aa r X i v : . [ qu a n t - ph ] F e b Necessary Adiabatic Run Times in Quantum Optimization
Lucas T. Brady
Department of Physics, University of California, Santa Barbara, CA 93106-5110, USA
Wim van Dam
Department of Computer Science, Department of Physics,University of California, Santa Barbara, CA 93106-5110, USA (Dated: August 27, 2018)Quantum annealing is guaranteed to find the ground state of optimization problems provided itoperates in the adiabatic limit. Recent work [Phys. Rev. X 6, 031010 (2016)] has found that forsome barrier tunneling problems, quantum annealing can be run much faster than is adiabaticallyrequired. Specifically, an n -qubit optimization problem was presented for which a non-adiabatic, ordiabatic, annealing algorithm requires only constant runtime, while an adiabatic annealing algorithmrequires a runtime polynomial in n .Here we show that this non-adiabatic speedup is a direct result of a specific symmetry in thestudied problem. In the more general case, no such non-adiabatic speedup occurs and we showwhy the special case achieves this speedup compared to the general case. We also prove that theadiabatic annealing algorithm has a necessary and sufficient runtime that is quadratically betterthan the standard quantum adiabatic condition suggests. We conclude with an observation aboutthe required precision in timing for the diabatic algorithm. Recent work in quantum adiabatic optimization [1] hasfocused on a class of Hamming-symmetric problems thatexhibits extremely strong non-adiabatic speedups overa slower adiabatic approach. Numerical evidence pre-sented by Muthukrishnan, Albash, and Lidar [2] showsthat for several barrier tunneling problems on n qubits,a well-calibrated constant time evolution of the quantumannealing Hamiltonian is sufficient. Thus, this algorithmsignificantly improves upon the slower adiabatic evolu-tion of the Hamiltonian, which could take polynomial oreven exponential time in n . Muthukrishnan et al. at-tribute this speedup to a diabatic cascade in which theground state is quickly depopulated in favor of higher ex-cited states and then repopulated right at the end of thediabatic evolution.Usually the sufficient run time of quantum adiabaticoptimization is estimated using the standard adiabaticcondition. This condition says that adiabaticity is en-sured if the running time grows as τ ≫ max s ∈ [0 , (cid:13)(cid:13)(cid:13) ∂ ˆ H ( s ) ∂s (cid:13)(cid:13)(cid:13) /g ( s ) , (1)for the spectral gap g ( s ). More accurate versions of thiscondition have been proven [3], but all of them dependlinearly on the matrix norm of ˆ H ( s ) or its derivativeswith respect to s divided by a low degree polynomialfunction of the gap g ( s ).The condition in Eq. 1 is merely a sufficient condi-tion, and it is possible to have adiabatic evolutions withshorter running times than Eq. 1 describes. Furthermoreit is also possible to have a non-adiabatic evolution thatsucceeds in solving the optimization problem at hand.It is such a non-adiabatic speedup that is described byMuthukrishnan et al. [2]. A non-adiabatic speedup is obviously significant fornear-term quantum computers where quantum anneal-ing is a potential application. Kong and Crosson [4]have studied these diabatic transitions, and more recentlythe current authors presented complementary findings[5]. These recent results indicates that this non-adiabaticspeedup can provide an alternate and efficient way ofsolving an important class of Hamming-symmetric bar-rier tunneling problems that are being used as toy models[2, 4, 6–11] to study the more general properties of quan-tum annealing in the presence of a barrier.Here we present results that indicate that even slightlymore generalized versions of symmetric barrier tunnelingproblems do not exhibit this fast non-adiabatic speedup.The base Hamiltonian used to study this class of prob-lems exists in a Hilbert space of n qubits and is givenbyˆ H ( s ) = − (1 − s ) n X i =1 σ ( i ) x + s " n X i =1 σ ( i ) z + b n X i =1 σ ( i ) z ! , (2)where b ( h ) is some localized barrier or perturbation and s = t/τ is a normalized time variable representing thelinear progression of time, t , from t = 0 to the algorithmstopping time τ . Current numerical evidence [2] suggeststhat the non-adiabatic speedup exists for many classes,shapes, and sizes of localized barriers b ( h ). This articlegeneralizes the problem slightly (ignoring b ( h ) for themoment): ˆ H ( s ) = − (1 − s ) n X i =1 σ ( i ) x + sµ n X i =1 σ ( i ) z , (3)by introducing a positive slope parameter µ and wefind that for the generic case µ = 1, the non-adiabatic p τ µ = 0 . µ = 1 µ = 2 FIG. 1: The single qubit success probability, p , as a functionof the total runtime for several µ values. The blue, dashed, µ = 1 line corresponds to the model that has been studiedin previous articles. Notice that the µ = 1 curve has severalspecial properties, including that it goes to p = 1 at finite τ ,resulting in the non-adiabatic speedup noted in other papers.The µ = 1 curves do not exhibit this p = 1 behavior. speedup no longer exists. We call µ a slope as it re-lates linearly the energy of the system with the Hammingweight P i σ ( i ) z of the n qubits.Since this Hamiltonian describes a simple toy model,it is unlikely that a physical system will exhibit the ex-act µ = 1 behavior, leading us to the conclusion that forrealizable problems, this diabatic speedup will not exist.In this article, we will focus on the b ( h ) = 0 case sinceit decouples all the qubits, allowing us to extract infor-mation about the system by studying the evolution of asingle qubit Hamiltonian. Since µ = 1 disrupts the non-adiabatic speedup even in this b ( h ) = 0 case, we fullyexpect similar disruption to occur for more complicatedbarriers and perturbations.We first need to define our criteria for an optimal run-time. If an algorithm on n qubits runs for time τ andhas a probability of success of p n ( τ ) at the end of thattime, its expected running time is τ /p n ( τ ), and the op-timal running time is the τ n that minimizes τ /p n ( τ ) for n qubits. In our case, we have n independent qubits,each of which has a probability of success of p , hence p n = p n , which is where the n dependence comes intothe minimization.In the µ = 1 case, p goes to 1 for finite τ , as seenin Fig. 1, meaning that p n = 1 at this value, leading tothe non-adiabatic speedup noted in other studies. Fig. 1also shows µ = 0 . µ = 2 curves. Note that for thesecurves the success probability does not achieve p = 1at finite τ . Similar plots can be obtained for other µ =1 and, as we note below, this failure to reach p = 1for finite τ , ultimately leads to the breaking of the non-adiabatic speedup. Therefore, this speedup is restricted O p ti m a l τ n / ( p ) n n µ = 0 . µ = 0 . µ = 1 . µ = 1 . µ = 1 . FIG. 2: Optimal expected running time of quantum anneal-ing, τ n /p n ( τ n ), as a function of n for different µ values. Unlikethe µ = 1 case, τ n increases with n for these µ values. linesthrough the data are power law fits of the form τ n = An r , andthe fitted r values in the order µ = (0 . , . , . , . , .
6) are(0 . , . , . , . , . /
2. A scaling powerof 1 / µ = 1 caseas found in [2] and our results below while being quadraticallyfaster than the sufficient adiabatic condition. to the special case of µ = 1.To demonstrate the lack of a non-adiabatic speedupin the µ = 1 cases, consider Fig. 2, which shows theoptimal expected runtime, τ n /p n ( τ n ), as a function of n . All of the µ curves shown are increasing, meaningthat the running time increases with n , and there is nonon-adiabatic algorithm that runs in constant time. Thefitted curves are to power laws of the form τ n = An p ,and all of the fitted p values are close to 1 /
2, indicatinga running time of O ( √ n ).We can extract the √ n running time behavior from thecurves in Fig. 1 as well because the qubits in our problemare completely decoupled. For sufficiently large runningtimes τ , the curves of the single qubit success probability p as a function of τ shown in Fig. 1 are bounded aboveand below by envelopes of the form1 − c ℓ ( µ ) τ q < p < − c u ( µ ) τ q , (4)with constants c ℓ ( µ ) and c u ( µ ). This relationship is ex-tracted by performing numerical fits to the minima andmaxima in curves like those seen in Fig. 3, and for allour fits to different µ data, q is close to 2. Note that c u (1) = 0, which, as we will see, is one of the main rea-sons why the µ = 1 diabatic speedup can occur.Muthukrishnan et al. [2] showed that the lower enve-lope with c ℓ ( µ ) guarantees that the worst case running-time for the µ = 1 case scales as O ( √ n ). We will employtheir method to show that a relationship such as Eq. 4provides both the necessary and sufficient condition for p τ µ = 1 . Fits
FIG. 3: A single qubit success probability curve as a functionof total runtime τ for µ = 1 . p = 1 − cτ − q . The first two minima and maxima wereexcluded from this fit and others since they tend to be moreabnormal. In this case, the upper envelope has a fitted q =1 .
998 and the lower envelope has a fitted q = 1 . the running time. Muthukrishnan et al. also apply meth-ods created by Boixo and Somma [12] to show that atleast Ω( n / ) is necessary for adiabatic evolution.If for n qubits a total success probability of p is desiredfrom the algorithm, then Eq. 4 tells us that (cid:18) − c ℓ ( µ ) τ q (cid:19) n ≤ p ≤ (cid:18) − c u ( µ ) τ q (cid:19) n . (5)We can manipulate this inequality, performing an ex-pansion for small c ∗ ( µ ) /τ q since τ will be large. Theresult of these manipulations gives us a relationship be-tween the running time and n : (cid:18) c u ( µ )ln 1 /p n (cid:19) /q ≤ τ ≤ (cid:18) c ℓ ( µ )ln 1 /p n (cid:19) /q . (6)Therefore, since q = 2 in our cases, having a runningtime that scales as √ n is both a necessary and sufficientcondition to reaching a desired probability. Note thatwhen µ = 1, c u (1) = 0, so one side is no longer bounded,leading to the possibility of a non-adiabatic speedup.In the Hamming weight problem, the gap is constantwith n , and all matrix norms of the Hamiltonian andits derivatives will depend linearly on n . Therefore, theadiabatic condition, Eq. 1, would predict O ( n ) scaling;whereas, our results indicate that a faster O ( √ n ) runningtime is sufficient. This result was shown in [2] for µ = 1,and our results indicate that this quadratic speedup holdsfor general slopes µ . P o w e r L a w C o e f fi c i e n t µ c ℓ ( µ ) c u ( µ ) FIG. 4: The curves like the one in Fig. 3 are bounded aboveand below by curves of the form 1 − c/τ . We show the valuesof c for the upper, c u , and lower, c ℓ , bounding functions asobtained from numerical fits. These coefficients are a functionof µ , and all of the fits used to obtain this data were goodquality. In the main text, we show that these bounding curvesdirectly lead to a O ( √ n ) running time for the algorithm inall cases except the µ = 1 case where c u (1) = 0. While the standard adiabatic condition overestimatesthe running time, there are other derivations that ap-ply to our problem more specifically and that provide astricter bound that matches our results. Jansen, Ruskai,and Seiler [3] showed that for fixed Hamiltonians ˆ H andˆ H with time evolution ˆ H ( t ) = (1 − t/τ ) ˆ H + t/τ ˆ H , thesuccess probability p of remaining in the ground statethroughout 0 ≤ t ≤ τ is bounded by p = 1 − O ( τ − ) . (7)If we take this to be the probability of success for a sin-gle qubit case, our results in Eqs. 5 and 6 imply that τ ∈ O ( √ n ) is sufficient for an adiabatic evolution. Thisshows that the result from Jansen et al. provides a strictersufficient condition than the standard adiabatic conditionfor our optimization problem with decoupled qubits.In Fig. 4 we plot the coefficients c u ( µ ) and c ℓ ( µ ) ob-tained from numerical fits. The fits used to obtain thesevalues are akin to those shown in Fig. 3, making us con-fident in the 1 /τ scaling of the error. Notice that as weapproach the special case µ = 1 we see that c u ( µ ) → µ = 1 the coefficient c u ( µ )stays close to zero. Hence for µ approximately (but notexactly) 1, the non-adiabatic speedup will persists fora large range of n until the adiabatic running time of O ( √ n ) is required again at very large n .All of our work so far has shown that the optimal run-ning time of this algorithm is O ( √ n ), but this does notimply that the optimal running time results from adia-batic evolution. If we look at the occupancy of the energystates for these optimal runs, we in fact see the ground τ n µ = 0 . µ = 1 . µ = 1 . µ = 2 . FIG. 5: These plots shows the runtime, τ , needed to ensurethat state of the system is at least 75% in the ground stateover the entire s evolution. This growth of τ with n comesclosest to a true adiabatic evolution, and we can see that the τ ∈ O ( √ n ) behavior holds even in this case. Power law fits tothese data sets show that the exponent for these curves, in theorder µ = (0 . , . , . , . . , . , . , . τ /p n criteria we used. state being depopulated during the s evolution. There-fore, a remaining question to ask is whether this behavioralso holds if we require the system to stay within a cer-tain range of its ground state for the entire s ∈ [0 , τ , needed to ensure thatthe system has at least a 75% chance of being mea-sured in its ground state for the entire s ∈ [0 ,
1] evolu-tion. All of these curves exhibit power law relationships, τ = Bn r , with fitted r = (0 . , . , . , . µ = (0 . , . , . , .
0) respectively. A similar plot can beobtained if a stricter cutoff than 75% is used.Fig. 5 shows that the runtime relationships we observeare in fact indicative of how adiabatic evolution behavesas well. Therefore, we are led to the conclusion that forgeneral µ = 1, the runtime τ ∈ Θ( √ n ) is both necessaryand sufficient to ensure finding the true ground state.The µ = 1 case remains a special case that goes againstthis rule, allowing for an extreme speedup to a constantrunning time.Our last goal will be to understand the width of thesuccess probability spike of p in the unperturbed, µ =1 case when it reaches the optimal p = 1. We willshow that this narrowness implies that to be successfulfor large n , one has to be very precise in using the rightrunning time τ .We know that there is a critical runtime τ c such that p = 1 for a single qubit. For run times close to this τ c , the probability of success can be modeled by p = 1 − δ = 1 − k ( τ − τ c ) , δ ≪ , (8)where | τ − τ c | is the required stopping precision of thealgorithm.Scaling the system to n qubits, the probability of suc-cess is p n = p n since the qubits are uncoupled in theunperturbed case: p n = (cid:0) − k ( τ − τ c ) (cid:1) n ≈ − nk ( τ − τ c ) . (9)If we want the probability of failure to be less than ε , wemust have that1 − ε < − kn ( τ − τ c ) ⇒ | τ − τ c | < ( ε/kn ) / . (10)Thus, maintaining the same success probability as n in-creases requires the acceptable imprecision | τ − τ c | toshrink according to n − / .For perturbed problems with a barrier we have runsimulations using square barriers like those considered in[11]. For µ = 1, we found the same n − / narrowing ofthe spiked success probability p n around the critical τ c running time.Our conclusion is therefore that, while the µ = 1case does exhibit a surprising non-adiabatic speedup thatcould potentially be exploited, this diabatic speedup isnot a general feature of this class of quantum anneal-ing problems. Running these algorithms adiabaticallyremains the best and only option to achieve success ingeneral. Acknowledgements
This material is based upon work supported by theNational Science Foundation under Grants No. 1314969and No. 1620843. [1] E. Farhi, J. Goldstone, S. Gutmann, M. Sipser,arXiv:quant-ph/0001106 (2000).[2] S. Muthukrishnan, T. Albash, D. A. Lidar, Phys. Rev. X , 031010 (2016).[3] S. Jansen, M. Ruskai, R. Seiler, J. Math. Phys. ,102111 (2007).[4] L. Kong, E. Crosson, arXiv:1511.06991 [quant-ph](2015).[5] W. van Dam, L. T. Brady, presentation at AdiabaticQuantum Computing Conference 2016.[6] E. Farhi, J. Goldstone, S. Gutmann,arXiv:quant-ph/0201031 (2002).[7] B. W. Reichardt, in Proceedings of the 36th Annual ACMSymposium on Theory of Computing, ACM Press (2004).[8] E. Crosson, M. Deng, arXiv:1410.8484 [quant-ph] (2014).[9] E. Crosson, A. Harrow, Proc. of FOCS 2016, pp. 714-723(2016). [10] Z. Jiang, V. N. Smelyanskiy, S. V. Isakov, S. Boixo, G.Mazzola, M. Troyer, and H. Neven, quant-ph/1603.01293(2016). (2016). [11] L. Brady, W. van Dam, Phys. Rev. A , 032309 (2016).[12] S. Boixo, R. D. Somma, Phys. Rev. A81