Synchronously-pumped OPO coherent Ising machine: benchmarking and prospects

The coherent Ising machine (CIM) is a network of optical parametric oscillators (OPOs) that solves for the ground state of Ising problems through OPO bifurcation dynamics. Here, we present experimental results comparing the performance of the CIM to quantum annealers (QAs) on two classes of NP-hard optimization problems: ground state calculation of the Sherrington-Kirkpatrick (SK) model and MAX-CUT. While the two machines perform comparably on sparsely-connected problems such as cubic MAX-CUT, on problems with dense connectivity, the QA shows an exponential performance penalty relative to CIMs. We attribute this to the embedding overhead required to map dense problems onto the sparse hardware architecture of the QA, a problem that can be overcome in photonic architectures such as the CIM

4-clique network minor embedding for quantum annealers

Quantum annealing is a proposed algorithm for computing solutions to combinatorial optimization problems. Current quantum annealing hardware is relatively sparse and therefore requires graph minor embedding in order to map an arbitrarily structured problem onto the sparse, and relatively small, quantum annealing processor. This paper proposes a new minor embedding method called 4-clique minor embedding. This is in contrast to the standard minor embedding technique of using a path of linearly connected qubits in order to represent a logical variable state. The 4-clique minor embedding is possible because of Pegasus graph connectivity, which is the native hardware graph for some of the current D-Wave quantum annealers. The Pegasus hardware graph has many 4-cliques, and it is possible to form a graph composed entirely of paths of connected 4-cliques, on which a problem can be minor embedded. The 4-clique chains come at the cost of additional qubit usage on the hardware graph, but they allow for stronger coupling within each chain thereby increasing chain integrity and reducing chain breaks. This 4-clique minor embedding technique is described in detail, and is compared against the standard linear path minor embedding with some experiments on two D-Wave quantum annealing processors with Pegasus hardware graphs. 4-clique minor embeddings can use weak chain strengths while successfully carrying out the computation of minimizing random all-to-all spin glass problem instances, in contrast to the linear path minor embeddings which have high chain break frequencies for weak chain strengths. This work shows that non standard minor embedding methods could be useful. For future quantum annealing architectures, distributing minor embeddings over more densely connected regions of hardware instead of linear paths may provide more robust computations for minor embedding problems

Advanced unembedding techniques for quantum annealers

The D-Wave quantum annealers make it possible to obtain high quality solutions of NP-hard problems by mapping a problem in a QUBO (quadratic unconstrained binary optimization) or Ising form to the physical qubit connectivity structure on the D-Wave chip. However, the latter is restricted in that only a fraction of all pairwise couplers between physical qubits exists. Modeling the connectivity structure of a given problem instance thus necessitates the computation of a minor embedding of the variables in the problem specification onto the logical qubits, which consist of several physical qubits "chained" together to act as a logical one. After annealing, it is however not guaranteed that all chained qubits get the same value (-1 or +1 for an Ising model, and 0 or 1 for a QUBO), and several approaches exist to assign a final value to each logical qubit (a process called "unembedding"). In this work, we present tailored unembedding techniques for four important NP-hard problems: the Maximum Clique, Maximum Cut, Minimum Vertex Cover, and Graph Partitioning problems. Our techniques are simple and yet make use of structural properties of the problem being solved. Using Erd\H{o}s-R\'enyi random graphs as inputs, we compare our unembedding techniques to three popular ones (majority vote, random weighting, and minimize energy). We demonstrate that our proposed algorithms outperform the currently available ones in that they yield solutions of better quality, while being computationally equally efficient