Stochastic Simulated Quantum Annealing for Fast Solution of Combinatorial Optimization Problems

TL;DR

Stochastic simulated quantum annealing (SSQA) targets large-scale combinatorial optimization by emulating quantum annealing on classical hardware through quantum Monte Carlo with $R$ replicas coupled by $J_{\perp}$. The method updates spin states in parallel using integral stochastic computing, enabling rapid traversal of the energy landscape toward the global minimum, while preserving quantum-like correlations via the replica coupling. Evaluations on graph isomorphism problems up to $N\approx 2500$ spins show SSQA delivers an order-of-magnitude reduction in time-to-solution (TTS) compared with SSA and can solve problem sizes roughly two orders of magnitude larger than QA and about 25 times larger than SA for similar convergence probabilities. These results position SSQA as a scalable, hardware-friendly approach for fully connected Ising models, with potential for substantial speedups in real-world GI and related optimization tasks.

Abstract

In this paper, we introduce stochastic simulated quantum annealing (SSQA) for large-scale combinatorial optimization problems. SSQA is designed based on stochastic computing and quantum Monte Carlo, which can simulate quantum annealing (QA) by using multiple replicas of spins (probabilistic bits) in classical computing. The use of stochastic computing leads to an efficient parallel spin-state update algorithm, enabling quick search for a solution around the global minimum energy. Therefore, SSQA realizes quantum-like annealing for large-scale problems and can handle fully connected models in combinatorial optimization, unlike QA. The proposed method is evaluated in MATLAB on graph isomorphism problems, which are typical combinatorial optimization problems. The proposed method achieves a convergence speed an order of magnitude faster than a conventional stochastic simulaated annealing method. Additionally, it can handle a 100-times larger problem size compared to QA and a 25-times larger problem size compared to a traditional SA method, respectively, for similar convergence probabilities.

Figures (8)

• Figure 1: Simulated annealing (SA) based on a spin network that consists of spins, spin biases, and spin weights. Spin states can be flipped between '+1' and '-1' in an attempt to reach the global minimum energy of the Hamiltonian.
• Figure 2: Quantum Monte Carlo (QMC) method using spin network replicas for SSQA. Each spin network is coupled with the upper and the lower spin networks with $J_{\perp}$.
• Figure 3: Concept of annealing process of SSQA. Each replica of the spin network searches for the global minimum energy with increasing $J_{\perp}$, which can reach the global minimum based on the coupled spins.
• Figure 4: Example of a four-node graph isomorphism (GI) problem. (a) Two graphs are isomorphic and (b) $Q$ coefficients of the QUBO model corresponding to the two four-node graphs.
• Figure 5: Energy versus cycles in SSQA for a GI problem of $N=2,500$ with $R=25$: (a) 20th replica, (b) 10th replica, (c) 1st replica, (d) $J_{perp}$. In each iteration, $J_{\perp}$ is increased with $\tau=100$. At the 1st, 3rd, and 4th iterations, these replicas reach the global minimum energy with different timing.