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arXiv:1701.01524 · Scientific Reports

Advantages of Unfair Quantum Ground-State Sampling

Brian Hu Zhang, Gene Wagenbreth, Víctor Martín-Mayor, Itay Hen

Published Apr 13, 2017• 26 citations
arXivHTMLPDF

Abstract

The debate around the potential superiority of quantum annealers over their classical counterparts has been ongoing since the inception of the field. Recent technological breakthroughs, which have led to the manufacture of experimental prototypes of quantum annealing optimizers with sizes approaching the practical regime, have reignited this discussion. However, the demonstration of quantum annealing speedups remains to this day an elusive albeit coveted goal. We examine the power of quantum annealers to provide a different type of quantum enhancement of practical relevance, namely, their ability to serve as useful samplers from the ground-state manifolds of combinatorial optimization problems. We study, both numerically by simulating stoquastic and non-stoquastic quantum annealing processes, and experimentally, using a prototypical quantum annealing processor, the ability of quantum annealers to sample the ground-states of spin glasses differently than thermal samplers. We demonstrate that (i) quantum annealers sample the ground-state manifolds of spin glasses very differently than thermal optimizers (ii) the nature of the quantum fluctuations driving the annealing process has a decisive effect on the final distribution, and (iii) the experimental quantum annealer samples ground-state manifolds significantly differently than thermal and ideal quantum annealers. We illustrate how quantum annealers may serve as powerful tools when complementing standard sampling algorithms.

DOI
10.1038/s41598-017-01096-6

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The debate around the potential superiority of quantum annealers over their classical counterparts has been ongoing since the inception of the field. Recent technological breakthroughs, which have led to the manufacture of experimental prototypes of quantum annealing optimizers with sizes approaching the practical regime, have reignited this discussion. However, the demonstration of quantum annealing speedups remains to this day an elusive albeit coveted goal. We examine the power of quantum annealers to provide a different type of quantum enhancement of practical relevance, namely, their ability to serve as useful samplers from the ground-state manifolds of combinatorial optimization problems. We study, both numerically by simulating stoquastic and non-stoquastic quantum annealing processes, and experimentally, using a prototypical quantum annealing processor, the ability of quantum annealers to sample the ground-states of spin glasses differently than thermal samplers. We demonstrate that (i) quantum annealers sample the ground-state manifolds of spin glasses very differently than thermal optimizers (ii) the nature of the quantum fluctuations driving the annealing process has a decisive effect on the final distribution, and (iii) the experimental quantum annealer samples ground-state manifolds significantly differently than thermal and ideal quantum annealers. We illustrate how quantum annealers may serve as powerful tools when complementing standard sampling algorithms.
Quantum annealers are computing machines that utilize quantum effects to solve hard optimization problems. Understanding the circumstances under which quantum annealers are more efficient than other optimization methods is a challenging open problem. A clue to this conundrum comes from studying the complexity of quantum annealers. In general, the complexity of a physical system relates to how efficiently the system can be simulated by classical algorithms. One then asks: if some class of quantum annealers are hard to simulate classically, are they also more powerful optimization machines? Here, the authors study quantum annealers belonging to two different complexity classes: the so-called ``stoquastic'' systems, which can be efficiently simulated with classical algorithms, and ``nonstoquastic'' systems, currently with no efficient classical treatment. Applied to a prototypical optimization problem, the authors observe that, on average, the stoquastic systems perform better on easier instances of the problem, while the more complex nonstoquastic annealers show significant advantage when applied to the hardest instances. The authors conjecture that the performance of the nonstoquastic Hamiltonians is closely related to their internal structure during the annealing process and elucidate how properties such as magnetic frustration can play a key role in devising more powerful quantum annealers.
Quantum annealing (QA) is a promising approach for solving combinatorial optimization problems; however, it is known to exhibit unfair sampling, in which degenerate ground states are not sampled with equal probability even for sufficiently long annealing times. Fair sampling is important in applications such as solution diversity assessment and combinatorial counting, yet the mechanism of unfair sampling remains poorly understood, particularly in constrained combinatorial optimization problems. In this work, we investigate unfair sampling of QA in weighted graph bipartitioning problems (GBP), a representative constrained optimization problem. We study how the penalty coefficient in the penalty method affects sampling fairness. Through numerical simulations and experiments on the D-Wave Advantage2 system, we show that increasing the penalty coefficient reduces unfair sampling in a representative single instance, and that this qualitative behavior is also observed on actual hardware. A scaling analysis over randomly generated instances with up to 12 spins reveals that, while this trend does not hold universally, more than 70% of instances exhibit monotonically increasing sampling fairness as the penalty coefficient increases, even at the largest system size studied. These results show that increasing the penalty coefficient improves sampling fairness, though at the cost of ground-state probability under practical annealing conditions, and call for a deeper theoretical understanding of unfair sampling in constrained optimization problems.
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.
Quantum annealing is a promising tool for solving optimization problems, similar in some ways to the traditional (classical) simulated annealing of Kirkpatrick et al. Simulated annealing takes advantage of thermal fluctuations in order to explore the optimization landscape of the problem at hand, whereas quantum annealing employs quantum fluctuations. Intriguingly, quantum annealing has been proved to be more effective than its classical counterpart in many applications. We illustrate the theory and the practical implementation of both classical and quantum annealing – highlighting the crucial differences between these two methods – by means of results recently obtained in experiments, in simple toy-models, and more challenging combinatorial optimization problems (namely, Random Ising model and Travelling Salesman Problem). The techniques used to implement quantum and classical annealing are either deterministic evolutions, for the simplest models, or Monte Carlo approaches, for harder optimization tasks. We discuss the pro and cons of these approaches and their possible connections to the landscape of the problem addressed.
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.
Reaching a degenerate ground-state manifold and sampling it uniformly are distinct properties of a quantum annealer. We quantify this distinction using an Ising ring with additional couplings between maximally separated sites. For antiferromagnetic benchmark couplings and N≥5N \geq 5N≥5, we derive the exact ground-state energy, degeneracy, and defect structure for each congruence class of NNN modulo four. This classification separates ground-state hit probability from conditional sampling bias. We embed instances with up to 162 logical spins on a D-Wave Advantage processor using length-two chains and compare its output with simulated annealing on the embedded and logical problems. For a frustrated N=83N=83N=83 instance, the processor visits 164 of the 166 exact ground states, yet the conditional distributions over states and defect sectors remain nonuniform. For N=160N=160N=160, the processor has a larger ground-state hit probability than logical simulated annealing (0.869 versus 0.519), but also a larger total-variation distance from uniformity (0.317 versus 0.136). A pause during the anneal raises the ground-state hit probability from 0.42 to 0.86 without restoring uniform sampling. These results identify a relaxation window and demonstrate that optimization success and sampling quality need not improve together, providing an exactly characterized benchmark for open-system quantum annealing.