The pitfalls of planar spin-glass benchmarks: Raising the bar for quantum annealers (again)

TL;DR

This paper reveals that planar spin-glass benchmarks, thought to be challenging for quantum annealers, can be efficiently solved by classical algorithms, highlighting the need for more robust benchmarking methods.

Contribution

The paper demonstrates that classical algorithms can efficiently solve planar gadget-based spin-glass problems, exposing limitations of current benchmarks for quantum annealers.

Findings

Classical algorithms like minimum-weight perfect matching solve these problems exactly.

Planar gadget problems are solvable in polynomial time by classical methods.

Current benchmarks may overestimate quantum annealers' performance.

Abstract

In an effort to overcome the limitations of random spin-glass benchmarks for quantum annealers, focus has shifted to carefully-crafted gadget-based problems whose logical structure has typically a planar topology. Recent experiments on these gadget problems using a commercially-available quantum annealer have demonstrated an impressive performance over a selection of commonly-used classical optimization heuristics. Here we show that efficient classical optimization techniques, such as minimum-weight perfect matching, can solve these gadget problems exactly and in polynomial time. We present approaches on how to mitigate this shortcoming of commonly-used benchmark problems based on planar logical topologies.

Equations

$${\mathcal{H}}_{\mathrm{P}}=\sum _{i=1}^n{J}_{ij}{\sigma }_i{\sigma }_j+\sum _{i=1}^n{h}_i{\sigma }_i,$$

$${\text{TTS}}_1=\frac{T}{p}.$$

$${\text{TTS}}_2=T\frac{\log (0.01)}{\log (1-p)}.$$

Full text

Section I Introduction

The quest for quantum speedup using analog quantum annealing machines with a transverse field remains elusive. There have been multiple attempts Rønnow et al. (2014); Boixo et al. (2014); Katzgraber et al. (2015); Venturelli et al. (2015); Denchev et al. (2016) to demonstrate that the D-Wave Systems Inc. quantum annealers can outperform classical optimization methods. Unfortunately, it has been relatively straightforward for classical optimization algorithms to stay ahead in this race Mandrà et al. (2016). Either the random spin-glass benchmark problems were too easy on the quasi-planar topology of the D-Wave quantum annealer Rønnow et al. (2014); Katzgraber et al. (2014, 2015), or the logical structure of carefully-crafted problems designed to give the annealer an advantage have a trivial structure Denchev et al. (2016).

The notable advances made by D-Wave Systems Inc. in the development of medium-scale quantum annealing technologies has inspired multiple corporations (e.g., Microsoft, Google, and IBM) to further invest in quantum computing, in addition to large-scale government-funded projects. The D-Wave 2000Q  device is a special-purpose quantum optimization machine specialized in minimizing quadratic unconstrained binary cost functions by means of quenching quantum fluctuations induced by a transverse field. The generic cost function to be minimized is given by

[TABLE]

where the $n$ variables $\sigma_{i}\in\{\pm 1\}$ are Boolean and the couplings $J_{ij}\in{\mathbb{R}}$ and biases $h_{i}\in{\mathbb{R}}$ are the parameters that define the problem to be minimized. In the case of the D-Wave chip, these variables are arranged in the so-called Chimera topology Bunyk et al. (2014). Real-world applications then require the embedding of the problems onto this topology, thus typically resulting in an embedding overhead that results in logical problems with less sites than the native topology of the chip. As such, it is desirable to find classes of problems for benchmarking that ideally use the complete set of variables on the chip, while not being a trivial optimization problem Rønnow et al. (2014).

There have been multiple approaches to harden the benchmark problems to be solved on different generations of the D-Wave device, ranging from post-selection methods based on statistical-physics metrics Katzgraber et al. (2015) to planting of solutions Hen et al. (2015) and the engineering of problems based on the classical algorithmic complexity of the Hamze–de Freitas–Selby Hamze and de Freitas (2004); Selby (2014) algorithm Marshall et al. (2016). Although these approaches generated harder problems for different generations of the D-Wave devices and there were some suggestions that there is some level of “quantumness” in the device Katzgraber et al. (2015), only studies tailored to explicitly demonstrate quantumness, as well as attempt to determine (quantum) speedup Rønnow et al. (2014); Mandrà et al. (2016) pushed the field forward noticeably. Both the studies of Denchev et al. Denchev et al. (2016) and King et al.King et al. (2017) focused on the generation of logical problems designed to elucidate the value of quantum fluctuations, as well as to “fool” archetypal classical optimization algorithms (e.g., simulated annealing Kirkpatrick et al. (1983), the classical pendant to quantum annealing Kadowaki and Nishimori (1998); Farhi et al. (2000)) to become stuck in the carefully-designed energy landscape of the problems. In Ref. Mandrà et al. (2016) it was subsequently shown that the use of state-of-the-art optimization techniques beyond simulated annealing for the benchmarks designed in Ref. Denchev et al. (2016) resulted in time-to-solutions scaling considerably better than the D-Wave device, as well as simulated quantum annealing. In this work we demonstrate that if the logical problems to be optimized on the D-Wave device have a planar structure, a quantum annealer would have to scale polynomially in the number of (logical) variables (i.e., be exponentially faster) to compete with the current classical state-of-the-art for frustrated problems on planar topologies, such as the minimum-weight perfect-matching (MWPM) exact algorithm Hartmann and Rieger (2001, 2004). We emphasize that both the benchmark instances designed by Denchev et al. Denchev et al. (2016) and King et al.King et al. (2017) suffer from the same problem. Namely, they can both be solved in polynomial time. Although one could, in principle, compare the quantum annealer against fast heuristics such as the Hamze–de Freitas–Selby Hamze and de Freitas (2004); Selby (2014) algorithm Marshall et al. (2016) or parallel tempering Monte Carlo with isoenergetic cluster moves Zhu et al. (2015), if claims of speedups over many orders of magnitude against classical algorithms are made, then the true state-of-the-art for planar topologies should be included in the study.

Although one might argue that exploiting the logical structure of the problem could be seen as “cheating,” combining MWPM techniques with simple cluster-finding and/or decimation techniques that are also polynomial in the size of the input would still scale exponentially faster than the D-Wave device. However, there would be no more guarantee for an exact result and the cluster-detecting MWPM algorithm could, at best, be seen as a heuristic that scales polynomial in the size of the input.

The paper is structured as follows. In Section II we describe the crafted benchmark problem designed by D-Wave Systems Inc. King et al. (2017) and in Section III we describe the classical algorithms and methods we used in our analysis of these problems. Results are summarized in Section IV, followed by a discussion that also includes different strategies to design problems on quantum annealing machines that might have potential for quantum speedup and cannot be solved with polynomial algorithms for planar technologies.

Section II D-Wave’s crafted problems

Given its hardware limitations, not all possible couplings $\{J_{ij}\}$ between two arbitrary qubits $i$ and $j$ can be set in the D-Wave quantum annealer. Indeed, only those couplings belonging to the native Chimera topology can be independently tuned within the range $[-1,\,+1]$, while the remaining are set to zero. The Chimera topology Bunyk et al. (2014) is composed of $k\times k$ unit cells, each containing a $\text{K}_{4,4}$ fully-connected bipartite graph of $8$ qubits. The unit cells are coupled together so that only adjacent unit cells share couplings. Despite the somewhat restrictive structure of the lattice, it can be shown that, in principle, any topology can be embedded, albeit at the cost of using multiple physical variables to define a logical variable.

In Ref. King et al. (2017), the latest incarnation of the quantum annealer, namely the D-Wave 2000Q  with over $2000$ quantum bits, is tested using a set of carefully crafted optimization problems also referred to as the “frustrated cluster loop” (FCL) problems. One of the main characteristics which makes the FCL problems appealing for benchmarking is that many classical heuristics struggle with minimizing the value of the cost function, even though the optimal configuration can be deduced by exploiting the actual structure of the problem Hen et al. (2015); King et al. (2017).

Although the FCL problems can, in principle, be directly generated for the Chimera topology Hen et al. (2015), the D-Wave Systems Inc. group has chosen a slightly different strategy, divided into two steps, which assists in elucidating the effects of the landscape ruggedness:

All couplings inside a $\text{K}_{4,4}$ unit cell are set to be ferromagnetic, i.e., $J_{ij}=-1$, $\forall i,j\in$ $\text{K}_{4,4}$ . because the unit cells are fully connected, all the “physical” qubits within a single cell are forced to behave as a single “logical” qubit. This process generates a two-dimensional lattice with open boundary conditions of these logical variables. 2. 2.

The FCL instances are then generated on the logical topology with a varying level of ruggedness of the energy landscape and parameters $\alpha$ (clause-to-variable ratio) and $\rho$ (precision), as defined in Ref. King et al. (2015). Note that for the ruggedness $R$ we expect $R\geq\rho$.

This local ruggedness $R$ then makes the problems hard to treat for typical classical optimization techniques with the interactions for the logical qubits being in the range $[-R,\,+R]$. Disconnected graphs are discarded in the study.

It is important to stress that, despite the fact that D-Wave 2000Q  can optimize problems on the Chimera topology, the benchmark problems used in Ref. King et al. (2017) are defined on the logical topology of the machine, namely on a two-dimensional lattice with open boundaries. Therefore, a fair comparison requires that the D-Wave 2000Q  benchmarking be performed against heuristics which are optimized for the logical problem, rather than on the Chimera topology. It is noteworthy, however, that the D-Wave 2000Q  solves problems on the full Chimera lattice, i.e., the machine seems to be able to efficiently overcome local energy barriers.

Section III Methods

In this Section, we briefly outline the algorithms used, as well as the definition of time-to-solution used in the benchmarks. Reference Zhu et al. (2015) provides the necessary details for the isoenergetic cluster move (ICM) heuristic.

III.1 Minimum-weight perfect matching algorithm

The minimum-weight perfect matching (MWPM) algorithm is an exact classical algorithm designed to find optimal configurations for planar two-dimensional spin-glass-like optimization problems without biases (i.e., $h_{i}=0$, $\forall i\in n$). The algorithm is polynomial in the size of the input $n$. The MWPM algorithm consists of three steps:

The planar spin-glass problem is mapped onto a minimum-weight perfect matching problem. 2. 2.

The minimum weight-perfect matching problem is solved exactly using the deterministic Blossom algorithm Kolmogorov (2009) that scales polynomially in the size of the input. 3. 3.

The perfect matching solution is then translated to the optimal configuration for the spin-glass problem.

III.2 Definition of time-to-solution

Heuristic methods, such as simulated annealing, simulated quantum annealing, the D-Wave 2000Q  quantum annealer or isoenergetic cluster moves, can only determine the optimum of the cost function up to a probability $p$. If the optimization procedure requires a certain time $T$, given that the optimum is only obtained with a probability $p$, it is necessary to define a time-to-solution (TTS). A simple (yet naive) possibility consists of making the observation that, on average, one needs to repeat the computation $\approx 1/p$ times in order to observe one optimal result. Therefore, for a computational time $T$, a possible definition of the TTS is

[TABLE]

A commonly-used, more accurate definition of the TTS that incorporates the cost of having uncertainty in the heuristic results is given as follows: Let $k$ be the number of (unknown) iterations required to have a probability of success of at least $99\%$, i.e., $s=0.99$. The probability that all $k$ attempts fail to find the correct answer is $P_{\text{wrong}}=(1-p)^{k}$. Because an overall probability of success $s$ is needed, it is required that $P_{\text{wrong}}<s$. Therefore, $k$ must be at least $k>\log(1-s)/\log(1-p)$. Assuming that each attempt require $T$ times, the TTS can be defined as

[TABLE]

Note that $\text{TTS}_{2}<\text{TTS}_{1}$ at fixed $p$ and $T$. However, in general, $\text{TTS}_{2}$ is preferred, because it gives a lower bound to the overall probability. Reference King et al. (2017) used the definition shown in Eq. (2). Using the raw data from Ref. King et al. (2017), we have converted their results into the more commonly used TTS shown in Eq. (3).

Section IV Results

Figure 1 summarizes our results where we compare the performance of the D-Wave 2000Q  quantum annealer to both ICM and MWPM. The simulations were performed on a single core of an Intel(R) Xeon(R) CPU (E5-1650v2 with 3.50GHz clock speed). While both the D-Wave 2000Q  quantum annealer and ICM scale exponentially, MWPM scales polynomially with the size of the input. To show this in more detail, we have generated artificial problems on perfect Chimera lattices of up to $256\times 256$ logical variables. Although for small systems the D-Wave 2000Q  chip is remarkably fast, only by doubling the largest number of logical variables on the chip results in MWPM outperforming the quantum annealer by approximately three orders of magnitude. One has to remember, however, that the D-Wave 2000Q  quantum annealer is a special purpose machine designed to minimize binary problems, whereas classical CMOS technologies require other processes to run, such as the operating system, kernel and other concurrent processes.

Section V Discussion

Although the D-Wave chip shows remarkable promise, in this work we show that benchmarks which encode the logical problem on a planar topology is bound to fail in reaching the crown of quantum speedup. The quantum annealer would have to perform exponentially faster, in order to outperform the exact polynomial algorithm.

So how can we eventually prove the value of quantum annealing topologies? First and foremost, encode the problems in nonplanar logical topologies to ensure that no exact polynomial algorithms can be used. One possible approach we call “anticlusters” (see Fig. 2) is to contract the links between the $\text{K}_{4,4}$ cells to become logical variables. This would results in a nontrivial nonplanar topology where each logical variable has $10$ neighbors, except for the logical variables on the edges of the lattice which only have $5$ neighbors. For a Chimera lattice with $c\times c$ $\text{K}_{4,4}$ cells (i.e., $8c^{2}$ physical qubits), the corresponding anticluster lattice would have $4c(c+1)$ logical qubits arranged on a square-lattice-like structure of linear dimensions $c\times c$. The large connectivity of the anticluster lattice, as well as the large number of logical variables allows for the generation of nontrivial benchmarking problems. For example, overlaying this topology that resembles the offspring of a farm fence with a square lattice with frustrated cluster loops or post-selected spin-glass problems, should generate hard benchmarks for classical algorithms.

A second alternative to demonstrate the capabilities of the D-Wave 2000Q  is to use the machine as a physical simulator to study nontrivial quantum physics Hamiltonians Harris et al. (2017). Because these are very hard to simulate already for small numbers of variables, the D-Wave 2000Q  might be able to outperform classical simulation techniques in the near future.

Acknowledgements.

We would like to thank the D-Wave team and, in particular, J. King for sharing their problem instances and raw data, as well as comments on the manuscript. We are also indebted to Richard Harris at D-Wave for sharing potentially groundbreaking, yet hopefully soon-to-be-published results. We thank H. Munoz-Bauza for rendering the anticluster problems. H. G. K. would like to thank Zheng Zhu for sharing his implementation of the ICM heuristic, as well as F. Hamze and G. Rosenberg for multiple discussions. He also thanks Franziskaner Naturtrüb for inspiration. H. G. K. acknowledges support from the NSF (Grant No. DMR-1151387). S. M. acknowledges Eleanor G. Rieffel for the useful discussions and the careful reading of the manuscript. This research is based upon work supported in part by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via MIT Lincoln Laboratory Air Force Contract No. FA8721-05-C-0002. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purpose notwithstanding any copyright annotation thereon.