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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