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 compare only 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, such as 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.

Spin systems with frustration and disorder are notoriously difficult to study both analytically and numerically. While the simulation of ferromagnetic statistical mechanical models benefits greatly from cluster algorithms, these accelerated dynamics methods remain elusive for generic spin-glass-like systems. Here we present a cluster algorithm for Ising spin glasses that works in any space dimension and speeds up thermalization by at least one order of magnitude at temperatures where thermalization is typically difficult. Our isoenergetic cluster moves are based on the Houdayer cluster algorithm for two-dimensional spin glasses and lead to a speedup over conventional state-of-the-art methods that increases with the system size. We illustrate the benefits of the isoenergetic cluster moves in two and three space dimensions, as well as the nonplanar chimera topology found in the D-Wave Inc. quantum annealing machine. PACS numbers: 75.50.Lk, 75.40.Mg, 05.50.+q, A plethora of problems across disciplines map onto spinglass-like Hamiltonians [1]. Despite decades of intense analytical and numerical scrutiny, a deep understanding of these paradigmatic models of disordered systems remains elusive. Given the inherent difficulties of studying these Hamiltonians analytically beyond mean-field theory as well as the continuous increase of computer power, progress in this field has benefited noticeably from numerical studies. The development of efficient Monte Carlo methods such as parallel tempering [2] and population annealing [3] has helped in understanding these systems at a much deeper level; however, most numerical studies are still plagued by corrections to finite-size scaling due to the small system sizes currently available [4].In contrast, simulations of spin Hamiltonians without disorder and frustration are comparably simple: Ferromagnetic systems have greatly benefited from the development of cluster algorithms [5,6] that help in overcoming critical slowing down close to phase transitions. Therefore, the holy grail of spin-glass simulations is to introduce accelerated cluster dynamics that improve upon the benefits of efficient simulation methods such as population annealing or parallel tempering Monte Carlo. In 2001 Houdayer introduced a seminal rejection-free cluster algorithm tailored to work for twodimensional Ising spin glasses [7]. The method updates large patches of spins at once, therefore effectively randomizing the configurations and efficiently overcoming large barriers in the free-energy landscape. Furthermore, the energy of the system remains unchanged when performing a cluster move. This means that the numerical overhead is very small because the rejection rate is zero and there is no need to, for example, compute any random numbers for a cluster update. The use of these cluster moves made it possible to obtain a speedup of several orders of magnitude in two-dimensional systems, therefore allowing us to simulate considerably larger system sizes.While cluster algorithms such as the Swendsen-Wang and Wol...

Recent tests performed on the D-Wave Two quantum annealer have revealed no clear evidence of speedup over conventional silicon-based technologies. Here, we present results from classical parallel-tempering Monte Carlo simulations combined with isoenergetic cluster moves of the archetypal benchmark problem-an Ising spin glass-on the native chip topology. Using realistic uncorrelated noise models for the D-Wave Two quantum annealer, we study the best-case resilience, i.e., the probability that the ground-state configuration is not affected by random fields and random-bond fluctuations found on the chip. We thus compute classical upper-bound success probabilities for different types of disorder used in the benchmarks and predict that an increase in the number of qubits will require either error correction schemes or a drastic reduction of the intrinsic noise found in these devices. We restrict this study to the exact ground state, however, the approach can be trivially extended to the inclusion of excited states if the success metric is relaxed. We outline strategies to develop robust, as well as hard benchmarks for quantum annealing devices, as well as any other (black box) computing paradigm affected by noise.

Recently, it was demonstrated both theoretically and experimentally on the D-Wave quantum annealer that transverse-field quantum annealing does not find all ground states with equal probability. In particular, it was proposed that more complex driver Hamiltonians beyond transverse fields might mitigate this shortcoming.Here, we investigate the mechanisms of (un)fair sampling in quantum annealing. While higher-order terms can improve the sampling for selected small problems, we present multiple counterexamples where driver Hamiltonians that go beyond transverse fields do not remove the sampling bias. Using perturbation theory we explain why this is the case. In addition, we present large-scale quantum Monte Carlo simulations for spin glasses with known degeneracy in two space dimensions and demonstrate that the fair-sampling performance of quadratic driver terms is comparable to standard transverse-field drivers. Our results suggest that quantum annealing machines are not well suited for sampling applications, unless post-processing techniques to improve the sampling are applied.

We study the problem to infer the ground state of a spin-glass Hamiltonian using data from another Hamiltonian with interactions disturbed by noise from the original Hamiltonian, motivated by the ground-state inference in quantum annealing on a noisy device. It is shown that the average Hamming distance between the inferred spin configuration and the true ground state is minimized when the temperature of the noisy system is kept at a finite value, and not at zero temperature. We present a spin-glass generalization of a well-established result that the ground state of a purely ferromagnetic Hamiltonian is best inferred at a finite temperature in the sense of smallest Hamming distance when the original ferromagnetic interactions are disturbed by noise. We use the numerical transfer-matrix method to establish the existence of an optimal finite temperature in one-and twodimensional systems. Our numerical results are supported by mean-field calculations, which give an explicit expression of the optimal temperature to infer the spin-glass ground state as a function of variances of the distributions of the original interactions and the noise. The mean-field prediction is in qualitative agreement with numerical data. Implications on postprocessing of quantum annealing on a noisy device are discussed.