Quantum Annealers Chain Strengths: A Simple Heuristic to Set Them All

TL;DR

The paper addresses solving Ising and max-cut problems with quantum annealers when the problem graph does not map directly to the hardware topology, highlighting minor embedding and the critical role of chain strengths. It analyzes how different logical-qubit encodings affect the minimum spectral gap $Δ_{\min}$ and shows that denser encodings require lower chain strengths, while coupler rescaling reduces $Δ_{\min}$ by the same factor. It introduces a simple per-instance chain-strength heuristic that uses only a small number of preprocessing shots and often outperforms the default method, delivering up to 17.2% improvement on tested max-cut instances. The work underscores embedding choice and hardware-aware tuning as practical levers to improve solution quality and time-to-solution on quantum annealers.

Abstract

Quantum annealers (QA), such as D-Wave systems, become increasingly efficient and competitive at solving combinatorial optimization problems. However, solving problems that do not directly map the chip topology remains challenging for this type of quantum computer. The creation of logical qubits as sets of interconnected physical qubits overcomes limitations imposed by the sparsity of the chip at the expense of increasing the problem size and adding new parameters to optimize. This paper explores the advantages and drawbacks provided by the structure of the logical qubits and the impact of the rescaling of coupler strength on the minimum spectral gap of Ising models. We show that densely connected logical qubits require a lower chain strength to maintain the ferromagnetic coupling. We also analyze the optimal chain strength variations considering different minor embeddings of the same instance. This experimental study suggests that the chain strength can be optimized for each instance. We design a heuristic that optimizes the chain strength using a very low number of shots during the pre-processing step. This heuristic outperforms the default method used to initialize the chain strength on D-Wave systems, increasing the quality of the best solution by up to 17.2% for tested instances on the max-cut problem.

Figures (4)

• Figure 1: Minimum spectral gap evaluations considering different types of logical qubits encoding a) Native Ising problem instance b) Same instance as in a. with the red qubit encoded as a chain of physical qubits c) Same instance as in a. with the red qubit encoded as a cycle of physical qubits d) Same instance as in a. with the red qubit encoded with a clique of physical qubits. Black edges represent logical couplers and red edges represent ferromagnetic couplers parametrized by the chain strength. The auto-coupler strength $h_6$ is uniformly spread on each physical qubit. e) Evolution of $\Delta_\mathrm{min}$ considering the whole annealing schedule for different values of the global chain strength $|F_\phi|$. f) Spectral gap evolution of each encoding type programmed with the corresponding optimal chain strength indicated by dashed lines in e.
• Figure 2: Heatmaps showing the average percentage overhead of the number of qubits used to embed similar instances using the CMR method compared to CME method. Each score is an average done over 100 instances. The score calculation in each cell is detailed in equation \ref{['embedding_ratio']}. a) and b) are embeddings generated for Advantage6.4 topology for Erdős-Rényi and d-regular graphs. b) and c) are embeddings generated for Advantage2_prototype2.2 topology using the same instances. Advantage6.4 and Advantage2_prototype2.2 can embed complete graphs of maximum size 82 and 174.
• Figure 3: Statistics on the breaking chain rate (see equation \ref{['eqn:break_rate']}) averaged over 30 instances of Erdős-Rényi graphs of size $n=80$ and density $p=0.3$. a) resp. b) show the average chain length repartition of embedding obtained with CMR resp. CME methods. Blue bars show the average frequency of each chain length. Orange bars show the average breaking chain rate with a black error bar for the standard deviation. c) shows the average and median frequency of corrupted ferromagnetic couplings on CMR embeddings. d) shows the number of different ferromagnetic coupling that are corrupted at least once during the 1024 shots on CMR embeddings.
• Figure 4: Maximum cut size obtained with a chain scan on four different embeddings of the same instance. The uniform_torque_compensation is run once for each embedding (hence, it is independent of the chain scan). The CME and upper right CMR embedding have the same number of physical qubits ($\pm 1\%$). The two other embeddings have $10\%$ more and $10\%$ less qubits than the CME embedding. The red violins show the average breaking chain rate $\bar{\epsilon_b}$ related to the chain scan curve.