Benchmarking Quantum Annealers with Near-Optimal Minor-Embedded Instances

TL;DR

This work tackles benchmarking quantum annealers by isolating the embedding bottleneck: it introduces a protocol to generate problem instances with near-optimal minor-embedding mappings to a D-Wave QA. The method starts from a complete-graph embedding on the chip topology via CME, then iteratively splits chains to produce source graphs of varying densities while keeping the target graph fixed, yielding near-optimal mappings $\phi_{nopt}$ that approach the theoretical lower bound on physical qubits. Empirical results show that the quantum processor (Advantage6.4) is competitive with, and sometimes superior to, classical solvers on very sparse instances of unweighted and weighted max-cut (densities $d<0.1$), but struggles with denser or constrained problems like MIS due to embedding overhead and penalty terms. The study provides a principled way to generate favorable benchmark instances and highlights the potential quantum advantage in specific, sparse combinatorial regimes while outlining limitations for more complex encodings.

Abstract

Benchmarking Quantum Process Units (QPU) at an application level usually requires considering the whole programming stack of the quantum computer. One critical task is the minor-embedding (resp. transpilation) step, which involves space-time overheads for annealing-based (resp. gate-based) quantum computers. This paper establishes a new protocol to generate graph instances with their associated near-optimal minor-embedding mappings to D-Wave Quantum Annealers (QA). This set of favorable mappings is used to generate a wide diversity of optimization problem instances. We use this method to benchmark QA on large instances of unconstrained and constrained optimization problems and compare the performance of the QPU with efficient classical solvers. The benchmark aims to evaluate and quantify the key characteristics of instances that could benefit from the use of a quantum computer. In this context, existing QA seem best suited for unconstrained problems on instances with densities less than $10\%$. For constrained problems, the penalty terms used to encode the hard constraints restrict the performance of QA and suggest that these QPU will be less efficient on these problems of comparable size.

Figures (5)

• Figure 1: Instance generation with iterative chain split. a) The algorithm starts with a complete graph embedding using CME method on the graph $G_\mathrm{t\_QPU}$. Each node in the source graph $G_\mathrm{s}$ is mapped to a chain in the target graph $G_\mathrm{t}$. b) A random logical node $v \in V_\mathrm{s}$ is selected. c) The corresponding chain $\phi_\mathrm{CME}(v)$ is split in two parts to create a new logical node in the source graph. It changes the mapping function to $\phi_\mathrm{nopt}$. Steps b. and c. are repeated on $G_\mathrm{s}'$ until the desired density in the logical graph is reached.
• Figure 2: Performance comparison of mapping functions found by our method $\phi_\mathrm{nopt}$ and the best among 100 tries of CMR method $\phi_\mathrm{100\_CMR}$. 30 instances are generated for each density. a) Comparison using the overhead ratio (see \ref{['eqn:overhead_ratio']}). Error bars show the standard deviations b) Ratio of the best cut size obtained for each mapping for each instance.
• Figure 3: QPU access time repartition for different annealing times per shot with a total running time limit of $1s$. Each color represents the fraction of time used for programming the QPU, annealing the $n$ shots, reading the results of the $n$ shots, and delays between the $n$ shots.
• Figure 4: Annealing time scan with a quantum processing time limit of $1s$ for three types of problems with various densities. a) Unweighted maxcut instances b) Weighted maxcut instances c) Weighted MIS.
• Figure 5: Benchmark of D-Wave Advantage6.4 and Tabu search (1s time limit). The results are expressed as ratio of the best result found by Gurobi (60s runtime limit). The size of each instance is provided in \ref{['table:embedding']} a) Unweighted max-cut b) Weighted max-cut c) MIS problem with a random solver.