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ACM International Conference on Computing Frontiers (CF)

On the limitations of the chimera graph topology in using analog quantum computers

Daniel Vert, Renaud Sirdey, Stéphane Louise

Published Apr 30, 2019• 20 citations
Publisher

Abstract

This paper investigates the possibility of using an analog quantum computer as commercialized by D-Wave to solve large QUBO problems by means of a single invocation of the quantum annealer. Indeed this machine solves a spin glass problem with programmable coefficients but subject to quite strong topology restrictions on the set of non-zero coefficients. Rather than mapping problem variables onto multiple qbits, an approach which requires many invocations of the annealer to solve small size problems, it is tempting to investigate the existence of sparse relaxations compliant with the qbits interconnection topology of the machine, hence solvable in one invocation of the annealing oracle, but still providing good-quality solutions to the original problem. This paper provides an experimental setup which aims to determine whether or not such convenient relaxations do exist or, rather, are easy to find. Our experiments suggest that it is not the case and, therefore, that solving even moderate size arbitrary problems with a single call to a quantum annealer is not possible at least within the constraints of the so-called Chimera topology. We conclude the paper with a number of perspectives that this results imply on the design of heuristics taking profit of a quantum annealing oracle to solve large scale problems.

DOI
10.1145/3310273.3322830

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This paper investigates the possibility of using an analog quantum computer as commercialized by D-Wave to solve large QUBO problems by means of a single invocation of the quantum annealer. Indeed this machine solves a spin glass problem with programmable coefficients but subject to quite strong topology restrictions on the set of non-zero coefficients. Rather than mapping problem variables onto multiple qbits, an approach which requires many invocations of the annealer to solve small size problems, it is tempting to investigate the existence of sparse relaxations compliant with the qbits interconnection topology of the machine, hence solvable in one invocation of the annealing oracle, but still providing good-quality solutions to the original problem. This paper provides an experimental setup which aims to determine whether or not such convenient relaxations do exist or, rather, are easy to find. Our experiments suggest that it is not the case and, therefore, that solving even moderate size arbitrary problems with a single call to a quantum annealer is not possible at least within the constraints of the so-called Chimera topology. We conclude the paper with a number of perspectives that this results imply on the design of heuristics taking profit of a quantum annealing oracle to solve large scale problems.
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.
Quadratic Unconstraint Binary Optimization (QUBO) is a combinatorial optimization problem such that ann×nn\times nn×nupper triangle matrixWWWis given and the objective is to find an n-bit vectorXXXthat minimizes the energy valueE(X)=XTWXE(X)=X^{T}WXE(X)=XTWX. A QUBO instanceWWWis sparse if instanceWWWhas few non-zero elements. The D-Wave 2000QQQis a quantum annealer that can solve 2048-bit sparse QUBO instances represented as a Chimera graph topology. We present a sparse QUBO solver running on GPUs for 2048-bit sparse QUBO with a Chimera graph topology. We have evaluated the performance of our sparse QUBO solver and the D-Wave 2000Q for solving 2048-bit QUBO instances with various resolutions. The experimental results show that our sparse QUBO solver running on a GPU cloud server with 8 NVIDIA A100 GPUs can find optimal solutions in less than 3ms for all instances while the D-Wave 2000QQQcannot find them in 996.7ms. Hence, our QUBO solver can find better solutions than the D-Wave 2000Q in less than 1/300 running time. We can think that our QUBO solver is a quantum annealer simulator with better performance in terms of the accuracy of solutions and the running time. Our result implies that quantum annealer D-Wave 2000QQQdoes not achieve quantum supremacy yet.
Quantum annealing is a generic solver for optimization problems that uses fictitious quantum fluctuation. The most groundbreaking progress in the research field of quantum annealing is its hardware implementation, i.e., the so-called quantum annealer, using artificial spins. However, the connectivity between the artificial spins is sparse and limited on a special network known as the chimera graph. Several embedding techniques have been proposed, but the number of logical spins, which represents the optimization problems to be solved, is drastically reduced. In particular, an optimization problem including fully or even partly connected spins suffers from low embeddable size on the chimera graph. In the present study, we propose an alternative approach to solve a large-scale optimization problem on the chimera graph via a well-known method in statistical mechanics called the Hubbard-Stratonovich transformation or its variants. The proposed method can be used to deal with a fully connected Ising model without embedding on the chimera graph and leads to nontrivial results of the optimization problem. We tested the proposed method with a number of partition problems involving solving linear equations and the traffic flow optimization problem in Sendai and Kyoto cities in Japan.
Quantum annealing is getting increasing attention in combinatorial optimization. The quantum processing unit by D-Wave is constructed to approximately solve Ising models on so-called Chimera graphs. Ising models are equivalent to quadratic unconstrained binary optimization (QUBO) problems and maximum cut problems on the associated graphs. We have tailored branch-and-cut as well as semidefinite programming algorithms for solving Ising models for Chimera graphs to provable optimality and use the strength of these approaches for comparing our solution values to those obtained on the current quantum annealing machine, D-Wave 2000Q. This allows for the assessment of the quality of solutions produced by the D-Wave hardware. In addition, we also evaluate the performance of a heuristic by Selby. It has been a matter of discussion in the literature how well the D-Wave hardware performs at its native task, and our experiments shed some more light on this issue. In particular, we examine how reliably the D-Wave computer can deliver true optimum solutions and present some surprising results.
The exploitation of quantum mechanical principles seems to provide a decisive advantage over classical computation technologies. In the recent years there has been big progress in realizing different quantum computer architectures, especially by the Canadian company D-Wave Systems, which developed the first commercially available adiabatic quantum annealer. By adiabatic evolution of quantum mechanical systems it can solve NP-hard discrete optimization problems probabilistically. Due to the restrictions of the hardware architecture the directly mappable problems are limited to quadratic unconstrained binary optimization (QUBO) problems over a so called Chimera interaction graph. After briefly introducing the physical and quantum computational background including the diffculties in determining the run time and success probability of the D-Wave machine, the presentation will focus on the mapping of more general problems onto their specific problem structure. In addition to the reductions to QUBOs the main part here is the minor embedding with the distribution of the weight of the original node onto several hardware nodes.
Abstract Quantum annealing promises to solve complex combinatorial optimization problems faster than current transistor-based computer technologies. Although to date only one commercially-available quantum annealer is procurable, one can already start to map out the application scope of these novel optimization machines. These mid-scale programmable analog special-purpose devices could, potentially, revolutionize optimization. However, their disruptive application domain remains to be found. While the commercial analog quantum optimization machine by D-Wave Systems Inc. already exceeds 1000 qubits, here it is argued that maybe smaller devices with better quality qubits, higher connectivity, and more tunability might be better suited to answer if quantum annealing will ever truly outperform specialized silicon technology combined with efficient heuristics for optimization and sampling applications.