Quick design of feasible tensor networks for constrained combinatorial optimization

TL;DR

This paper introduces a new method for designing feasible tensor networks for constrained combinatorial optimization, using elementary mathematics to create efficient solutions without relying on complex physics concepts.

Contribution

It presents a novel approach to construct feasible tensor networks algebraically, enabling faster and more versatile solutions for constrained optimization problems.

Findings

Feasible tensor networks can be constructed using nilpotent-matrix manipulation.

The method allows for faster construction compared to traditional approaches.

Feasible solutions, including optimal ones, can be reliably obtained through imaginary time evolution.

Abstract

Quantum computers are expected to enable fast solving of large-scale combinatorial optimization problems. However, their limitations in fidelity and the number of qubits prevent them from handling real-world problems. Recently, a quantum-inspired solver using tensor networks has been proposed, which works on classical computers. Particularly, tensor networks have been applied to constrained combinatorial optimization problems for practical applications. By preparing a specific tensor network to sample states that satisfy constraints, feasible solutions can be searched for without the method of penalty functions. Previous studies have been based on profound physics, such as U(1) gauge schemes and high-dimensional lattice models. In this study, we devise to design feasible tensor networks using elementary mathematics without such a specific knowledge. One approach is to construct tensor networks with nilpotent-matrix manipulation. The second is to algebraically determine tensor parameters. We showed mathematically that such feasible tensor networks can be constructed to accommodate various types of constraints. For the principle verification, we numerically constructed a feasible tensor network for facility location problem, to find much faster construction than conventional methods. Then, by performing imaginary time evolution, feasible solutions were always obtained, ultimately leading to the optimal solution.