Quantum annealing (QA) can be used to quickly obtain near-optimal solutions for quadratic unconstrained binary optimization (QUBO) problems. In QA hardware, each decision variable of a QUBO should be mapped to one or more adjacent qubits in such a way that pairs of variables defining a quadratic term in the objective function are mapped to some pair of adjacent qubits. However, qubits have limited connectivity in existing QA hardware. This has spurred work on preprocessing algorithms for embedding the graph representing problem variables with quadratic terms into the hardware graph representing qubits adjacencies, such as the Chimera graph in hardware produced by D-Wave Systems. In this paper, we use integer linear programming to search for an embedding of the problem graph into certain classes of minors of the Chimera graph, which we call template embeddings. One of these classes corresponds to complete bipartite graphs, for which we show the limitation of the existing approach based on minimum odd cycle transversals (OCTs). One of the formulations presented is exact and thus can be used to certify the absence of a minor embedding using that template. On an extensive test set consisting of random graphs from five different classes of varying size and sparsity, we can embed more graphs than a state-of-the-art OCT-based approach, our approach scales better with the hardware size, and the runtime is generally orders of magnitude smaller. Summary of Contribution: Our work combines classical and quantum computing for operations research by showing that integer linear programming can be successfully used as a preprocessing step for adiabatic quantum optimization. We use it to determine how a quadratic unconstrained binary optimization problem can be solved by a quantum annealer in which the qubits are coupled as in a Chimera graph, such as in the quantum annealers currently produced by D-Wave Systems. The paper also provides a timely introduction to adiabatic quantum computing and related work on minor embeddings.
Quantum Annealing (QA) offers a promising framework for solving NP-hard optimization problems, but its effectiveness is constrained by the topology of the underlying quantum hardware. Solving an optimization problem
P via QA involves a hardware-aware circuit compilation which requires representing
P as a graph
GP and embedding it into the hardware connectivity graph
GQ that defines how qubits connect to each other in a QA-based quantum processing unit (QPU).
Minor Embedding (ME) is a possible operational form of this hardware-aware compilation. ME heuristically builds a map that associates each node of
GP -- the logical variables of
P -- to a chain of adjacent nodes in
GQ by means of one of its minors, so that the arcs of
GP are preserved as physical connections among qubits in
GQ.
The static topology of hardwired qubits can clearly lead to inefficient compilations because
GQ cannot be a clique, currently. We propose a methodology and a set of criteria to evaluate how the hardware topology
GQ can negatively affect the embedded problem, thus making the quantum optimization more sensible to noise.
We evaluate the result of ME across two QPU topologies: Zephyr graphs (used in current D-Wave systems) and Havel-Hakimi graphs, which allow controlled variation of the average node degree. This enables us to study how the ratio `number of nodes/number of incident arcs per node' affects ME success rates to map
GP into a minor of
GQ.
Our findings, obtained through ME executed on classical, i.e. non-quantum, architectures, suggest that Havel-Hakimi-based topologies, on average, require shorter qubit chains in the minor of
GP, exhibiting smoother scaling of the largest embeddable
GP as the QPU size increases. These characteristics indicate their potential as alternative designs for QA-based QPUs.
Abstract Quantum annealing (QA) is a novel type of analog computation that aims to use quantum mechanical fluctuations to search for optimal solutions of Ising problems. QA in the transverse Ising model, implemented on D-Wave quantum processing units, are available as cloud computing resources. In this study we report concise benchmarks across three generations of D-Wave quantum annealers, consisting of four different devices, for the NP-hard discrete combinatorial optimization problems unweighted maximum clique and unweighted maximum cut on random graphs. The Ising, or equivalently quadratic unconstrained binary optimization, formulation of these problems do not require auxiliary variables for order reduction, and their overall structure and weights are not highly variable, which makes these problems simple test cases to understand the sampling capability of current D-Wave quantum annealers. All-to-all minor embeddings of size 52, with relatively uniform chain lengths, are used for a direct comparison across the Chimera, Pegasus, and Zephyr device topologies. A grid-search over annealing times and the minor embedding chain strengths is performed in order to determine the level of reasonable performance for each device and problem type. Experiment metrics that are reported are approximation ratios for non-broken chain samples, chain break proportions, and time-to-solution for the maximum clique problem instances. How fairly the quantum annealers sample optimal maximum cliques, for instances which contain multiple maximum cliques, is quantified using entropy of the measured ground state distributions. The newest generation of quantum annealing hardware, which has a Zephyr hardware connectivity, performed the best overall with respect to approximation ratios and chain break frequencies.
Quantum annealing provides a promising route for the development of quantum optimization devices, but the usefulness of such devices will be limited in part by the range of implementable problems as dictated by hardware constraints. To overcome constraints imposed by restricted connectivity between qubits, a larger set of interactions can be approximated using minor embedding techniques whereby several physical qubits are used to represent a single logical qubit. However, minor embedding introduces new types of errors due to its approximate nature. We introduce and study quantum annealing correction schemes designed to improve the performance of quantum annealers in conjunction with minor embedding, thus leading to a hybrid scheme defined over an encoded graph. We argue that this scheme can be efficiently decoded using an energy minimization technique provided the density of errors does not exceed the per-site percolation threshold of the encoded graph. We test the hybrid scheme using a D-Wave Two processor on problems for which the encoded graph is a two-level grid and the Ising model is known to be NP-hard. The problems we consider are frustrated Ising model problem instances with ``planted'' (a priori known) solutions. Applied in conjunction with optimized energy penalties and decoding techniques, we find that this approach enables the quantum annealer to solve minor embedded instances with significantly higher success probability than it would without error correction. Our work demonstrates that quantum annealing correction can and should be used to improve the robustness of quantum annealing not only for natively embeddable problems but also when minor embedding is used to extend the connectivity of physical devices.
Minor embedding is a required compilation step for quantum annealing, mapping logical problem graphs onto sparse hardware topologies. Despite its central role in determining solution quality, no standardized benchmark exists for comparing embedding algorithms: prior studies use incompatible graph libraries, inconsistent metrics, and non-reproducible experimental setups, making cross-algorithm comparisons unreliable. We present Ember (Embedding Minor Benchmark for Evaluative Reproducibility), an open-source benchmarking framework addressing this gap. Ember provides a standardized algorithm interface with seeded, reproducible execution infrastructure; a diverse graph library of 24,016 instances spanning structured, random, and physics-motivated problem types not previously used in embedding benchmarks; and a unified analysis pipeline supporting all three current D-Wave hardware topologies (Chimera, Pegasus, Zephyr). We evaluate five algorithms across the full library on Chimera and find that no algorithm dominates universally: rankings vary systematically with graph structure, and the best algorithm depends on the family being embedded. We also examine the effects of hardware topology (including Pegasus and Zephyr), qubit error rates, and evaluate a reinforcement-learning approach (CHARME) within a narrower test set. Ember is available at https://github.com/zachmacsmith/ember and is installable via pip install ember-qc.
Quantum annealers such as the D-Wave 2X computer are designed to natively solve Quadratic Unconstrained Binary Optimization (QUBO) problems to optimality or near optimality. Most NP-hard problems, which are hard for classical computers, can be naturally described as quadratic binary problems that contain a quadratic binary objective function and one or more constraints. Since a QUBO cannot have constraints, each such constraint has to be added to the objective function as a penalty, in order to solve on D-Wave. For a minimization problem, for instance, such penalty can be a quadratic term that gets a value zero, if the constraint is satisfied, and a large value, if it is not. In many cases, however, the penalty can significantly increase the number of quadratic terms in the resulting QUBO and make it too large to embed into the D-Wave hardware. In this paper, we develop an alternative method for formulating and embedding constraints of the type Σi=1sxi=1, which is much more scalable than the existing ones, and analyze the properties of the resulting embeddings.