Quantum-Inspired Mean Field Probabilistic Model for Combinatorial Optimization Problems

Abstract: Combinatorial optimization problems are pivotal across many fields. Among these, Quadratic Unconstrained Binary Optimization (QUBO) problems, central to fields like portfolio optimization, network design, and computational biology, are NP-hard and require exponential computational resources. To address these challenges, we develop a novel Quantum-Inspired Mean Field (QIMF) probabilistic model that approximates solutions to QUBO problems with enhanced accuracy and efficiency. The QIMF model draws inspiration from quantum measurement principles and leverages the mean field probabilistic model. We incorporate a measurement grouping technique and an amplitude-based shot allocation strategy, both critical for optimizing cost functions with a polynomial speedup over traditional methods. Our extensive empirical studies demonstrate significant improvements in solution evaluation for large-scale problems of portfolio selection, the weighted maxcut problem, and the Ising model. Specifically, using S&P 500 data from 2022 and 2023, QIMF improves cost values by 152.8% and 12.5%, respectively, compared to the state-of-the-art baselines. Furthermore, when evaluated on increasingly larger datasets for QUBO problems, QIMF's scalability demonstrates its potential for large-scale QUBO challenges.