Comparing Three Generations of D-Wave Quantum Annealers for Minor Embedded Combinatorial Optimization Problems

TL;DR

This work benchmarks four D-Wave quantum annealers across Chimera, Pegasus, and Zephyr topologies on fixed $N=52$ minor embeddings for two NP-hard graph problems, Maximum Clique and Maximum Cut, using 1000 anneal-readout cycles per setting. By grid-searching over annealing times and chain strengths without post-processing, it reports approximation ratios, chain-break rates, and fair-sampling entropy, highlighting that the newest Zephyr-connected hardware delivers the best overall sampling. The results show a clear progression with hardware topology: Pegasus devices perform strongly on clique instances, while the latest device Advantage2_prototype1.1 achieves the best overall performance for both problems, aided by denser connectivity and shorter chains. These findings inform practical usage and parameter tuning for moderate-sized minor-embedded problems on cloud-based quantum annealers and demonstrate tangible hardware improvements over short development times.

Abstract

Quantum annealing is a novel type of analog computation that aims to use quantum mechanical fluctuations to search for optimal solutions of Ising problems. Quantum annealing in the Transverse Ising model, implemented on D-Wave QPUs, are available as cloud computing resources. In this article we report concise benchmarks across three generations of D-Wave quantum annealers, consisting of four different devices, for the NP-Hard combinatorial optimization problems unweighted maximum clique and unweighted maximum cut on random graphs. The Ising, or equivalently QUBO, formulation of these problems do not require auxiliary variables for order reduction, and their overall structure and weights are not highly complex, 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 and chain break proportions. How fairly the quantum annealers sample optimal maximum cliques, for instances which contain multiple maximum cliques, is also 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.

Figures (11)

• Figure 1: Example random $52$ node graph with $254$ edges. The left plot shows the single Maximum Clique of the graph, which has size $5$. The right plot shows a Maximum Cut partition of the same graph, where the two partition node and edges are green and cyan, and the shared edges between the bi-partition are black.
• Figure 2: Maximum clique mean approximation ratios (y-axis) vs chain strength (x-axis) for the $200$$G(n,p)$ random graphs. The aggregated results are shown in the form of $10$ lines, per plot, representing the mean approximation ratio for $10$ linearly spaced graph density intervals from $0.05$ to $0.95$, where the color of each line encodes the mean graph density for that interval. The color coding is shown in the colorbar below the plots. Problem QUBOs executed on DW_2000Q_6 (left column), Advantage_system4.1 (center-left column), Advantage_system6.1 (center-right column), and Advantage2_prototype1.1 (right column). The annealing time in microseconds are varied across $2000$ microseconds (top row), $1000$ microseconds (second row), $100$ microseconds (third row), $10$ microseconds (fourth row), and $1$ microsecond (bottom row). The x-axis and y-axis labels are consistent and shared between all of the sub-figures.
• Figure 3: Chain break proportions (y-axis) vs chain strength (x-axis) for each of the maximum clique instances on the $200$$G(n,p)$ random graphs. The aggregated results are shown in the form of $10$ lines, per plot, representing the mean chain break proportion for $10$ linearly spaced graph density intervals from $0.05$ to $0.95$, where the color of each line encodes the mean graph density for that interval. The line colors encode the problem graph density using the same colorscale from Figure \ref{['fig:maximum_clique_approx_ratio']}. The D-Wave devices are DW_2000Q_6 (left column), Advantage_system4.1 (center-left column), Advantage_system6.1 (center-right column), and Advantage2_prototype1.1 (right column). The annealing time in microseconds are varied across $2000$ microseconds (top row), $1000$ microseconds (second row), $100$ microseconds (third row), $10$ microseconds (fourth row), and $1$ microsecond (bottom row).
• Figure 4: Maximum cut mean approximation ratios (y-axis) vs chain strength (x-axis) for each of the $150$$G(n,p)$ random graphs. The aggregated results are shown in the form of $10$ lines per plot representing the mean approximation ratio for $10$ linearly spaced graph density intervals from $0.05$ to $0.95$, where the color of each line encodes the mean graph density for that interval. The color coding is shown in the colorbar below the plots. Problem QUBOs sampled using DW_2000Q_6 (left column), Advantage_system4.1 (center-left column), Advantage_system6.1 (center-right column), and Advantage2_prototype1.1 (right column). The annealing time in microseconds are varied across $2000$ microseconds (first row), $1000$ microseconds (second row), $100$ microseconds (third row), $10$ microseconds (fourth row), and $1$ microsecond (bottom row).
• Figure 5: Chain break proportions (y-axis) vs chain strength (x-axis) for each of the maximum cut Isings from the $150$$G(n,p)$ random graphs. The aggregated results are shown in the form of $10$ lines per plot representing the mean approximation ratio for $10$ linearly spaced graph density intervals from $0.05$ to $0.95$, where the color of each line encodes the mean graph density for that interval. The line colors encode the problem graph density using the same colorscale from Figure \ref{['fig:maximum_cut_approx_ratio']}. The D-Wave devices are DW_2000Q_6 (left column), Advantage_system4.1 (center-left column), Advantage_system6.1 (center-right column), and Advantage2_prototype1.1 (right column). The annealing time in microseconds are varied across $2000$ microseconds (first row), $1000$ microseconds (second row), $100$ microseconds (third row), $10$ microseconds (fourth row), and $1$ microsecond (bottom row).