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arXiv:1411.5693 · Science

Quantum versus classical annealing of Ising spin glasses

Bettina Heim, Troels F. Rønnow, Sergei V. Isakov, Matthias Troyer

Published Mar 12, 2015• 218 citations
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Abstract

Quantum annealers use quantum fluctuations to escape local minima and find low-energy configurations of a physical system. Strong evidence for superiority of quantum annealing (QA) has come from comparing QA implemented through quantum Monte Carlo (QMC) simulations to classical annealing. Motivated by recent experiments, we revisit the question of when quantum speedup may be expected. Although a better scaling is seen for QA in two-dimensional Ising spin glasses, this advantage is due to time discretization artifacts and measurements that are not possible on a physical quantum annealer. Simulations in the physically relevant continuous time limit, on the other hand, do not show superiority. Our results imply that care must be taken when using QMC simulations to assess the potential for quantum speedup.

DOI
10.1126/science.aaa4170

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Quantum annealers use quantum fluctuations to escape local minima and find low-energy configurations of a physical system. Strong evidence for superiority of quantum annealing (QA) has come from comparing QA implemented through quantum Monte Carlo (QMC) simulations to classical annealing. Motivated by recent experiments, we revisit the question of when quantum speedup may be expected. Although a better scaling is seen for QA in two-dimensional Ising spin glasses, this advantage is due to time discretization artifacts and measurements that are not possible on a physical quantum annealer. Simulations in the physically relevant continuous time limit, on the other hand, do not show superiority. Our results imply that care must be taken when using QMC simulations to assess the potential for quantum speedup.
Probing the lowest energy configuration of a complex system by quantum annealing was recently found to be more effective than its classical, thermal counterpart. By comparing classical and quantum Monte Carlo annealing protocols on the two-dimensional random Ising model (a prototype spin glass), we confirm the superiority of quantum annealing relative to classical annealing. We also propose a theory of quantum annealing based on a cascade of Landau-Zener tunneling events. For both classical and quantum annealing, the residual energy after annealing is inversely proportional to a power of the logarithm of the annealing time, but the quantum case has a larger power that makes it faster.
We discuss an Ising spin glass where each S=1/2 spin is coupled antiferromagnetically to three other spins (3-regular graphs). Inducing quantum fluctuations by a time-dependent transverse field, we use out-of-equilibrium quantum Monte Carlo simulations to study dynamic scaling at the quantum glass transition. Comparing the dynamic exponent and other critical exponents with those of the classical (temperature-driven) transition, we conclude that quantum annealing is less efficient than classical simulated annealing in bringing the system into the glass phase. Quantum computing based on the quantum annealing paradigm is therefore inferior to classical simulated annealing for this class of problems. We also comment on previous simulations where a parameter is changed with the simulation time, which is very different from the true Hamiltonian dynamics simulated here.
Recent theoretical and experimental studies have suggested that quantum Monte Carlo (QMC) simulation can behave similarly to quantum annealing (QA). The theoretical analysis was based on calculating transition rates between local minima, in the large spin limit using WentzelKramers-Brillouin (WKB) approximation, for highly symmetric systems of ferromagnetically coupled qubits. The rate of transition was observed to scale the same in QMC and incoherent quantum tunneling, implying that there might be no quantum advantage of QA compared to QMC other than a prefactor. Quantum annealing is believed to provide quantum advantage through large scale superposition and entanglement and not just incoherent tunneling. Even for incoherent tunneling, the scaling similarity with QMC observed above does not hold in general. Here, we compare incoherent tunneling and QMC escape using perturbation theory, which has much wider validity than WKB approximation. We show that the two do not scale the same way when there are multiple homotopy-inequivalent paths for tunneling. We demonstrate through examples that frustration can generate an exponential number of tunneling paths, which under certain conditions can lead to an exponential advantage for incoherent tunneling over classical QMC escape. We provide analytical and numerical evidence for such an advantage and show that it holds beyond perturbation theory.
There has been considerable progress in the design and construction of quantum annealing devices. However, a conclusive detection of quantum speedup over traditional silicon-based machines remains elusive, despite multiple careful studies. In this work we outline strategies to design hard tunable benchmark instances based on insights from the study of spin glasses - the archetypal random benchmark problem for novel algorithms and optimization devices. We propose to complement head-to-head scaling studies that compare quantum annealing machines to state-of-the-art classical codes with an approach that compares the performance of different algorithms and/or computing architectures on different classes of computationally hard tunable spin-glass instances. The advantage of such an approach lies in having to only compare the performance hit felt by a given algorithm and/or architecture when the instance complexity is increased. Furthermore, we propose a methodology that might not directly translate into the detection of quantum speedup, but might elucidate whether quantum annealing has a "`quantum advantage" over corresponding classical algorithms like simulated annealing. Our results on a 496 qubit D-Wave Two quantum annealing device are compared to recently-used state-of-the-art thermal simulated annealing codes.
Recently, Heim, R\o nnow, Isakov and Troyer [{\it Science} \textbf{348} (2015) 215] have reported that Monte Carlo simulations for the Ising spin glass model on the square lattice in the physically relevant continuous-imaginary-time limit do not show superiority of quantum annealing (QA) using transverse field against classical annealing (CA). Although the QA schedule that they had used has been using conventionally, however the QA schedule mathematically has no guarantee that the used schedule is the best QA schedule for performance of optimization. We propose a new QA schedule for studying transverse-field-based quantum versus classical annealing of the Ising model. The present QA schedule is made based on the concept of perturbation theory. This QA schedule utilizes a smallest effective transverse field derived in this article. As a case study, we study QA of the Ising spin glass model on the square lattice at low but finite temperature. A new Monte Carlo algorithm using the physically relevant continuous-imaginary-time limit is performed. As the simulation results, we show superiority of QA against CA when the annealing time is spent enough.
Quantum annealing is a heuristic optimization algorithm that exploits quantum evolution to find low-energy states. Quantum annealers have scaled up in recent years to tackle increasingly larger and more highly connected discrete optimization and quantum simulation problems. Nevertheless, a computational quantum advantage in exact optimization using quantum annealing hardware has so far remained elusive. Here, we present evidence for a quantum annealing scaling advantage in approximate optimization. The advantage is relative to the top classical heuristic algorithm: parallel tempering with isoenergetic cluster moves (PT-ICM). The setting is a family of 2D spin-glass problems with high-precision spin-spin interactions. To achieve this advantage, we implement quantum annealing correction (QAC): an embedding of a bit-flip error-correcting code with energy penalties that leverages the properties of the D-Wave Advantage quantum annealer to yield over 1,300 error-suppressed logical qubits on a degree-5 interaction graph. We generate random spin-glass instances on this graph and benchmark their time-to-epsilon, a generalization of the time-to-solution metric for low-energy states. We demonstrate that, with QAC, quantum annealing exhibits a scaling advantage over PT-ICM at sampling low-energy states with an optimality gap of at least 1.0%. This amounts to the first demonstration of an algorithmic quantum speedup in approximate optimization.