I describe how real quantum annealers may be used to perform local (in state
space) searches around specified states, rather than the global searches
traditionally implemented in the quantum annealing algorithm. The quantum
annealing algorithm is an analogue of simulated annealing, a classical
numerical technique which is now obsolete. Hence, I explore strategies to use
an annealer in a way which takes advantage of modern classical optimization
algorithms, and additionally should be less sensitive to problem
mis-specification then the traditional quantum annealing algorithm.Comment: In Proceedings PC 2016, arXiv:1606.06513. An extended version of this
  contribution will appear on arXiv soon which will describe more detailed
  algorithms, comment more on robustness to problem mis-specification, comment
  on thermal sampling applications, and discuss applications on real device

Chancellor, Nicholas

Electronic Proceedings in Theoretical Computer Science

English

arXiv

I describe how real quantum annealers may be used to perform local (in state space) searches around specified states, rather than the global searches traditionally implemented in the quantum annealing algorithm. The quantum annealing algorithm is an analogue of simulated annealing, a classical numerical technique which is now obsolete. Hence, I explore strategies to use an annealer in a way which takes advantage of modern classical optimization algorithms, and additionally should be less sensitive to problem mis-specification then the traditional quantum annealing algorithm

Nicholas Chancellor

Directory of Open Access Journals

A.A. Abbott and D.C. Horsman (Eds.): 7th InternationalWorkshop on Physics and Computation (PC 2016).EPTCS 214, 2016, pp. 16–21, doi:10.4204/EPTCS.214.4c© N. ChancellorThis work is licensed under theCreative Commons Attribution License.An Overview of Approaches to Modernize QuantumAnnealing Using Local SearchesNicholas ChancellorDepartment of Physics, Durham UniversitySouth Road, Durham, UKnicholas.chancellor@durham.ac.ukI describe how real quantum annealers may be used to perform local (in state space) searches aroundspecified states, rather than the global searches traditionally implemented in the quantum annealingalgorithm. The quantum annealing algorithm is an analogue of simulated annealing, a classicalnumerical technique which is now obsolete. Hence, I explorestrategies to use an annealer in a waywhich takes advantage of modern classical optimization algorithms, and additionally should be lesssensitive to problem mis-specification then the traditional qu ntum annealing algorithm.1 Quantum Annealing Algorithm vs. Local SearchRecently, there has been much interest in using the Quantum Annealing Algorithm (QAA) [2, 3, 20]which utilizes quantum tunnelling to aid in solving commercially interesting problems. A complete listof all potential applications would be too long to give here.However, applications have been studiedin such diverse fields as finance [23], computer science [17],machine learning [1, 4], communications[6, 10, 11, 22], graph theory [9], and aeronautics [18], illustrating the importance of such algorithms toreal world problems.The archetypal model for quantum annealing, because of its connection to condensed matter physicsas well as the fact that it can be implemented on real devices is the transverse field Ising model, withHamiltonianH(s) given byH(s) =−A(s) ∑iσ xi +B(s)HProblem, HProblem=−∑ihiσ zi − ∑i, j∈χJi j σ zi σzj . (1)HProblemencodes the problem of interest,χ is the hardware graph, andA(s) andB(s) are the anneal-ing schedule, which determines how the energy scales of the transverse and longitudinal terms changewith the annealing parameter,s∈ [0,1]. The problem is encoded by specifying the values ofhi andJi j .For the QAA,A(0) ≫ B(0) andA(1) ≪ B(1), andA(s) decreases monotonically whileB(s) increasesmonotonically with increasings. Applying the QAA consists of monotonically increasings with timesuch that the ground state of the system changes over time betw en he (known) ground state of the trans-verse part of the Hamiltonian (∑i σ xi ) to the solution of the (classical) problem to be solved, Eq.(1). Thesearch space of the transverse Ising model is a hypercube wher each vertex corresponds to a bitstring,the dimension is equal to the number of qubits, and the Hamming distance between classical states corre-sponds to the number of edges which must be traversed betweenh states. This structure is independentof the interaction graph defined byJi j which, along withhi , determine the energy at each vertex.I choose to focus on the transverse field Ising model for concreteness, and because the action ofthe transverse field is a quantum analogue of single bit flip updates in classical Monte Carlo methods.However, the arguments presented in this paper should hold for most other search spaces as well, withN. Chancellor 17the Hamming distance replaced with a more general notion of search space distance. Because the rele-vant effect of problem mis-specification is the energy difference in the states which are searched, localsearches can remain valid even if the global space is corrupted by a mis-specification.The QAA can be though of as analogous to classical Simulated Annealing (SA) in which quantumfluctuations mediated by the addition of non-commuting terms to a classical Hamiltonian, play the rolewhich temperature plays in SA. Simple SA, however, has been super eded by more sophisticated algo-rithms, such as parallel tempering [19, 25], population anne li g [21, 24, 26], and isoenergetic clusterupdates [28] to name a few. This then begs the question of whether quantum annealing hardware can beused in a clever way to gain the advantages of these modern classical algorithms, by using a hybrid algo-rithm employing both quantum and classical search techniques, or by using multiple quantum searchesin a sequential way to make algorithmic gains.The QAA, as it is currently designed, is not amenable to such adaptations. It is a global search,and there is no obvious way to insert information, from either a classical algorithm or previous runs ofthe QAA, in a meaningful way to improve the performance. Furthermore, the QAA is fundamentallydifferent from classical annealing in that, due to the famous no-cloning theorem [27] of quantum me-chanics, we cannot determine exactly what the intermediatest t of the system is part way though theanneal. This is in direct contrast to SA, where every intermediat state is known, and can be manipulatedarbitrarily to build better algorithms. For example, classical gains can be made by running many runsin parallel and probabilistically replacing poor performing copies with those which are performing well(population annealing), or raising the temperature for those which perform poorly and lowering it forthose which perform well (parallel tempering).In order to build quantum versions, let us consider a subroutine similar to QAA, but which performsa local search of a region of phase space with a controllable size around a user selected initial state. Theinput and output of a single step of this algorithm is completely classical, so the no-cloning theorem is nolonger a barrier and these local quantum searches can be combined arbitrarily with both other quantumsearches and classical searches.Moreover, an effective temperature which can be used to construct analogues to parallel temper-ing and population annealing. To construct this we first diagonalize the Hamiltonian of a single qubitunder quantum annealingH1(s′) = −A(s′)σ x+B(s′)σ z to obtain the ground state ratio of probabilityamplitudesψ(1)ψ(2)=√A(s′)2+B(s′)2A(s′)+B(s′)A(s′).From this ratio an effective temperature can be derived by comparing to a Boltzmann distribution,Te f f(s′) = 2[ln(∣∣∣∣ψ(1)ψ(2)∣∣∣∣2)]−1.For details of these algorithms as well as more discussion oftolerance to problem mis-specification, usein thermal sampling, and feasibility in real devices, see [14].2 Local Search on an AnnealerFor a useful local search we desire two properties, firstly the search should be local in the sense that itonly explores a fraction of the states in the state space and secondly the search should seek out moreoptimal (lower energy) solutions over less optimal ones. Consider a protocol to search the phase space18 An Overview of Approaches to Modernize Quantum Annealing Using Local Searchesnear a chosen classical state in the presence of a low temperature bath. The system is first initialized ats= 1 in a state which specifies the starting point of the algorithm and therefore the region to be searched.Local search with a controllable range is then performed by decreasing the annealing parameters inEq. (1) to a prescribed values′ (thereby turning on a transverse field), possibly waiting for a period oftime, and then returning tos= 1 and reading out the final state normally. The low temperature bath willmoderate transitions between states, with detailed balance cting as a guarantee that more optimal stateswill be favoured in the search.One model which has been able to successfully predict experimental results [5,7,8,16] is to assumedecoherence acts in the energy eigenbasis. In this model, which arises from a perturbative expansionin coupling strength [12, 13], coherence can be lost rapidlybetween energy eigenstates and transitionsbetween these states can be mediated by the bath but the eigenstates themselves are not disrupted by thebath. Because the eigenstates themselves will generally behighly quantum objects, even a completelyincoherent superposition of them can still support quantumeffects.Solving problems using tunnelling mediated by open quantumsystem effects means that even ifthe system is initialized in an excited state, interactionswith the environment will cause probabilitytransitions to other eigenstates. Detailed balance implies that for a bath with finite temperature thetransitions will occur preferentially toward lower energystates. Furthermore, ifA(s) is appropriatelysmall compared toB(s) in Eq. (1) then the quantum fluctuations can be viewed as localfluctuationsaround a classical state, the strongerA(s) is compared toB(s), the less local this search will be. Considerthe perturbative expansion around a (non-degenerate) classical tate|C(s= 0)〉 which can be written as,|C(s)〉=1N∞∑n=0(A(s)B(s))nDn(∑iσ xi)n|C(0)〉 (2)whereDn is a diagonal matrix which depends on the spectrum ofHProblem andN is a normalizationfactor. If we assume dephasing noise, then the tunnelling rate between two perturbed classical states,|C(s)〉 and|C′(s)〉 will be proportional to〈C(s) |∑i σ zi |C′(s)〉. By inserting the state given in Eq. (2), wesee that〈C(s)∣∣∣∣∣∑iσ zi∣∣∣∣∣C′(s)〉∝(A(s)B(s))H (C(0),C′(0))+ · · · (3)whereH (C(0),C′(0)) is the Hamming distance (number of edges required to traverse on the hypercube)between the two classical states and. . . indicates higher powers ofA(s)B(s) . For smallA(s)B(s) , tunnelling be-tween perturbed classical states is thereforeexponentiallysuppressed in the Hamming distance betweenthe states. As an eigenstate of a transverse field Ising model|C(s 6= 0)〉 is a fundamentally quantum objectwhich will exhibit quantum entanglement and therefore willbe able to mediate tunnelling between clas-sical states quantum mechanically. We therefore expect a quantum advantage to be preserved within thelocal search. By using quantum searches only locally we haveremoved the possibility of gaining a quan-tum advantage for long range searches beyond the range of each loc l search. However, for this price wegain a major advantage, the classical long range search can be done using state-of-the-art techniques suchas parallel tempering or population annealing, therefore asm ll quantum advantage in the local searchstill results in an improvement over the underlying classical algorithm. By contrast, traditional quantumannealing only represents algorithmic improvement over classical methods if the quantum advantage isat leastas large as the advantage which state-of-the-art classicalte hniques such as parallel temperinghave over simulated annealing.The question is now how one could program the initial state. The initial states required for the localsearch protocol are completely classical, and therefore could be programmed directly by manipulatingN. Chancellor 19Figure 1: Schematic representation of a single annealing cycle to perform a local search. First thestandard QAA is implemented with the problem Hamiltonian given in Eq. (1) to initialize the qubitsin the desired state. After this the Hamiltonian is reprogrammed and the device is annealed tos′ andoptionally allowed to remain at that point for a timeτ . The device is then annealed tos= 1 and read out.the qubits in a classical way. Another completely classicalmethod would be to prepare a simple energylandscape where the desired state has the lowest energy and first heat and than cool the system, thuspreparing it by classical thermal annealing. Both of these methods would require additional controlsor degrees of freedom which may not be accessible on a real device. For this reason I will insteadfocus on preparing the initial state using the standard QAA,which an annealer is able to performbydefinition. This is accomplished by running the QAA with a simple Hamiltonian to guarantee that thesystem is initialized in a desired statey ( y(i) ∈ {−1,1} ) with a high probability, for exampleHinit (y) =−∑i y(i)σ zi −∑i, j∈χ y(i)y( j)σzi σzj , which is a gauge transform of an unfrustrated ferromagnetic systemin a strong field, and will have a very simple energy landscapeand a relatively large energy differencebetween the lowest energy and first excited state. Annealingru s with this Hamiltonian therefore shouldtherefore reach the target statey with a high probability. After this step, one needs to be ableto r programHProblem in Eq. (1) to be the Hamiltonian of the problem in which we are int rested.Once we have the desired initial state and problem Hamiltonian programmed, we simply need toturn on a desired strength of transverse field, controlled bythe values′ shown in Fig. 1. It may also bedesirable to wait for a timeτ before turning the field off again and reading out the state. The readoutdoes not need to be any different than what is used with the standard QAA. It is worth pointing out that inthe limit each run is effectively a noisy continuous time quant m random walk with a localized startingcondition and a finite temperature bath. In a related work I will examine the relationship between QAAand quantum random walks in the context of global rather thanlocal searches [15].AcknowledgmentsThe author was supported by EPSRC (grant ref: EP/L022303/1), and thanks Viv Kendon for severalcritical readings of [14] and useful discussions. The author further thanks Trevor Lanting, Helmut Katz-graber, Gabriel Aeppli, Andrew G. Green, and Paul A. Warburton for useful discussions.20 An Overview of Approaches to Modernize Quantum Annealing Using Local SearchesReferences[1] S. H. Adachi & M. P. Henderson (2015):Application of Quantum Annealing to Training of Deep NeuralNetworks. Available athttps://arxiv.org/abs/1510.06356.[2] E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren& D. Preda (2001):A quantum adiabatic evolutionalgorithm applied to random instances of an NP-complete problem. Science292, pp. 472–475, doi:10.1126/science.1057726.[3] J. Brooke, D. Bitko, T. F. Rosenbaum & G. Aeppli (1999):Quantum annealing of a disordered magnet.Science284, pp. 779–781, doi:10.1126/science.284.5415.779.[4] M. H. Amin, E. Andriyash, J. Rolfe, B. Kulchytskyy & R. Melko, (2016):Quantum Boltzmann Machine.Available athttps://arxiv.org/abs/1601.02036.[5] M. W. Johnsonet al. (2011):Quantum annealing with manufactured spins. Nature473, pp. 194–198, doi:10.1038/nature10012.[6] N. Chancellor, S. Szoke, W. Vinci, G. Aeppli & P. A. Warburton, (2016):Maximum-Entropy Inference witha Programmable Annealer. Scientific Reports6(22318), doi:10.1038/srep22318.[7] N. G. Dicksonet al. (2013):Thermally assisted quantum annealing of a 16-qubit problem. Nature Commu-nications4(1903), doi:10.1038/ncomms2920.[8] S. Boixo et al. (2016):Computational Role of Multiqubit Tunneling in a Quantum Annealer. Nature Com-munications7(10327), doi:10.1038/ncomms10327.[9] W. Vinci, K. Markström, S. Boixo, A. Roy, F. M. Spedalier, P. A. Warburton & S. Severini (2014):Hearingthe Shape of the Ising Model with a Programmable Superconducting-Flux Annealer. Scientific Reports4(5703), doi:10.1038/srep05703.[10] Y. Otsubo, J. I. Inoue, K. Nagata & M. Okada (2012):Effect of quantum fluctuation in error-correcting codes.Physical Review E86(051138), doi:10.1103/PhysRevE.86.051138.[11] Y. Otsubo, J. I. Inoue, K. Nagata & M. Okada (2014):Code-division multiple-access multiuser demodulatorby using quantum fluctuations. Physical Review E90(012126), doi:10.1103/PhysRevE.90.012126.[12] T. Albash, S. Boixo, D. A. Lidar & P. Zanardi (2014):Quantum Adiabatic Markovian Master Equations.New Journal of Physics14(123016), doi:10.1088/1367-2630/14/12/123016.[13] H. P. Breuer & F. Petruccione (2002):The Theory of Open Quantum Systems. Oxford University Press.[14] N. Chancellor (to appear)Modernizing Quantum Annealing using Local Searches.[15] N. Chancellor, J. Morley, V. Kendon & S. Bose (in preparation): Adiabatic and quantum walk search algo-rithms as quantum annealing extremes.[16] N. Chancellor, P. A. Warburton & G. Aeppli (2016):Experimental Freezing of mid-Evolution Fluctuationswith a Programmable Annealer. Available athttp://arxiv.org/abs/arXiv:1605.07549.[17] V. Choi (2010): Adiabatic Quantum Algorithms for the NP-Complete Maximum-Weight Independent Set,Exact Cover and 3SAT Problems. Available athttp://arxiv.org/abs/1004.2226.[18] G. E. Coxson, C. R. Hill & J. C. Russo (2014):Adiabatic quantum computing for finding low-peak-sidelobecodes. High Performance Extreme Computing conference, IEEE7, pp. 3910–3916, doi:10.1039/b509983h.[19] D. J. Earl & M. W. Deem (2005):Parallel Tempering: Theory, Applications, and New Perspectiv s. PhysicalChemistry Chemical Physics7, pp. 3910–3916, doi:10.1039/b509983h.[20] A. B. Finilla, M. A. Gomez, C. Sebenik & J. D. Doll (1994):Quantum annealing: A new methodfor minimizing multidimensional functions. Chemical Physics Letters219, pp. 343–348, doi:10.1016/0009-2614(94)00117-0.[21] K. Hukushima & Y. Iba (2003):The Monte Carlo Method in the Physical Sciences: Celebrating the 50thAnniversary of the Metropolis Algorithm. 690, AIP.N. Chancellor 21[22] S. P. Jordan, E. Farhi & P. W. Shor (2006):Error-correcting codes for adiabatic quantum computation.Physical Review A74(052322), doi:10.1103/PhysRevA.74.052322.[23] M. Marzec (2014):Portfolio Optimization: Applications in Quantum Computing. Social Science ResearchNetwork Technical Report, doi:10.2139/ssrn.2278729.[24] J. Matcha (2010):Population Annealing with Weighted Averages: A Monte CarloMethod for Rough FreeEnergy Landscapes. Physical Review E82(026704), doi:10.1103/PhysRevE.82.026704.[25] R. H. Swendsen & J. S. Wang (1986):Replica Monte Carlo Simulation of Spin-Glasses. Physical ReviewLetters57(2607), doi:10.1103/PhysRevLett.57.2607.[26] W. Wang, J. Matcha & H. G. Katzgraber (2015):Population annealing: Theory and application in spinglasses. Physical Review E92(063307), doi:10.1103/PhysRevE.92.063307.[27] W. K. Wootters & W. H. Zurek (1982):A single quantum cannot be cloned. Nature299, pp. 802–803, doi:10.1038/299802a0.[28] Z. Zhu, A. H. Ochoa & H. G. Katzgraber (2015):Efficient Cluster Algorithm for Spin Glasses in Any SpaceDimension. Physical Review Letters115(077201), doi:10.1103/PhysRevLett.115.077201.

An Overview of Approaches to Modernize Quantum Annealing Using Local Searches

Abstract

Similar works

Full text

Available Versions

Directory of Open Access Journals