Recent experimental works have studied how the quantum annealing approach, as implemented in D-Wave devices, deals with pathological instances of the (polynomial) bipartite matching problem known to be hard for classical simulated annealing algorithms and shown that the quantum approach also performed rather poorly on such instances. In this paper, we investigate the extent to which the qubits interconnection topologies explains these latter experimental results. In particular, we provide evidences that the sparsity of these topologies which, as such, lead to QUBO problems of artificially inflated sizes can partly explain the aforementioned disappointing observations. Therefore, this paper hints that denser interconnection topologies are necessary to unleash the potential of the quantum annealing approach.
Recent experimental works have studied how the quantum annealing approach, as implemented in D-Wave devices, deals with pathological instances of the (polynomial) bipartite matching problem known to be hard for classical simulated annealing algorithms and shown that the quantum approach also performed rather poorly on such instances. In this paper, we investigate the extent to which the qubits interconnection topologies explains these latter experimental results. In particular, we provide evidences that the sparsity of these topologies which, as such, lead to QUBO problems of artificially inflated sizes can partly explain the aforementioned disappointing observations. Therefore, this paper hints that denser interconnection topologies are necessary to unleash the potential of the quantum annealing approach.
Abstract This paper experimentally investigates the behavior of analog quantum computers as commercialized by D-Wave when confronted to instances of the maximum cardinality matching problem which is specifically designed to be hard to solve by means of simulated annealing. We benchmark a D-Wave “Washington” (2X) with 1098 operational qubits on various sizes of such instances and observe that for all but the most trivially small of these it fails to obtain an optimal solution. Thus, our results suggest that quantum annealing, at least as implemented in a D-Wave device, falls in the same pitfalls as simulated annealing and hence provides additional evidences suggesting that there exist polynomial-time problems that such a machine cannot solve efficiently to optimality. Additionally, we investigate the extent to which the qubits interconnection topologies explains these latter experimental results. In particular, we provide evidences that the sparsity of these topologies which, as such, lead to QUBO problems of artificially inflated sizes can partly explain the aforementioned disappointing observations. Therefore, this paper hints that denser interconnection topologies are necessary to unleash the potential of the quantum annealing approach.
Abstract Quantum annealing is a type of analog computation that aims to use quantum mechanical fluctuations in search of optimal solutions of QUBO (quadratic unconstrained binary optimization) or, equivalently, Ising problems. Since NP-hard problems can in general be mapped to Ising and QUBO formulations, the quantum annealing paradigm has the potential to help solve various NP-hard problems. Current quantum annealers, such as those manufactured by D-Wave Systems, Inc. have various practical limitations including the size (number of qubits) of the problem that can be solved, the qubit connectivity, and error due to the environment or system calibration, which can reduce the quality of the solutions. Typically, for an arbitrary problem instance, the corresponding QUBO (or Ising) structure will not natively embed onto the available qubit architecture on the quantum chip. Thus, in these cases, a minor embedding of the problem structure onto the device is necessary. However, minor embeddings on these devices do not always make use of the full sparse chip hardware graph, and a large portion of the available qubits stay unused during quantum annealing. In this work, we embed a disjoint random QUBO on the unused parts of the chip alongside the QUBO to be solved, which acts as an indicator of the solution quality of the device over time. Using experiments on three different D-Wave quantum annealers, we demonstrate that (i) long term trends in solution quality exist on the D-Wave device, and (ii) the unused qubits can be used to measure the current level of noise of the quantum system.
The observation of an unequivocal quantum speedup remains an elusive objective for quantum computing. The D-Wave quantum annealing processors have been at the forefront of experimental attempts to address this goal, given their relatively large numbers of qubits and programmability. A complete determination of the optimal time-to-solution (TTS) using these processors has not been possible to date, preventing definitive conclusions about the presence of a scaling advantage. The main technical obstacle has been the inability to verify an optimal annealing time within the available range. Here we overcome this obstacle and present a class of problem instances for which we observe an optimal annealing time using a D-Wave 2000Q processor over a range spanning up to more than 2000 qubits. This allows us to perform an optimal TTS benchmarking analysis and perform a comparison to several classical algorithms, including simulated annealing, spin-vector Monte Carlo, and discrete-time simulated quantum annealing. We establish the first example of a scaling advantage for an experimental quantum annealer over classical simulated annealing: we find that the D-Wave device exhibits certifiably better scaling than simulated annealing, with 95% confidence, over the range of problem sizes that we can test. However, we do not find evidence for a quantum speedup: simulated quantum annealing exhibits the best scaling by a significant margin. Our construction of instance classes with verifiably optimal annealing times opens up the possibility of generating many new such classes, paving the way for further definitive assessments of scaling advantages using current and future quantum annealing devices.
We benchmark Quantum Annealing (QA) vs. Simulated Annealing (SA) with a focus on the impact of the embedding of problems onto the different topologies of the D-Wave quantum annealers. The series of problems we study are especially designed instances of the maximum cardinality matching problem that are easy to solve classically but difficult for SA and, as found experimentally, not easy for QA either. In addition to using several D-Wave processors, we simulate the QA process by numerically solving the time-dependent Schrödinger equation. We find that the embedded problems can be significantly more difficult than the unembedded problems, and some parameters, such as the chain strength, can be very impactful for finding the optimal solution. Thus, finding a good embedding and optimal parameter values can improve the results considerably. Interestingly, we find that although SA succeeds for the unembedded problems, the SA results obtained for the embedded version scale quite poorly in comparison with what we can achieve on the D-Wave quantum annealers.
This paper assesses the performance of the D-Wave 2X (DW) quantum annealer for finding a maximum clique in a graph, one of the most fundamental and important NP-hard problems. Because the size of the largest graphs DW can directly solve is quite small (usually around 45 vertices), we also consider decomposition algorithms intended for larger graphs and analyze their performance. For smaller graphs that fit DW, we provide formulations of the maximum clique problem as a quadratic unconstrained binary optimization (QUBO) problem, which is one of the two input types (together with the Ising model) acceptable by the machine, and compare several quantum implementations to current classical algorithms such as simulated annealing, Gurobi, and third-party clique finding heuristics. We further estimate the contributions of the quantum phase of the quantum annealer and the classical post-processing phase typically used to enhance each solution returned by DW. We demonstrate that on random graphs that fit DW, no quantum speedup can be observed compared with the classical algorithms. On the other hand, for instances specifically designed to fit well the DW qubit interconnection network, we observe substantial speed-ups in computing time over classical approaches.
This paper assesses the performance of the D-Wave 2X (DW) quantum annealer for finding a maximum clique in a graph, one of the most fundamental and important NP-hard problems. Because the size of the largest graphs DW can directly solve is quite small (usually around 45 vertices), we also consider decomposition algorithms intended for larger graphs and analyze their performance. For smaller graphs that fit DW, we provide formulations of the maximum clique problem as a quadratic unconstrained binary optimization (QUBO) problem, which is one of the two input types (together with the Ising model) acceptable by the machine, and compare several quantum implementations to current classical algorithms such as simulated annealing, Gurobi, and third-party clique finding heuristics. We further estimate the contributions of the quantum phase of the quantum annealer and the classical post-processing phase typically used to enhance each solution returned by DW. We demonstrate that on random graphs that fit DW, no quantum speedup can be observed compared with the classical algorithms. On the other hand, for instances specifically designed to fit well the DW qubit interconnection network, we observe substantial speed-ups in computing time over classical approaches.