Quantum-Inspired Optimization through Qudit-Based Imaginary Time Evolution

2025-12-04

AI Summary

This paper addresses combinatorial optimization problems with integer decision variables. Method: We propose a quantum-inspired imaginary-time evolution algorithm that directly encodes integer variables using multilevel quantum systems (qudits), inherently satisfying single-assignment constraints and substantially reducing the number of required variables. A gradient-adaptive mechanism is designed to construct Hermitian evolution operators, ensuring the system remains in efficiently simulatable product states throughout evolution. The algorithm integrates imaginary-time evolution approximation, classical gradient-based optimization, and iterative updates of unitary operator sequences. Results: Experiments on the constrained Min-d-Cut problem demonstrate that our method significantly outperforms Gurobi’s penalty-function approach—particularly for larger values of d—achieving faster convergence and higher-quality solutions.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Quantum Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Imaginary-time evolution has been shown to be a promising framework for tackling combinatorial optimization problems on quantum hardware. In this work, we propose a classical quantum-inspired strategy for solving combinatorial optimization problems with integer-valued decision variables by encoding decision variables into multi-level quantum states known as qudits. This method results in a reduced number of decision variables compared to binary formulations while inherently incorporating single-association constraints. Efficient classical simulation is enabled by constraining the system to remain in a product state throughout optimization. The qudit states are optimized by applying a sequence of unitary operators that iteratively approximate the dynamics of imaginary time evolution. Unlike previous studies, we propose a gradient-based method of adaptively choosing the Hermitian operators used to generate the state evolution at each optimization step, as a means to improve the convergence properties of the algorithm. The proposed algorithm demonstrates promising results on Min-d-Cut problem with constraints, outperforming Gurobi on penalized constraint formulation, particularly for larger values of d.

Problem

• Develops a classical quantum-inspired optimization method using qudits
• Solves combinatorial optimization with integer variables via imaginary time evolution
• Improves convergence with adaptive gradient-based operator selection

Innovation

• Uses qudit encoding to reduce decision variables and enforce constraints
• Employs product state approximation for efficient classical simulation
• Applies adaptive gradient-based operators to improve algorithm convergence