Canadian Summer School on Quantum Information (CSSQI) (10th : 2010)

Accurate and decoherence-protected adiabatic quantum computation Lidar, Daniel 2010-07-23

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Accurate and decoherence-protected adiabatic quantum computation Daniel Lidar, USCWorkshop on Quantum Algorithms, Computational Models, and Foundations of Quantum MechanicsVancouver, July 23, 2010PRL 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 systemsize, 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 systemsize. 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 systemsize, 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 systemsize. 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 analyticand has N zero derivatives at the initial and �nal times. Apply dynamical decoupling pulses generated by global magnetic �elds inthe x and z directions. The shorter the pulses and pulse intervals, thehigher 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 nx xσ σ⊗jXzzσ σ⊗jZ1 ni+1 j+1x xσ σ⊗ijXXzzσ σ⊗ijZ ZScheme for dynamical-decoupling-protected universal adiabatic QC switch interactions adiabatically, locally1 2 j+1 nx xσ σ⊗jXzzσ σ⊗jZ1 ni+1 j+1x xσ σ⊗ijXXzzσ σ⊗ijZ ZzBzBzBzBxBxBxBScheme for dynamical-decoupling-protected universal adiabatic QC switch fields rapidly but globallyProof Strategy (and talk outline)LetS  distance between desired ground stateand actual system state, at �nal time TLemma 1: the distance inequality:S �decoherence distance� �adiabatic distance�due to open system + due to closed systemnon-unitary evolution| {z }dDdeviations from adiabaticity| {z }adand show that both these distances can be made arbitrarily small.How? Lemma 2 � adiabatic distance ad: For slow enough evolution, can bemade arbitrarily small using analytic interpolation and almost-constantboundary conditions (or using onlyC2 interpolation provided evolving moreslowly, for the same error) Lemma 3 (with the help of Lemmas 4 & 5) � decoherence distancedD: Can be made arbitrarily small using dynamical decoupling (with �nite-width pulses and �nite pulse intervals)Tools: Distance Measure and Operator NormDistance between states: trace distanceD[1;2]  12k1 2k1kAk1  TrjAj = Xsing.val.(A);jAj pAyAWhen applied to pure states i = j iih ij I�ll write D[ 1; 2].Operator normkAk  supkj ik=1qh jAyAj i= maxsing.val.(A)Distancesad ad0ad() () ()                     ( )BTT TTφφ ρρ⊗=()Tρ0ad:0:  perfectly adiabaticSBHXX=Sd, system+bathd =Distances0()Tρ()TρaddDd, system+bathd =0ad:0:  perfectly adiabaticSBHXX=Sdad ad0ad() () ()                     ( )BTT TTφφ ρρ⊗=DistancesBTrBTr0()Tρ()Tρ0ad ad S,ad() () ()TT Tφφ ρ=S()TρaddDdSδ, system+bathd =, system onlyδ =0ad:0:  perfectly adiabaticSBHXX=Sdad ad0ad() () ()                     ( )BTT TTφφ ρρ⊗=DistancesBTrBTrBTrBTrad()Tφ()Tψ0()Tρ()TρS()TρaddadδDdSδ, system+bathd =, system onlyδ =0ad:0:  perfectly adiabaticSBHXX=Sdad ad0ad() () ()                     ( )BTT TTφφ ρρ⊗=0ad ad S,ad() () ()TT Tφφ ρ=DistancesBTrBTrBTrBTrad()Tφ()Tψ0()Tρ()TρS()TρaddadδDdSδ, system+bathd =, system onlyδ =0ad:0:  perfectly adiabaticSBHXX=SdS ad Ddδ δ≤ +0ad ad S,ad() () ()TT Tφφ ρ=ad ad0ad() () ()                     ( )BTT TTφφ ρρ⊗=DistancesBTrBTrBTrBTrad()Tφ()Tψ0()Tρ()TρS()TρaddadδDdSδ, system+bathd =, system onlyδ =0ad:0:  perfectly adiabaticSBHXX=Sdclosed system:reduce via improved adiabatic theoremopen system:reduce via dynamical decouplingS ad Ddδ δ≤ +ad ad0ad() () ()                     ( )BTT TTφφ ρρ⊗=0ad ad S,ad() () ()TT Tφφ ρ=Part 1: Closed System AQCDimensionless time: = t=T 2 [0;1]; T = �nal timeTrue �nal state j (1)i is the solution of the (rescaled) Schr�dinger equation:dj id = iTHadj iGoal of AQC: simultaneously minimize T(n) and the errorad D[ (1);ad(1)]:Questions... What determines T(n)? How to make ad small?Textbook criterion:to have jh (1)jad(1)ij2  12need T  1 maxsjhexcited()j_Had()jad()ijmin gap2() :Gap dependence on n determines T(n). For e� cient AQC algorithms:(n)  0nz; z = dynamical critical exponentUnfortunately this criterion is not quite right...Adiabatic Distance for Closed SystemsDepending on the di�erentiability of Had one can prove di�erent versions of the adiabatictheorem (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) = 0Then (norm is operator norm):T rk_Hadk23 =) ad <r2DAL, A. Rezakhani, A, Hamma, JMP 50, 102106 (2009):Assume Had(τ):F has a non-degenerate and gapped ground stateF is analytic in a strip of width γ around [0,1]F its first N derivatives vanish at τ = 0,1Let r> 1. ThenT = rNk_Hadk23 =) ad < (N + 1)+1rN=) Closed-system AQC is resilient against control errors which preserve gapbut causeHad(s) to deviate from its intended path, as long as �nal Hamiltonianis correct.This is a form of inherent fault tolerance to control errors not shared by thecircuit model!How to accomplish this? Design Hamiltonian according to criteria above.Part 2: Open System AQCJoint System-Bath EvolutionH(t) = HS(t)IB +IS HB| {z }H0+HSB;HS(t) = Had(t)| {z }implements AQC+ HC(t)| {z }implements DDHSB = XSB[We�ll see later how to enforce [Had(t);HC(t0)] = 0 8t;t0:]Joint system-bath propagator and joint state:U(t) = T exp[iZ t0H(t0)dt0]; (t) = U(t)(0)U(t)yDecoupled joint system-bath state: 0(t) = U0(t)(0)U0(t)yDecoupling 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-bathcoupling. Implemented via HC(t).The sequence ZOO, in increasing order of performance quality:PDD = a periodic repetition of a basic sequenceRDD = a random pulse sequenceCDD = a concatenated sequence (recursively structured)UDD = a sequence optimized to cancel pure qubit dephasing with the smallestpossible number of pulsesQDD = a sequence optimized to cancel general qubit decoherence with thesmallest 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 iiHH gHg≡=∑(1)eff6Dynamical Decoupling Conditionexp( )SBiH τ≡ −f†††22†11()()()exp( )iSBiNiNgg gggg gHi gτ≈−∑"ffffirst order Magnus expansionDynamical Decoupling TheoryDD Parameters:K = no. of pulsesw = pulse width = pulse intervalc = K(w+) = cycle timeL = number of PDD cyclesT = Lc = total timeTransform to interaction picture de�ned by Had +HB, i.e.,~U(t) = Uyad(t)UyB(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=jc= I, i.e., He(jc) = 0:Deviation from ideality at �nal time is thus quanti�ed by�error phase�:  TkHe(T)kHow is the �error phase�related to the �decoupling distance�dD?�error phase�:  TkHe(T)kThen 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)]LetJ  kHSBk  kHad +HBkand assume Jc < (absolute convergence condition of Magnus expansion).Then (Lemma 5):(T)  JTw +w| {z }error due to�nite pulse width+ (JT)2L +JT min[1;(exp(2c)12c 1)]| {z }error due to terms not removedby �rst order DD procedure:Joint AQC-DD Optimization for Local HamiltoniansFor local Hamiltonians Had and HB: = kHad +HBk O(n2)Recall closed system adiabaticity theorem, ensuring error small if N is large:T = rNk_Hadk23 =) ad < (N + 1)+1rN rN 130n3z+4 KL( +w)where we used gap condition for e� cient AQC algorithms:(n)  0nz; z = dynamical critical exponentGiven and �xed parameters of the problem are J, 0, and z.Need to ensure that each of the terms upper bounding (T) vanishes as afunction 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)=Jwith 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 arbitrarilyaccurate 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 afunction 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)=Jwith 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 arbitrarilyaccurate 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 lawin the system size, [using only 2-local Hamiltonians, in the presence of 1-local noise],assuming access to (dynamical decoupling) pulses whose width andintervals shrink as a power law in the system size. The latter powerlaw exponent is linear in the dynamical critical exponent of the closedsystem.Part 3: Seamlessly Combining AQC & DDNeed to show how to achieve �non-interference�condition[Had(t);HC(t0)] = 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 NoiseA 2-local Hamiltonian that is universal for AQC (J.D. Biamonte and P.J. Love,arXiv:0704.1287):Hunivad (t) = Xi;2fx;zghi (t)i + Xi;j;2fx;zgJij(t)ij:1-local noise (linear decoherence model):HlinSB = X=x;y;znXj=1j BjDecoupling group that decouples HlinSB = P=x;y;zPnj=1j Bj :G = fI;X;Y;Zg; X =nOj=1xj, etc.This requires only global pulses.G is the stabilizer of an [[n;n  2;2]] stabilizer code C (n even, x =evenweight binary string):C = fj xi = (jxi+jnotxi)=p2gE.g. n = 4:j00iL = (j0000i+j1111i)=p2; j10iL = (j0011i+j1100i)=p2j01iL = (j0101i+j1010i)=p2; j11iL = (j1001i+j0110i)=p2Encoded single-qubit generators for C = fj xi = (jxi+jnotxi)=p2g:Xj = x1xj+1 Zj = zj+1znEncoded two-qubit generators:Xi Xj = xi+1xj+1 ZiZj = zi+1zj+1Thus universal AQC can be combined with DD using only 2-local xixj andzizj interactions over C.1 2 j+1 nx xσ σ⊗jXzzσ σ⊗jZUniversality for the [[n,n-2,2]] code1 ni+1 j+1x xσ σ⊗ijXXzzσ σ⊗ijZ ZzBzBzBzBxBxBxBPhysical examples where X, Z (as pulses for DD) and xixj;zizj (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 thesystem sizen, using only 2-local Hamiltonians, in the presence of 1-local noise, assumingaccess to (dynamical decoupling) pulses whose width and intervals shrink as a power lawin the system size. The latter power law exponent is linear in the dynamical criticalexponent 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 thesystem sizen, using only 2-local Hamiltonians, in the presence of 1-local noise, assumingaccess to (dynamical decoupling) pulses whose width and intervals shrink as a power lawin the system size. The latter power law exponent is linear in the dynamical criticalexponent 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 EndsEnergy gap stabilization Jordan, Farhi & Shor (PRA 74, 052322 (2006)) introduced a scheme thatcan protect AQC against 1-local noise using a energy gap against localexcitations. Such gaps can be engineered into the Hamiltonians using error detectingcodes 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 wasdesigned forWe used dynamical decoupling (open-loop, feedback-free) to deal with couplingto 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 upMaybe: 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 matterOpen System EvolutionConsider the uncoupled setting HSB = 0, denoted by the superscript 0.ideal adiabatic state: 0S;ad(t) = jad(t)ihad(t)jactual state under Had(t) : 0S(t) = j (t)ih (t)jstate under HC(t) : 0C(t)state under HB(t) : 0B(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)] = adProof of the Distance InequalityPartial trace can only decrease distance:D[S(T);0S;ad(T)]| {z }SD[(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-Zτ τFID – T2*EchoSignalPulsesT(|0〉 + |1〉)(|0〉 + |1〉)Dynamical DecouplingDynamical Decoupling (DD) = a sequence of pulses applied to the system,sometimes forming a group G, designed to reduce the e�ective system-bathcoupling. Implemented via HC(t).The sequence ZOO, in increasing order of performance quality:PDD = a periodic repetition of a basic sequenceRDD = a random pulse sequenceCDD = a concatenated sequence (recursively structured)UDD = a sequence optimized to cancel pure qubit dephasing with the smallestpossible number of pulsesQDD = a sequence optimized to cancel general qubit decoherence with thesmallest possible number of pulsesSubsystem Code Construction for [Had(t);HC(t0)] = 0The decoupling groupG induces a decomposition of the system Hilbert spaceHS via its groupalgebra CG and its commutant CG0:HS = MJCnJ CdJ;CG = MJInJ MdJ; CG0 = MJMnJ IdJ:nJ = multiplicity of irrep J; dJ = dimensionAdiabatic state is encoded into a left factorCJ  CnJ: annJ-dimensional codeCJ storinglognJ 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 LJJ;InJ IdJ.Non-interference condition is satis�ed because [CG;CG0] = 0.

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