Workshop on Quantum Algorithms, Computational Models and Foundations of Quantum Mechanics

Accurate and decoherence-protected adiabatic quantum computation 2010

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Accurate and decoherence-protected adiabatic quantum computation Daniel Lidar, USC Workshop on Quantum Algorithms, Computational Models, and Foundations of Quantum Mechanics Vancouver, July 23, 2010 PRL 100, 160506 (2008) JMP 50, 102106 (2009) $: Motivation  Protect Adiabatic Quantum Computation (AQC) against decoherence and control errors  Find rigorous bounds on …delity of AQC in presence of coupling to environment  Main result of this talk: Theorem: AQC can be performed with a …delity approaching 1 as a power law in the system size, using only 2-local Hamiltonians, in the presence of 1-local noise, assuming access to (dynamical decoupling) pulses whose width and intervals shrink as a power law in the system size. The power law exponent is linear in the dynamical critical exponent of the closed system. Main reference: DAL, “Towards Fault Tolerant Adiabatic Quantum Computation”, Phys. Rev. Lett. 100, 160506 (2008) Motivation  Protect Adiabatic Quantum Computation (AQC) against decoherence and control errors  Find rigorous bounds on …delity of AQC in presence of coupling to environment  Main result of this talk: Theorem: AQC can be performed with a …delity approaching 1 as a power law in the system size, using only 2-local Hamiltonians, in the presence of 1-local noise, assuming access to (dynamical decoupling) pulses whose width and intervals shrink as a power law in the system size. The power law exponent is linear in the dynamical critical exponent of the closed system. Main reference: DAL, “Towards Fault Tolerant Adiabatic Quantum Computation”, Phys. Rev. Lett. 100, 160506 (2008) In other words: You can get arbitrarily accurate open-system AQC for the following price:  Design your adiabatic evolution Hamiltonian (2-local) so that it is analytic and has N zero derivatives at the initial and …nal times.  Apply dynamical decoupling pulses generated by global magnetic …elds in the x and z directions. The shorter the pulses and pulse intervals, the higher the …delity. This result doesn’t depend on bath temperature.  To ensure that adiabatic evolution and dynamical decoupling are compat- ible, encode into a simple (distance-2) quantum error detection code. 1 2 j+1 n x xσ σ⊗ jX z zσ σ⊗ jZ 1 ni+1 j+1 x xσ σ⊗ i jX X z zσ σ⊗ i jZ Z Scheme for dynamical-decoupling-protected universal adiabatic QC switch interactions adiabatically, locally 1 2 j+1 n x xσ σ⊗ jX z zσ σ⊗ jZ 1 ni+1 j+1 x xσ σ⊗ i jX X z zσ σ⊗ i jZ Z zB zB zBzB xB xB xB Scheme for dynamical-decoupling-protected universal adiabatic QC switch fields rapidly but globally Proof Strategy (and talk outline) Let S  distance between desired ground state and actual system state, at …nal time T Lemma 1: the distance inequality : S  “decoherence distance” “adiabatic distance” due to open system + due to closed system non-unitary evolution| {z } dD deviations from adiabaticity| {z } ad and show that both these distances can be made arbitrarily small. How?  Lemma 2 — adiabatic distance ad: For slow enough evolution, can be made arbitrarily small using analytic interpolation and almost-constant boundary conditions (or using onlyC2 interpolation provided evolving more slowly, for the same error)  Lemma 3 (with the help of Lemmas 4 & 5) — decoherence distance dD: Can be made arbitrarily small using dynamical decoupling (with …nite- width pulses and …nite pulse intervals) Tools: Distance Measure and Operator Norm Distance between states: trace distance D[1; 2]  1 2 k1  2k1 kAk1  TrjAj = X sing.val.(A); jAj  p AyA When applied to pure states i = j iih ij I’ll write D[ 1;  2]. Operator norm kAk  sup kj ik=1 q h jAyAj i = max sing.val.(A) Distances ad ad 0 ad ( ) ( ) ( )                      ( ) BT T T T φ φ ρ ρ ⊗ = ( )Tρ 0 ad : 0 :  perfectly adiabatic SBHX X = Sd , system+bathd = Distances 0( )Tρ ( )Tρ add Dd , system+bathd = 0 ad : 0 :  perfectly adiabatic SBHX X = Sd ad ad 0 ad ( ) ( ) ( )                      ( ) BT T T T φ φ ρ ρ ⊗ = Distances BTr B Tr 0( )Tρ ( )Tρ 0 ad ad S,ad( ) ( ) ( )T T Tφ φ ρ= S( )Tρ add Dd Sδ , system+bathd = , system onlyδ = 0 ad : 0 :  perfectly adiabatic SBHX X = Sd ad ad 0 ad ( ) ( ) ( )                      ( ) BT T T T φ φ ρ ρ ⊗ = Distances BTr B Tr BTr BTr ad( )Tφ ( )Tψ 0( )Tρ ( )Tρ S( )Tρ add adδ Dd Sδ , system+bathd = , system onlyδ = 0 ad : 0 :  perfectly adiabatic SBHX X = Sd ad ad 0 ad ( ) ( ) ( )                      ( ) BT T T T φ φ ρ ρ ⊗ = 0 ad ad S,ad( ) ( ) ( )T T Tφ φ ρ= Distances BTr B Tr BTr BTr ad( )Tφ ( )Tψ 0( )Tρ ( )Tρ S( )Tρ add adδ Dd Sδ , system+bathd = , system onlyδ = 0 ad : 0 :  perfectly adiabatic SBHX X = Sd S ad Ddδ δ≤ + 0 ad ad S,ad( ) ( ) ( )T T Tφ φ ρ= ad ad 0 ad ( ) ( ) ( )                      ( ) BT T T T φ φ ρ ρ ⊗ = Distances BTr B Tr BTr BTr ad( )Tφ ( )Tψ 0( )Tρ ( )Tρ S( )Tρ add adδ Dd Sδ , system+bathd = , system onlyδ = 0 ad : 0 :  perfectly adiabatic SBHX X = Sd closed system: reduce via improved adiabatic theorem open system: reduce via dynamical decoupling S ad Ddδ δ≤ + ad ad 0 ad ( ) ( ) ( )                      ( ) BT T T T φ φ ρ ρ ⊗ = 0 ad ad S,ad( ) ( ) ( )T T Tφ φ ρ= Part 1: Closed System AQC Dimensionless time:  = t=T 2 [0; 1]; T = …nal time True …nal state j (1)i is the solution of the (rescaled) Schrödinger equation: dj i d = iTHadj i Goal of AQC: simultaneously minimize T (n) and the error ad  D[ (1); ad(1)]: Questions...  What determines T (n)?  How to make ad small? Textbook criterion: to have jh (1)jad(1)ij2  1 2 need T  1  maxs jhexcited()j _Had()jad()ij min gap2() : Gap dependence on n determines T (n). For e¢ cient AQC algorithms: (n)  0nz; z = dynamical critical exponent Unfortunately this criterion is not quite right... Adiabatic Distance for Closed Systems Depending on the di¤erentiability of Had one can prove di¤erent versions of the adiabatic theorem (Lemma 2).  Jansen, Ruskai, & Seiler [J. Math. Phys. 48, 102111 (2007)]. Assume: – the ground state manifold of Had is gapped. – Had() is twice di¤erentiable on [0; 1] – _Had(0) = _Had(1) = 0 Then (norm is operator norm): T  rk _Hadk2 3 =) ad < r2 DAL, A. Rezakhani, A, Hamma, JMP 50, 102106 (2009): Assume Had(τ): F has a non-degenerate and gapped ground state F is analytic in a strip of width γ around [0,1] F its first N derivatives vanish at τ = 0, 1 Let r > 1. Then T = r  N k _Hadk2 3 =) ad < (N + 1)
+1rN =) Closed-system AQC is resilient against control errors which preserve gap but causeHad(s) to deviate from its intended path, as long as …nal Hamiltonian is correct. This is a form of inherent fault tolerance to control errors not shared by the circuit model! How to accomplish this? Design Hamiltonian according to criteria above. Part 2: Open System AQC Joint System-Bath Evolution H(t) = HS(t)  IB + IS HB| {z } H0 +HSB; HS(t) = Had(t)| {z } implements AQC + HC(t)| {z } implements DD HSB = X  S B [We’ll see later how to enforce [Had(t); HC(t 0)] = 0 8t; t0:] Joint system-bath propagator and joint state: U(t) = T exp[i Z t 0 H(t0)dt0 ]; (t) = U(t)(0)U(t)y Decoupled joint system-bath state: 0(t) = U0(t)(0)U0(t)y Decoupling distance: dD  D[(T ); 0(T )] S  dD + ad: Already saw ad can be made arbitrarily small using analytic interpolation. Goal: Simultaneously minimize dD using dynamical decoupling. This is an optimization problem: generically decoherence worsens with increas- ing T , while closed-system adiabaticity improves. Dynamical Decoupling (DD) = a sequence of pulses applied to the system, sometimes forming a group G, designed to reduce the e¤ective system-bath coupling. Implemented via HC(t). The sequence ZOO, in increasing order of performance quality: PDD = a periodic repetition of a basic sequence RDD = a random pulse sequence CDD = a concatenated sequence (recursively structured) UDD = a sequence optimized to cancel pure qubit dephasing with the smallest possible number of pulses QDD = a sequence optimized to cancel general qubit decoherence with the smallest possible number of pulses “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 DD Parameters: K = no. of pulses w = pulse width  = pulse interval c = K(w + ) = cycle time L = number of PDD cycles T = Lc = total time Transform to interaction picture de…ned by Had +HB, i.e., ~U(t) = U y ad(t)  U y B(t)U(t) The “e¤ective error Hamiltonian”: ~U(t)  eitHe(t) He(t) can be calculated using Dyson or (better) Magnus expansion. Note that in decoupled limit: ~U(t) HSB!0! UC(t) t=j c = I, i.e., He(j c) = 0: Deviation from ideality at …nal time is thus quanti…ed by “error phase”:   TkHe(T )k How is the “error phase” related to the “decoupling distance”dD? “error phase”:   TkHe(T )k Then distance between desired and actual (Lemma 4): D[(T ); 0(T )] = dD  min[1; (e  1)=2]   if   1: Minimize error phase ) mininize decoupling distance. Error Phase Bound for Periodic DD [K. Khodjasteh & DAL, Phys. Rev. A 78, 012355 (2008)] Let J  kHSBk   kHad +HBk and assume J c <  (absolute convergence condition of Magnus expansion). Then (Lemma 5): (T )  JTw  + w| {z } error due to …nite pulse width + (JT )2 L + JT min[1; ( exp(2c) 1 2c  1)]| {z } error due to terms not removed by …rst order DD procedure : Joint AQC-DD Optimization for Local Hamiltonians For local Hamiltonians Had and HB:  = kHad +HBk  O(n2) Recall closed system adiabaticity theorem, ensuring error small if N is large: T = r  N k _Hadk2 3 =) ad < (N + 1)
+1rN  r  N 1 30 n3z+4  KL( + w) where we used gap condition for e¢ cient AQC algorithms: (n)  0nz; z = dynamical critical exponent Given and …xed parameters of the problem are J , 0, and z. Need to ensure that each of the terms upper bounding (T ) vanishes as a function of n. This is the case that if pulse interval  and pulse width w scale as   n(3z+3+1)=0; w  n(6z+5+1+2)=J with 1 > 0 and 2 > 0. For then, using Lemmas 4 & 5 we have proven Lemma 3: dD . n2 + (J=0)2n(1+1) + (J=0)n1 n!1! 0 =) Using PDD with properly chosen parameters we can obtain arbitrarily accurate AQC. Shortcoming: pulse intervals and widths must shrink with n as a power law... Could perhaps be remedied by employing concatenated DD. Need to ensure that each of the terms upper bounding (T ) vanishes as a function of n. This is the case that if pulse interval  and pulse width w scale as   n(3z+3+1)=0; w  n(6z+5+1+2)=J with 1 > 0 and 2 > 0. For then, using Lemmas 4 & 5 we have proven Lemma 3: dD . n2 + (J=0)2n(1+1) + (J=0)n1 n!1! 0 =) Using PDD with properly chosen parameters we can obtain arbitrarily accurate AQC. Shortcoming: pulse intervals and widths must shrink with n as a power law... Could perhaps be remedied by employing concatenated DD. In conclusion we’ve (almost) proven the Theorem: AQC can be performed with a …delity approaching 1 as a power law in the system size, [using only 2-local Hamiltonians, in the presence of 1-local noise], assuming access to (dynamical decoupling) pulses whose width and intervals shrink as a power law in the system size. The latter power law exponent is linear in the dynamical critical exponent of the closed system. Part 3: Seamlessly Combining AQC & DD Need to show how to achieve “non-interference” condition [Had(t); HC(t 0)] = 0 8t; t0: Can be done using encoding. Example — Stabilizer-normalizer construction: DD pulses are stabilizer elements. AQC implemented via normalizer elements. 2-Local Universal AQC Resistant Against 1-local Noise A 2-local Hamiltonian that is universal for AQC (J.D. Biamonte and P.J. Love, arXiv:0704.1287): Hunivad (t) = X i;2fx;zg hi (t)  i + X i;j;2fx;zg Jij(t)  i   j : 1-local noise (linear decoherence model): H linSB = X =x;y;z nX j=1 j Bj Decoupling group that decouples H linSB = P =x;y;z Pn j=1   j Bj : G = fI;X; Y; Zg; X = nO j=1 xj , etc. This requires only global pulses. G is the stabilizer of an [[n; n  2; 2]] stabilizer code C (n even, x =even weight binary string): C = fj xi = (jxi+ jnotxi) = p 2g E.g. n = 4: j00iL = (j0000i+ j1111i) = p 2; j10iL = (j0011i+ j1100i) = p 2 j01iL = (j0101i+ j1010i) = p 2; j11iL = (j1001i+ j0110i) = p 2 Encoded single-qubit generators for C = fj xi = (jxi+ jnotxi) = p 2g: Xj =  x 1 x j+1 Zj =  z j+1 z n Encoded two-qubit generators: Xi Xj =  x i+1 x j+1 Zi Zj =  z i+1 z j+1 Thus universal AQC can be combined with DD using only 2-local xi  x j and zi z j interactions over C. 1 2 j+1 n x xσ σ⊗ jX z zσ σ⊗ jZ Universality for the [[n,n-2,2]] code 1 ni+1 j+1 x xσ σ⊗ i jX X z zσ σ⊗ i jZ Z zB zB zBzB xB xB xB Physical examples where X, Z (as pulses for DD) and xi  x j ;  z i z j (as Hamil- tonians for AQC) are available and controllable:  Capacitively coupled ‡ux qubits (D.V. Averin and C. Bruder, Phys. Rev. Lett. 91, 057003 (2003))  Spin models implemented with polar molecules (A. Micheli, G. Brennen, and P. Zoller, Nature Phys. 2, 341 (2006)) Conclusions  Theorem: AQC can be performed with a …delity approaching 1 as a power law in the system size n, using only 2-local Hamiltonians, in the presence of 1-local noise, assuming access to (dynamical decoupling) pulses whose width and intervals shrink as a power law in the system size. The latter power law exponent is linear in the dynamical critical exponent of the closed system.  Open questions: - Can a similar result be shown with n-independent pulse width and interval? - The distance bound is rather crude because of use of the triangle inequality; can it be tightened by directly treating adiabatic evolution in the open system? - A fault-tolerance threshold for AQC? Conclusions  Theorem: AQC can be performed with a …delity approaching 1 as a power law in the system size n, using only 2-local Hamiltonians, in the presence of 1-local noise, assuming access to (dynamical decoupling) pulses whose width and intervals shrink as a power law in the system size. The latter power law exponent is linear in the dynamical critical exponent of the closed system.  Open questions: - Can a similar result be shown with n-independent pulse width and interval? - The distance bound is rather crude because of use of the triangle inequality; can it be tightened by directly treating adiabatic evolution in the open system? - A fault-tolerance threshold for AQC? Odds and Ends Energy gap stabilization  Jordan, Farhi & Shor (PRA 74, 052322 (2006)) introduced a scheme that can protect AQC against 1-local noise using a energy gap against local excitations.  Such gaps can be engineered into the Hamiltonians using error detecting codes  However, the resulting universal Hamiltonians are at least 4-local  Their scheme is fully compatible with the DD scheme presented here     Control Error ● Suppose final Hamiltonian is slightly off: ● Resulting ground state is: ● Gap against local operators ensures  denominator  Why not Use Standard Quantum Error Correction? Not known how to embed fault-tolerant QEC into adiabatic evolution:  Requires feedback, which may break adiabaticity (A. Allahverdyan & G. Mahler, arXiv:0804.1643)  Embedded FT-QEC circuit will face di¤erent error model than what it was designed for We used dynamical decoupling (open-loop, feedback-free) to deal with coupling to environment, and showed it can deal with the relevant error model. And, we used an analytic interpolation to boost …delity of closed-system adia- batic evolution. Is AQC “Better” than Other QC Paradigms? No:  Computationally equivalent to circuit model (Aharonov et al., :quant-ph/0405098; A. Mizel, DAL, M. Mitchell, PRL 99, 070502 (2007))  Doesn’t have a fault tolerance theory to back it up Maybe:  Perhaps more easily implemented in certain systems, in particular solid state (quantum dots, superconducting)  Inherently protected against leakage if kBT is smaller than gap  Inherently protected against path deformations: only end points matter Open System Evolution Consider the uncoupled setting HSB = 0, denoted by the superscript 0. ideal adiabatic state: 0S;ad(t) = jad(t)ihad(t)j actual state under Had(t) :  0 S(t) = j (t)ih (t)j state under HC(t) :  0 C(t) state under HB(t) :  0 B(t) decoupled joint state: 0(t) = 0S(t)  0C(t)  0B(t) ideal adiabatic joint state: 0ad(t)  0S;ad(t)  0C(t)  0B(t) Let d () denote distances in the joint (system) Hilbert space. target distance: S  D[S(T ); 0S;ad(T )] decoupling distance: dD  D[(T ); 0(T )] adiabatic distance: dad  D[0(T ); 0ad(T )] = ad Proof of the Distance Inequality Partial trace can only decrease distance: D[S(T );  0 S;ad(T )]| {z } S  D[(T ); 0ad(T )] Triangle inequality: D[(T ); 0ad(T )]  D[(T ); 0(T )]| {z } dD +D[0(T ); 0ad(T )]| {z } ad =) desired distance inequality (Lemma 1): S  dD + ad: Inspired by Hahn spin echo (1950) π/2-Y π-Z τ τ FID – T2* Echo Signal Pulses T(|0〉  + |1〉) (|0〉  + |1〉) Dynamical Decoupling Dynamical Decoupling (DD) = a sequence of pulses applied to the system, sometimes forming a group G, designed to reduce the e¤ective system-bath coupling. Implemented via HC(t). The sequence ZOO, in increasing order of performance quality: PDD = a periodic repetition of a basic sequence RDD = a random pulse sequence CDD = a concatenated sequence (recursively structured) UDD = a sequence optimized to cancel pure qubit dephasing with the smallest possible number of pulses QDD = a sequence optimized to cancel general qubit decoherence with the smallest possible number of pulses Subsystem Code Construction for [Had(t); HC(t0)] = 0 The decoupling group G induces a decomposition of the system Hilbert spaceHS via its group algebra CG and its commutant CG0: HS = M J CnJ  CdJ ; CG = M J InJ MdJ ; CG0 = M J MnJ  IdJ : nJ = multiplicity of irrep J ; dJ = dimension Adiabatic state is encoded into a left factorCJ  CnJ : an nJ -dimensional codeCJ storing lognJ qubits. AQC is enacted via CG0. DD pulses act on the right factors, enacted via the elements of CG. DD pulses project each S in system-bath Hamiltonian to L J J;InJ  IdJ . Non-interference condition is satis…ed because [CG;CG0] = 0.

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