aa r X i v : . [ qu a n t - ph ] M a y Towards Fault Tolerant Adiabatic Quantum Computation
Daniel A. Lidar
Departments of Chemistry, Electrical Engineering, and Physics,Center for Quantum Information Science & Technology,University of Southern California, Los Angeles, CA 90089
I show how to protect adiabatic quantum computation (AQC) against decoherence and certaincontrol errors, using a hybrid methodology involving dynamical decoupling, subsystem and stabilizercodes, and energy gaps. Corresponding error bounds are derived. As an example I show how toperform decoherence-protected AQC against local noise using at most two-body interactions.
Adiabatic quantum computation (AQC), originally de-veloped to solve optimization problems [1], offers a fas-cinating alternative to the standard circuit model [2] towhich it is computationally equivalent [3]. The effectsof decoherence on AQC were studied in several works[4, 5, 6]. Unlike the circuit model, for which an elabo-rate theory of fault tolerant QC exists along with a noisethreshold for fault tolerance [7], it is not yet known howto make AQC fault tolerant. Here I show how AQC canbe can be protected against decoherence and certain con-trol errors. To do so I devise a hybrid method that involv-ing dynamical decoupling (DD) [8], subsystem [9, 10, 11]and stabilizer codes [12], and energy gaps [13, 14].Viewed as a closed system, AQC proceeds via slow evo-lution on a timescale set by the system’s minimal energygap ∆ from the ground state [1, 3]. In the presence ofthe system-bath interaction H SB this gap can be signifi-cantly reduced because the interaction will cause energylevel splittings, or an effective broadening of system en-ergy levels; when these levels overlap adiabaticity breaksdown and so does AQC, even at zero temperature [5]. Abath at finite temperature presents another problem: inthe universality proofs [3] the system energy gap scalesas an inverse polynomial in the problem size, so that thetemperature too must be lowered polynomially to preventthermal excitations. All of the problems listed above aredue to the presence of H SB . Clearly, if H SB can be effec-tively eliminated or reduced, this will enhance the fidelityof AQC. The main tool I shall use to this end is dynam-ical decoupling, which involves the application of strongand fast pulses. Perhaps surprisingly, this can be donewithout interfering with the slow adiabatic evolution. Distance measure and operator norm .— As a dis-tance measure between states I use the trace distance D [ ρ , ρ ] ≡ k ρ − ρ k , where k A k ≡ Tr | A | , | A | ≡√ A † A [2]. When applied to pure states ρ i = | ψ i ih ψ i | I shall write D [ ψ , ψ ]. The operator norm is k A k ≡ sup k| ψ ik =1 k A | ψ ik = max i λ i , where λ i ∈ Spec( | A | ). Closed-system adiabatic error .— Let s = t/T ∈ [0 , T the final time. Letthe system Hamiltonian that implements AQC, H ad ( s ),act on n qubits. In AQC the ground state | φ ad ( s ) i of H ad ( s ) at the final time s = 1 encodes the solu-tion to the computational problem [1]. The actual final state | ψ (1) i is the solution of the Schr¨odinger equation d | ψ i /ds = − iT H ad | ψ i ( ~ = 1 units are used throughout).In AQC one is therefore interested in minimizing the er-ror δ ad ≡ D [ ψ (1) , φ ad (1)]. Most of the known AQC algo-rithms interpolate between initial and final local Hamil-tonians, H and H , via H ad ( s ) = (1 − f ( s )) H + f ( s ) H ,where f (0) = 0 and f (1) = 1, and exhibit a final timethat scales as a polynomial in the problem/system size n . Locality means that k H ad k ∼ ∆ O ( n ), where ∆ isthe energy scale. Thus k d j H ad /ds j k ∼ ∆ | d j f /ds j | O ( n ).Let { E i ( s ) } i =0 be the eigenvalues of H ad ( s ), and let∆ ≡ min i,s | E i ( s ) − E ( s ) | be the minimum gap fromthe instantaneous ground state energy E ( s ). Assumethat ∆( n ) ∼ ∆ n − z , where z > H ad ,and assuming that ˙ H ad (0) = ˙ H ad (1) = 0, one can provedifferent versions of the adiabatic theorem. For exam-ple, (i) [15]: if H ad ( s ) is twice differentiable on [0 ,
1] thenprovided T ∼ r k ˙ H ad k / ∆ the error can be made arbi-trarily small in the time dilation factor r > δ ad < r − .Or, (ii) [16]: if H ad ( s ) is infinitely differentiable on [0 , T ∼ rN k ˙ H ad k / ∆ , the error can be madeexponentially small in the order N of an asymptotic ex-pansion: δ ad < r − N . In both cases T ∼ n ζ / ∆ , (1)where ζ = 3 z + 2 for case (i) and ζ = 2 z + 1 for case(ii), and I omitted | d j f /ds j | . In AQC the interpolationfrom H ad (0) to H ad (1) can be chosen at will, in particularso as to satisfy the above conditions on H ad . This showsthat closed-system AQC is resilient against control errorswhich cause H ad ( s ) to deviate from its intended path, aslong as these do not modify the end point H ad (1). This isa form of inherent fault tolerance to control errors whichis not shared by the circuit model [17]. Open system evolution .— A description in terms of H ad alone neglects the fact that in reality the adiabatic quan-tum computer system is never perfectly isolated. The ac-tual Hamiltonian is H ( t ) = H S ( t ) ⊗ I B + I S ⊗ H B + H SB ,where I denotes the identity operator, H S = H ad + H C ( H B ) acts on the system (bath) alone, H C ( t ) is a controlHamiltonian, and H SB = P α S α ⊗ B α , where S α ( B α )acts on the system (bath). The role of H C is to imple-ment a DD procedure. The total propagator is U ( t ) = T exp[ − i R t H ( t ′ ) dt ′ ], where T denotes time ordering.The time evolved system state is ρ S ( t ) = Tr B ρ ( t ), where ρ ( t ) = U ( t ) ρ (0) U ( t ) † is the joint system-bath state. Be-low I explain how to choose H C ( t ) so that[ H ad ( t ) , H C ( t ′ )] = 0 ∀ t, t ′ . (2)It is this condition that will allow application of DD with-out interfering with the adiabatic evolution. Considerthe uncoupled setting H SB = 0, to be denoted by the su-perscript 0. The ideal, noise-free adiabatic system stateis ρ S, ad ( t ) = | φ ad ( t ) ih φ ad ( t ) | . Because the adiabatic,control, and bath Hamiltonians all commute we have ρ ( t ) = ρ S ( t ) ⊗ ρ ( t ) ⊗ ρ B ( t ), where ρ S ( t ) = | ψ ( t ) ih ψ ( t ) | [ ρ ( t ) = | ψ C ( t ) ih ψ C ( t ) | ] is the actual system evolutionunder H ad [ H C ], and ρ B ( t ) is the bath state evolved un-der H B . Let ρ ( t ) ≡ ρ S, ad ( t ) ⊗ ρ ( t ) ⊗ ρ B ( t ) denotethe “ideal adiabatic joint state,” with purely adiabaticevolution of the first factor. Note that ρ S (0) = ρ S, ad (0). General error bound .— Let d ( δ ) denote distances inthe joint (system) Hilbert space. To quantify the devia-tion of the actual evolution from the desired one, let: δ S ≡ D [ ρ S ( T ) , ρ S, ad ( T )] , d D ≡ D [ ρ ( T ) , ρ ( T )] d ad ≡ D [ ρ ( T ) , ρ ( T )] = δ ad , d tot ≡ D [ ρ ( T ) , ρ ( T )] . The overall objective is to minimize the distance δ S be-tween the actual system state and the ideal, noise-freeadiabatic system state. The distance between the un-coupled joint state and the ideal adiabatic joint state is d ad , which equals δ ad since k A ⊗ B k = k A k k B k and k ρ B k = k ρ k = 1 . The “decoupling distance” is d D :the distance between the joint state in the coupled anduncoupled settings. Minimization of this distance is thetarget of the DD procedure. Finally, d tot is the distancebetween the actual and ideal joint states.Because taking the partial trace can only decrease thedistance between states [2], we have δ S ≤ d tot . Using thetriangle inequality we have d tot ≤ d D + d ad . Therefore: δ S ≤ d D + δ ad . (3)This key inequality shows that the total system erroris bounded above by the sum of two errors: (i) due tothe system-bath interaction in the presence of decou-pling ( d D ); (ii) due to the deviations from adiabaticityin the closed system ( d ad ). I shall present a procedureintended to minimize d D jointly with d ad . This is anoptimization problem: generically decoherence (closed-system adiabaticity) worsens (improves) with increasing T . Dynamical decoupling .— I now show how to minimizethe decoupling error d D . To do so I propose to applystrong and fast dynamical decoupling (DD) pulses to thesystem on top of the adiabatic evolution. It is conve-nient to first transform to an interaction picture definedby H ad + H B , i.e., U ( t ) = U ad ( t ) ⊗ U B ( t ) ˜ U ( t ), where U X ( t ) = T exp[ − i R t H X ( t ′ ) dt ′ ], X ∈ { ad , B } . Then ˜ U satisfies the Schr¨odinger equation ∂ ˜ U /∂t = − i ˜ H ˜ U , with˜ H = U † B ⊗ U † ad [ H C + H SB ] U B ⊗ U ad = H C + ˜ H SB , wherethe second equality required Eq. (2). Define an effec-tive “error Hamiltonian” H eff ( t ) via ˜ U ( t ) = e − itH eff ( t ) ,which can be conveniently evaluated using the Mag-nus expansion [19]. Now consider a sequence of non-overlapping control Hamiltonians H ( k )DD ( t ) applied for du-ration w (pulse width) at pulse intervals τ , i.e., H C ( t ) = 0for t k ≤ t < t k +1 − w and H C ( t ) = H ( k )DD for t k +1 − w ≤ t < t k +1 , where t k = k ( τ + w ), k ∈ Z K . The se-quence { H ( k )DD } K − k =0 defines a “DD protocol” with cycletime T c = K ( τ + w ) and unitary pulses P k generatedby ˜ H ( t ) = H ( k )DD + ˜ H SB , t k +1 − w ≤ t < t k +1 . Inthe “ideal pulse limit” w = 0 one defines the “decou-pling group” G = { G k ≡ P K − · · · P k +1 P k } K − k =0 suchthat G = I S . Then the total propagator becomes˜ U ( T c ) = Q K − k =0 exp[ − iτ ( G † k ˜ H SB G k )] ≡ e − iT c H ideff , where H ideff denotes the resulting effective Hamiltonian, withMagnus series H ideff = P ∞ j =0 H id( j )eff [8]. To lowest order: H id(0)eff = 1 K K − X k =0 G † k ˜ H SB G k ≡ Π G ( ˜ H SB ) . (4)In the limit τ → H ideff = H id(0)eff , so that byproperly choosing G one can effectively eliminate H SB .Returning to non-ideal ( w >
0) pulses, we have shownby use of k [ A, B ] k ≤ k A kk B k and the Dyson ex-pansion that minimization of the “error phase” Φ( T ) ≡ T k H eff ( T ) k implies minimization of the decoupling dis-tance d D [20]: d D ≤ min[1 , ( e Φ − / ≤ Φ if Φ ≤ . (5)For single-qubit systems we and others have shown thatconcatenated DD pulse sequences can decrease Φ expo-nentially in the number of concatenation levels [21]. HereI focus on periodic pulse sequences for simplicity. In pe-riodic DD (PDD) one repeatedly applies the DD protocol { H ( k )DD } K − k =0 to the system, i.e., H C ( t + lK ) = H C ( t ), l ∈ Z L . The total time is thus T = L ( τ + w ), where the totalnumber of pulses is L and the number of cycles is L/K . Acalculation of the total error phase Φ( T ) proceeds in twosteps. First we find an upper bound Θ l on Φ l ( T c ) for the l th cycle, using the Magnus expansion. Then we upperbound Φ( T ) by ( L/K ) max l Θ l . Let J ≡ k H SB k (system-bath coupling strength), β ≡ k H ad + H B k ≤ β S + β B ,where β S = k H ad k and β B = k H B k , and α = O (1) aconstant. A worst case analysis yields [18]:Φ( T ) ≤ α ( JT ) L/K + JT wτ + w + JT ( exp(2 βT c ) − βT c − , (6)This bound is valid as long the third term is ≤ JT and theMagnus series is absolutely convergent over each cycle, asufficient condition for which is JT c < π [18, 19]. Joint AQC-DD optimization .— Recall Eq. (1) forclosed system adiabaticity. The given and fixed parame-ters of the problem are J , ∆ , and z (or ζ ). The task isto ensure that each of the terms in Eq. (6) vanishes as afunction of n . I show in [22] that if τ and w scale as τ ∼ n − ( ζ + ǫ ) / ∆ , w ∼ n − (2 ζ + ǫ + ǫ ) /J, (7)with ǫ > ǫ >
0, then d D . ( J/ ∆ ) n − ǫ + n − ǫ + ( J/ ∆ ) n − ǫ , (8)which is arbitrarily small in the large n limit. Combiningthis with the bounds above ( δ ad < r − or δ ad < r − N )and inequality (3), it follows that for an AQC algorithmwith time scaling as T = L ( τ + w ) ∼ ∆ − n ζ , the totalerror δ S can be made arbitrarily small. This is the firstmain result of this work: using PDD with properly chosenparameters we can obtain arbitrarily accurate AQC .However, there is a shortcoming: the pulse intervalsand widths must shrink with n as a power law, with anexponent dictated by the dynamical critical exponent z of the model [Eq. (7)]. I expect that this can be remediedby employing concatenated DD [18, 21]. Seamless AQC-DD .— The entire analysis relies so faron the “non-interference” condition (2). When can itbe satisfied? Fortunately, the general background theorywas worked out in [9, 10], though without any referenceto AQC. I review this theory and make the connectionto AQC explicit. The decoupling group G induces a de-composition of the system Hilbert space H S via its groupalgebra C G and its commutant C G ′ , as follows: H S ∼ = M J C n J ⊗ C d J , (9) C G ∼ = M J I n J ⊗ M d J , C G ′ ∼ = M J M n J ⊗ I d J . (10)Here n J and d J are, respectively, the multiplicity anddimension of the J th irreducible representation (irrep)of the unitary representation chosen for G , while I N and M N are, respectively, the N × N identity matrix and un-specified complex-valued N × N matrices. The adiabaticstate is encoded into (one of) the left factors C J ≡ C n J ,i.e., each such factor (with J fixed) represents an n J -dimensional code C J storing log d n J qu d its. The DDpulses act on the right factors. As shown in [9], the dy-namically decoupled evolution on each factor (code) C J will be noiseless in the ideal limit w, τ → G ( S α ) = L J λ J,α I n J ⊗ I d J for all system operators S α in H SB ,whence H id(0)eff = L J [( I n J ⊗ I d J )] S ⊗ [ P α λ J,α B α ] B .Thus, assuming the latter condition is met, under theaction of DD the action of H id(0)eff on the code C J is pro-portional to I n J , i.e., is harmless . Quantum logic, or AQC, is enacted by the elements of C G ′ . Dynamical de-coupling operations are enacted via the elements of C G . Condition (2) is satisfied because [ C G , C G ′ ] = 0. Stabilizer decoupling .— An important example of thegeneral C G / C G ′ construction is when G is the stabilizerof a quantum error correcting code and the commutantis the normalizer N of the code [12]. Because a stabilizergroup is Abelian its irreps are all one-dimensional. Astabilizer code encoding n qubits into n J = k has n − k generators, each of which has eigenvalues ±
1. Then J runs over the 2 n − k different binary vectors of eigenval-ues, meaning that H S ∼ = L J = {± ,..., ± } C k , and each ofthe subspaces in the sum is a valid code C J . Here the el-ements of N are viewed as Hamiltonians. For this reasononly the encoded single-qubit normalizer operations arerequired; encoded two-body interactions are constructedas tensor products of single-qubit ones. Energy-gap protection .— Application of DD pulses isthe main mechanism I propose for protection of AQC,but it has a shortcoming as noted above. Fortunately,the formulation presented here easily accommodates theAQC energy-gap protection strategy proposed in [13],which can be viewed as adding another layer of pro-tection for dealing with finite-resource-DD. Namely, ifthe decoupling group G is also a stabilizer group forcode C J , then for each Pauli error S α in H SB there isat least one element P j ∈ G such that { P j , S α } = 0,and otherwise [ P j , S α ] = 0 [12]. We can then addan energy penalty term H P = − E P P |G|− j =1 P j ∈ C G to H S , where E P > H id( j ≥ = 0. To lowest order, H id(1)eff = P α S α ⊗ B (1) α , and an “erred state” will beof the form | ψ ⊥ α i = S α | ψ i , where | ψ i = P j | ψ i ∈ C J ∀ j . Then H P | ψ ⊥ α i = { [ a − ( K − K − E P } | ψ ⊥ α i ,where a is the number of stabilizer elements that anti-commute with S α . Thus | ψ ⊥ α i is an eigenstate of H P and has a ( K − E P more energy than any state in thecode space. Ref. [13] showed, using a Markovian modelof qubits coupled to a photon bath, the important re-sult that this energy gap for erred states implies thatthe temperature need only shrink logarithmically ratherthan polynomially in the problem size. However, notethat to deal with generic system-bath interactions boththe stabilizer and normalizer elements must involve k -local interactions, with k > .— Firstrecall a recent universality result. The following sim-ple 2-local Hamiltonian allows for universal AQC [23]: H univad ( t ) = P i ; α ∈{ x,z } h αi ( t ) σ αi + P i,j ; α ∈{ x,z } J αij ( t ) σ αi σ αj .With this all the tools have been assembled to demon-strate the second main result of this work: a stabi-lizer decoupling procedure against 1-local noise that usesonly 2-local interactions. By 1-local noise I mean themain nemesis of quantum computing, namely the lin-ear decoherence model: H lin SB = P α = x,y,z P nj =1 σ αj ⊗ B αj ,where { B αj } are arbitrary bath operators. To beat H lin SB , use the Abelian “universal decoupling group” [8] G uni = { I, X, Y, Z } , where X ( Y, Z ) = N nj =1 σ x ( y,z ) j .It is simple to verify that Π G uni ( H lin SB ) = 0. Asnoted in Ref. [9], G uni is the stabilizer of an [[ n, n − , C , whose codewords are {| ψ x i =( | x i + | not x i ) / √ } , where x is an even-weight binarystring of length n , with n even. For example, for n = 4 we find: | i L = ( | i + | i ) / √ | i L =( | i + | i ) / √ | i L = ( | i + | i ) / √ | i L = ( | i + | i ) / √
2. Now universal AQC over C can be implemented using 2-local Hamiltonians. Tocompute over C we replace each Pauli matrix in H univad by its encoded partner. Encoded single-qubit operationsfor C are the 2-local ¯ X j = σ x σ xj +1 and ¯ Z j = σ zj +1 σ zn ,where j = 1 , ..., n −
2. The 2-local interactions σ xi σ xj and σ zi σ zj appearing in H ad are replaced by the 2-local¯ X i ¯ X j = σ xi +1 σ xj +1 and ¯ Z i ¯ Z j = σ zi +1 σ zj +1 . Thus we seethat universal AQC can be combined with DD using only2-local σ xi σ xj and σ zi σ zj interactions over C .Examples of promising QC implementations where X , Z (as pulses for DD) and σ xi σ xj , σ zi σ zj (as Hamiltonians forAQC) are available and controllable, are systems includ-ing capacitive coupling of flux qubits [24] and spin modelsimplemented with polar molecules [25]. Also note that inprinciple, as discussed above, we can create an additionalenergy gap [13] against single-qubit errors by adding apenalty term H P = − E P ( X + Y + Z ) to the systemHamiltonian. However, H P is an n -local interaction. Conclusions and outlook .— Using a combination ofvarious tools in the arsenal of decoherence control I haveshown how to protect AQC against decoherence. WhileI believe that the methods proposed here should signifi-cantly contribute towards the viability and robustness ofAQC, what is still missing is a threshold theorem for faulttolerant AQC. This will most likely require the incorpo-ration of feedback, in order to correct DD pulse imper-fections and other control noise [17]. One possibility fordoing so might be to perform syndrome measurements onthe commutant factor [ C d J in Eq. (9)] as in recent circuit-model fault tolerance work using subsystems codes [7]. Acknowledgements .— Important discussions with K.Khodjasteh, A. Hamma, and P. Zanardi are gratefully ac-knowledged. Supported under grant NSF CCF-0523675. [1] E. Farhi et al. , eprint quant-ph/0001106.[2] M.A. Nielsen and I.L. Chuang,
Quantum Computationand Quantum Information (Cambridge University Press,Cambridge, UK, 2000).[3] D. Aharonov et al. , SIAM J. Comp. , 166 (2007); M.S.Siu, Phys. Rev. A , 062314 (2005); A. Mizel, D.A.Lidar, M. Mitchell, Phys. Rev. Lett. , 070502 (2007).[4] A.M. Childs, E. Farhi, and J. Preskill, Phys. Rev. A , 012322 (2001).[5] M.S. Sarandy and D.A. Lidar, Phys. Rev. Lett. ,250503 (2005).[6] J. Roland and N.J. Cerf, Phys. Rev. A , 032330 (2005);M. Tiersch and R. Sch¨utzhold, Phys. Rev. A , 062313(2007).[7] P. Aliferis, A.W. Cross, Phys. Rev. Lett. , 220502(2007).[8] P. Zanardi, Phys. Lett. A , 77 (1999); L. Viola, E.Knill and S. Lloyd, Phys. Rev. Lett. , 2417 (1999).[9] L. Viola, E. Knill, and S. Lloyd, Phys. Rev. Lett. ,3520 (2000).[10] P. Zanardi, Phys. Rev. A , 012301 (2001).[11] J. Kempe et al. , Phys. Rev. A , 042307 (2001).[12] D. Gottesman, Phys. Rev. A , 127 (1998).[13] S.P. Jordan, E. Farhi, and P.W. Shor, Phys. Rev. A ,052322 (2006).[14] D. Bacon, K.R. Brown, K.B. Whaley, Phys. Rev. Lett. , 247902 (2001); Y.S. Weinstein and C.S. Hellberg,Phys. Rev. Lett. , 110501 (2007).[15] S. Jansen, M.-B. Ruskai, and R. Seiler, J. Math. Phys. , 102111 (2006).[16] A. Hamma and D.A. Lidar, eprint arXiv:0804.0604.[17] I assume that there is no other time scale such as in K.P.Marzlin and B.C. Sanders, Phys. Rev. Lett. , 160408(2004), or as would be the result of noise on f ( s ) which isindependent of T . Such noise is a potentially detrimentalsource of errors I do not account for – see Ref. [16].[18] K. Khodjasteh and D.A. Lidar, eprint arXiv:0803.4320.[19] F. Casas, J. Phys. A , 15001 (2007).[20] D.A. Lidar, P. Zanardi, and K. Khodjasteh, eprintarXiv:0803.4268.[21] K. Khodjasteh and D.A. Lidar, Phys. Rev. Lett. ,180501 (2005); K. Khodjasteh and D.A. Lidar, Phys.Rev. A , 062310 (2007); W. Yao, R.-B. Liu, and L. J.Sham, Phys. Rev. Lett. , 077602 (2007); W.M. Witzeland S. Das Sarma, Phys. Rev. B , 045218 (2007); W.Zhang et al. , eprint arXiv.org:0801.0992.[22] To derive Eq. (8) let L = Kn ζ + ǫ , and absorb α into T . The first term in Eq. (6) thus yields n − ǫ < (∆ /J ) ,which is satisfied for large enough n if (i) ǫ >
0. Us-ing T = L ( τ + w ), the second term yields w < / ( JL ),which we can satisfy with (ii) w = n − (2 ζ + ǫ + ǫ ) / ( JK ), ǫ >
0, and the first term yields (iii) w < / ( J √ LK ) − τ ;but w > τ < / ( J √ LK ). We can satisfy (iii) and (iv) by choosing (v) τ = n − ( ζ + ǫ ) / ( K ∆ ), since then T = L ( τ + w ) n ≫ → ∆ − n ζ as it should. As for the third term in Eq. (6),note that e x − x − ≤ ( e − x if x ≤
1. Indeed, x = 2 βT c ∼ β ( n − ( ζ + ǫ ) / ∆ + n − (2 ζ + ǫ + ǫ ) /J ) ≪ β S andβ B ∼ O ( n ) (local Hamiltonians; e.g., a sce-nario where each system particle is in contact with alocal (spin) bath). Therefore the third term is boundedabove by ( e − JT βT c n ≫ → O ( n )( J/ ∆ ) n − ǫ , so thatwe require ǫ > , 057003(2003).[25] A. Micheli, G. Brennen, and P. Zoller, Nature Phys.2