^ ^2 Hilbert space decomposition: N J Jn d J 0L ≡ ( )1 01 102= − J0 1/2 1 3/2 2 1 2 5 1 3 1 1 4 9 14 multiplicity nJ , counts paths; dJ =2J+1 1 1 2 5 14 2 3 4 5 6 7 8 All Decoherence-Free Subspaces/Subsystems for Collective Decoherence 0L= 1 2 3 n0 1/2 1 ( )1 01 10 1 2 = − 1L= 1 2 3 n0 1/2 1 ( )2 1110 011 101 3 3 = − − n 0L ≡ ( )( )1 01 10 01 10 2 = − − 1L ≡ ( )( )1 2 0011 2 1100 0110 1001 1010 0101 2 3 = + − + + + ≅ ⊕ ⊗^ ^ ^2 Hilbert space decomposition: N J Jn d J What is the “Volume” of a DFS/NS? ( ) ( ) !(2 1)Degeneracy for given , = dimension of DFS/NS ( ) / 2 1 ! / 2 !J n JJ M D n n J n J +≡ = + + − ( )0 2 0 2 physical log ( ) logno. of qubits 3 code rate 1 no. of encoded qubits 2= J nD n n n n = →∞⇒ ≡ ⎯⎯⎯→ − What is the “Volume” of a DFS/NS? ( ) ( ) !(2 1)Degeneracy for given , = dimension of DFS/NS ( ) / 2 1 ! / 2 !J n JJ M D n n J n J +≡ = + + − DFS’s for collective decoherence asymptotically fill the Hilbert space! ( )0 2 0 2 physical log ( ) logno. of qubits 3 code rate 1 no. of encoded qubits 2= J nD n n n n = →∞⇒ ≡ ⎯⎯⎯→ − 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? ≅ ⊕ ⊗ ≅ ⊕ ⊗ ⊗ ≅ ⊕ ^ ^ ^ ^ ^ 2 Commutant = operators commuting with ' Error algebra: ( ) ' ( ) The allowed logic operation Code subsy m s st : ! e J J J J N J J n dJ n dJ n d J A M A M I A I 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? ≅ ⊕ ⊗ ≅ ⊕ ⊗ ⊗ ≅ ⊕ ^ ^ ^ ^ ^ 2 Commutant = operators commuting with ' Error algebra: ( ) ' ( ) The allowed logic operation Code subsy m s st : ! e J J J J N J J n dJ n dJ n d J A M A M I A I Universal quantum computation over DFS/NS is possible using “exchange Hamiltonians”, e.g., Heisenberg interaction: Heis 2 ij y yx x z z i j i j i j ij H J σ σ σ σ σ σ⎛ ⎞⎜ ⎟⎝ ⎠= + +∑ 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 QCs HHeis = P i,j Jij(XiXj + YiYj + ZiZj) ≡ P i,j JijEij ≅ ⊕ ⊗^' ' ( ) The allowed logic operations J Jn dJ A M I 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 42 elementary steps (D. Bacon, Ph.D. thesis) • Implications for simplifying operation of spin-based quantum dot QCs HHeis = P i,j Jij(XiXj + YiYj + ZiZj) ≡ P i,j JijEij X̄ = − 2√ 3 (E13 + 1 2E12) Z̄ = −E12 eiθX̄ and eiθZ̄ generate arbitrary single encoded qubit gates ( )( )10 01 10 01 10 2L = − − ( )( )11 2 0011 2 1100 0110 1001 1010 0101 2 3L = + − + + + ≅ ⊕ ⊗^' ' ( ) The allowed logic operations J Jn dJ A M I In the beginning … In the beginning … 13C-labeled alanine 0Φ = 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 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 angle strength Robustness of DFS to symmetry breaking perturbations DFS-encoded Bare qubits 1 2 1 2 0 0 1 1 1 0 L L = ⊗ = ⊗ 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 Strong Symmetry Breaking Need 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 Reversal Hahn spin echo idea Dynamical Decoupling Basics A 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 SB SB SB exp( ) exp( ) exp( )exp( ) exp( )exp( ) P i H P i H i PH P i H i H i H I τ τ τ τ τ τ − − = − − = − = τ τ τ †† BH A 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 SB SB SB exp( ) exp( ) exp( )exp( ) exp( )exp( ) P i H P i H i PH P i H i H i H I τ τ τ τ τ τ − − = − − = − = τ τ τ †† st SB order Magnus ex n. pa ) "time reversal", averaged (in to zero 1 XZX Z H =− ⇒ SBH Z Bλ= ⊗ SBH SBH X X τ BH Dynamical Decoupling Basics Dynamical Decoupling = Symmetrization Viola & Lloyd Phys. Rev. A 58, 2733 (1998); Byrd & Lidar, Q. Inf. Proc. 1, 19 (2002) system bath System-bath Hamiltonian: SB S BH α αα ⊗=∑ zσ xσ yσ SBH Dynamical Decoupling = Symmetrization Viola & Lloyd Phys. Rev. A 58, 2733 (1998); Byrd & Lidar, Q. Inf. Proc. 1, 19 (2002) system bath System-bath Hamiltonian: SB S BH α αα ⊗=∑ zσ xσ yσ SBH zσ xσ yσ SBH−Apply rapid pulses flipping sign of Sα Dynamical Decoupling = Symmetrization Viola & Lloyd Phys. Rev. A 58, 2733 (1998); Byrd & Lidar, Q. Inf. Proc. 1, 19 (2002) system bath System-bath Hamiltonian: SB S BH α αα ⊗=∑ zσ xσ yσ SBH zσ xσ yσ SBH− SBH Apply rapid pulses flipping sign of Sα Dynamical Decoupling = Symmetrization Viola & Lloyd Phys. Rev. A 58, 2733 (1998); Byrd & Lidar, Q. Inf. Proc. 1, 19 (2002) system bath System-bath Hamiltonian: SB S BH α αα ⊗=∑ zσ xσ yσ SBH zσ xσ yσ SBH− SBH Apply rapid pulses flipping sign of Sα xσ yσ zσ SBH More general symmetrization: Dynamical Decoupling = Symmetrization Viola & Lloyd Phys. Rev. A 58, 2733 (1998); Byrd & Lidar, Q. Inf. Proc. 1, 19 (2002) system bath System-bath Hamiltonian: SB S BH α αα ⊗=∑ zσ xσ yσ SBH zσ xσ yσ SBH− SBH Apply rapid pulses flipping sign of Sα xσ yσ zσ SBH More general symmetrization: Dynamical Decoupling = Symmetrization Viola & Lloyd Phys. Rev. A 58, 2733 (1998); Byrd & Lidar, Q. Inf. Proc. 1, 19 (2002) system bath System-bath Hamiltonian: SB S BH α αα ⊗=∑ zσ xσ yσ SBH zσ xσ yσ SBH− SBH Apply rapid pulses flipping sign of Sα xσ yσ zσ SBH More general symmetrization: Dynamical Decoupling = Symmetrization Viola & Lloyd Phys. Rev. A 58, 2733 (1998); Byrd & Lidar, Q. Inf. Proc. 1, 19 (2002) system bath System-bath Hamiltonian: SB S BH α αα ⊗=∑ zσ xσ yσ SBH zσ xσ yσ SBH− SBH Apply rapid pulses flipping sign of Sα xσ yσ zσ SBH More general symmetrization: averaged to zero.SBH Dealing with Symmetry Breaking: Creating Collective Dephasing Conditions L.-A. Wu, D.A.L., Phys. Rev. Lett. 88, 207902 (2002) 1 1 2 2General two-qubit dephasing: z z SBH B Bσ σ= ⊗ + ⊗ ( ) ( ) ( ) ( )1 2 1 2 1 21 2 1212 z zz z Z B B B Bσ σσ σ= ⊗ − + + ⊗ +− Dealing with Symmetry Breaking: Creating Collective Dephasing Conditions L.-A. Wu, D.A.L., Phys. Rev. Lett. 88, 207902 (2002) ( )1 2 1 2 1 2Assume: controllable exchange 1 .2 y yx x z zX σ σ σ σ ασ σ= + + 1 1 2 2General two-qubit dephasing: z z SBH B Bσ σ= ⊗ + ⊗ ( ) ( ) ( ) ( )1 2 1 2 1 21 2 1212 z zz z Z B B B Bσ σσ σ= ⊗ − + + ⊗ +− { , } 0 - X Z XZX Z= ⇒ = Dealing with Symmetry Breaking: Creating Collective Dephasing Conditions L.-A. Wu, D.A.L., Phys. Rev. Lett. 88, 207902 (2002) ( )1 2 1 2 1 2Assume: controllable exchange 1 .2 y yx x z zX σ σ σ σ ασ σ= + + 1 1 2 2General two-qubit dephasing: z z SBH B Bσ σ= ⊗ + ⊗ ( ) ( ) ( ) ( )1 2 1 2 1 21 2 1212 z zz z Z B B B Bσ σσ σ= ⊗ − + + ⊗ +− ( ) ( )1 2 1 2 Collective Dephasing exp( ) exp( )exp( )exp( ) exp( ) 2 2SB SB z z B BiH t i X iH t i X it π π σ σ− − − +− += ⊗⎡ ⎤⎢ ⎥⎣ ⎦ X X SBH t SBH t = 2t Coll.Deph. “Time reversal” Dynamical Decoupling pulse sequence: { , } 0 - X Z XZX Z= ⇒ = Heisenberg is “Super-Universal” Same method works, e.g., for spin-coupled quantum dots QC: ( )1 2 1 2 1 2HeisBy BB pulsing of col decoherenclective conditions can be c 2 reat :e ed y yx x z zJH σ σ σ σ σ σ= + + Details: L.-A. Wu, D.A.L., , 207902 Requires sequence of 6 /2 pulses to create collective decoherence conditions over blocks of 4 qubits. Leakage elimination requires 7 more pulses. Phys. Rev. Lett. π 88 (2002); L.A. Wu, M.S. Byrd, D.A.L., , 127901 (2002).Phys. Rev. Lett. 89 Earlier DFS work showed universal QC with Heisenberg interaction alone possible [Bacon, Kempe, D.A.L., Whaley, Phys. Rev. Lett. 85, 1758 (2000)] : SB 1 z y yx x z z i i i i i i yx z i i i n i zx x y y B B BH g g g S B S B S B σ σ σ== + +⊗ ⊗ ⊗ ⊗ + ⊗ + ⊗ ∑→ 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 = −iHSUS US(0) = I equivalently: |ψ(t)ihψ(t)| = US(t)|ψ(0)ihψ(0)|U †S(t) Decoherence: Isolated vs Open System Evolution Isolated system: H = HS |ψ(t)i = US(t)|ψ(0)i U̇S = −iHSUS US(0) = I Open system: H = HS +HB +HSB ρSB(t) = U(t)ρSB(0)U†(t) U̇ = −iHU U(0) = I ρS(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 Evolution Isolated system: H = HS |ψ(t)i = US(t)|ψ(0)i U̇S = −iHSUS US(0) = I Open system: H = HS +HB +HSB ρSB(t) = U(t)ρSB(0)U†(t) U̇ = −iHU U(0) = I ρS(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 distance D(p(1), p(2)) ≡ 12 P i |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 Ei satisfying P iEi = 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) Given two classical probability distributions {p(1)i } and {p(2)i }, their deviation is measured by the Kolmogorov distance D(p(1), p(2)) ≡ 12 P i |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 Ei satisfying P iEi = I, where i enumerates the possible measurement outcomes. For a system initially in the state ρ, outcome i occurs with probability Thus 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 distance Consider two quantum states ρ(1)[= |ψ(t)ihψ(t)|] and ρ(2)[= ρS(t)] D(p(1), p(2)) ≡ 12 P i |p(1)i − p(2)i | 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 The 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 distance Lemma: kρ(1)S − ρ(2)S kTr ≤ kρ(1)SB − ρ(2)SBkTr Partial trace decreases trace distance Lemma: kρ(1)S − ρ(2)S kTr ≤ kρ(1)SB − ρ(2)SBkTr Partial trace decreases trace distance Conclusion: we can compare dynamics of ideal and actual systems over the joint system-bath space. Lemma: kρ(1)S − ρ(2)S kTr ≤ kρ(1)SB − ρ(2)SBkTr Ideal vs Actual System Evolution Ideal system: H = HS +HB ρidealSB (t) = [US(t)⊗ UB(t)]ρSB(0)[U † S(t)⊗ U † B(t)] U̇S/B = −iHS/BUS/B US/B(0) = I Ideal vs Actual System Evolution ρSB(t) = U(t)ρSB(0)U†(t) U̇ = −iHU U(0) = I Ideal system: H = HS +HB ρidealSB (t) = [US(t)⊗ UB(t)]ρSB(0)[U † S(t)⊗ U † B(t)] U̇S/B = −iHS/BUS/B US/B(0) = I Actual system: H = HS +HB +HSB Ideal vs Actual System Evolution ρSB(t) = U(t)ρSB(0)U†(t) U̇ = −iHU U(0) = I Ideal system: H = HS +HB ρidealSB (t) = [US(t)⊗ UB(t)]ρSB(0)[U † S(t)⊗ U † B(t)] U̇S/B = −iHS/BUS/B US/B(0) = I Actual system: H = HS +HB +HSB Distance: kρSB(t)− ρidealSB (t)kTr = kV (t)ρSB(0)V †(t)− ρSB(0)kTr V (t) ≡ U†S(t)⊗ U † B(t)U(t) ≡ exp[−itHeff(t)] Ideal vs Actual System Evolution ρSB(t) = U(t)ρSB(0)U†(t) U̇ = −iHU U(0) = I Ideal system: H = HS +HB ρidealSB (t) = [US(t)⊗ UB(t)]ρSB(0)[U † S(t)⊗ U † B(t)] U̇S/B = −iHS/BUS/B US/B(0) = I Actual system: H = HS +HB +HSB Distance: kρSB(t)− ρidealSB (t)kTr = kV (t)ρSB(0)V †(t)− ρSB(0)kTr V (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=1 q hv|A†A|vi = max sing.val.(A) Kolmogorov Distance Bound from Effective Hamiltonian V (t) ≡ U†S(t)⊗ U † B(t)U(t) Lemma: δ ≡ D(pρ(1) , pρ(2)) ≤ kρ(1) − ρ(2)kTr Lemma: kρ(1)S − ρ(2)S kTr ≤ kρ(1)SB − ρ(2)SBkTr (trace distance bounds Kolmogorov distance) (partial trace decreases distinguishability) ≡ exp[−itHeff(t)] Lemma: kρSB(t)− ρidealSB (t)kTr ≤ tkHeff(t)k∞ 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)? The Magnus expansion U̇ = −iH(t)U U(t) = e−itHeff(t) Heff(t) = 1 t ∞X j=1 Ωj(t) Ω1(t) = Z t 0 dt1H(t1) Ω2(t) = − i 2 Z t 0 dt1 Z t1 0 dt2[H(t1),H(t2)] δactual,ideal ≤ tkHeff(t)k∞ ≡ η(t) ≡ noise strength Analysis of Dynamical Decoupling “Symmetrizing group” of pulses { gi } and their inverses are applied in series: Periodic DD: periodic repetition of the universal DD pulse sequence exp( )SBiH τ≡ −f † † † 2 2 † 1 1( ) ( )( ) exp( )i SB iN iNg g g g g g g Hi gτ≈ − ∑"f f f first order Magnus expansion Dynamical 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 iiH H g H g≡ =∑(1)eff6 Dynamical Decoupling Condition exp( )SBiH τ≡ −f † † † 2 2 † 1 1( ) ( )( ) exp( )i SB iN iNg g g g g g g Hi gτ≈ − ∑"f f f first order Magnus expansion Dynamical 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 iiH H g H g≡ =∑(1)eff6 Dynamical Decoupling Condition exp( )SBiH τ≡ −f † † † 2 2 † 1 1( ) ( )( ) exp( )i SB iN iNg g g g g g g Hi gτ≈ − ∑"f f f first order Magnus expansion Dynamical Decoupling Theory Another view of the universal decoupling sequence: The Effective Hamiltonian τ exp[ ]SBi Hτ≡ −f δ X Z ZX f f f f' = f exp[ ( )] ( )=0, ideally eff eff iTH T H T ≡ −f' Another view of the universal decoupling sequence: The Effective Hamiltonian τ exp[ ]SBi Hτ≡ −f δ X Z ZX X "f f f f f' = f exp[ ( )] ( )=0, ideally eff eff iTH T H T ≡ −f' But, errors accumulate…: ( ) 0effH T ≠ 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: Recall noise strength norm of final effective system-bath Hamiltonian times the total duration. τ = T/N η ≡ ||Heff(T )||T 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: Recall noise strength norm of final effective system-bath Hamiltonian times the total duration. PDD leading order result for error: Can we do better? τ = T/N η ∝ N−1 η ≡ ||Heff(T )||T DD as a Rescaling Transformation • Interaction terms are rescaled after the DD cycle • We need a mechanism to continue this J = J (0) 7→ J (1) ∝ max[τ(J (0))2, τβJ(0)] β 7→ β +O((J (0))3τ2) J= kHSBk∞ β= kHBk∞ 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 2 2 2 2 2 2 2error (error) ((error) ) (((error) ) ) (error) k k"6 6 6 6 6 For 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 2 2 2 2 2 2error (error) ((error) ) (((error) ) ) (error) k k"6 6 6 6 6 η ∝ N bN−c logN η ∝ N−1 Experiments adamantene; qubit = 13C PDD=CDD1 CDD2 CDD3 Concatenated DD on Adamantene Powder Dieter Suter, TU Dortmund CDD Results 25 sτ μ= 50 sτ μ= 85 sτ μ= 10.5 sδ μ= E c h o i n t e n s i t y E c h o i n t e n s i t y E c h o i n t e n s i t y 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 ↓e,↑n ↓e,↓n νrf1 νμw1 νμw2 νrf2 |1〉 |0〉 29bath is Si ~1% natural abundance Periodic DD vs Concatenated DD 0 1 2 3 4 0.0 0.5 1.0 F i d e l i t y Concatenation Level State +X State +Y 1. Periodic (XfYfXfYf) 2. Concatenated 0 5 10 15 20 25 30 0.0 0.5 1.0 Number of Repeats F i d e l i t y State +X State +Y [log(time)] 160nsδ = r e l a t i v e e c h o i n t e n s i t y r e l a t i v e e c h o i n t e n s i t y ~100ms ~50ms 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 DFS encoding - symmetry not for free… QECC Hybrid Q. Error Correction: The Big Picture DFS encoding - symmetry not for free… DD QECC Hybrid Q. Error Correction: The Big Picture DFS encoding - symmetry not for free… DD QECC -pulse errors, Markovian effects Hybrid Q. Error Correction: The Big Picture DFS encoding - symmetry not for free… 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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