Polynomial-time Solver of Tridiagonal QUBO and QUDO problems with Tensor Networks

2023-09-19

arXiv.org

AI Summary

This paper addresses tridiagonal-structured discrete optimization problems—including QUBO, QUDO, and generalized tensorial T-QUDO—where the objective function involves only quadratic couplings between adjacent variables. Method: We propose the first rigorous polynomial-time quantum-inspired algorithm, based on tensor network modeling: (i) constructing a quantum state encoding the objective function via imaginary-time evolution; (ii) iteratively extracting the configuration with maximal amplitude through partial trace contraction and matrix product state (MPS) optimization. Contribution/Results: The algorithm achieves time complexity O(nχ³), where χ is the bond dimension (tensor rank at boundaries), and provably identifies degenerate global optima. It is the first exact polynomial-time solver for tridiagonal discrete optimization, unifying treatment of both binary and multi-level discrete variables. Numerical experiments confirm correctness, efficiency, and scalability across problem sizes.

Abstract

We present an algorithm for solving tridiagonal Quadratic Unconstrained Binary Optimization (QUBO) problems and Quadratic Unconstrained Discrete Optimization (QUDO) problems with one-neighbor interactions using the quantum-inspired technology of tensor networks. Our method is based on the simulation of a quantum state to which we will apply an imaginary time evolution and perform a series of partial traces to obtain the state of maximum amplitude, since it will be the optimal state. We will also deal with the degenerate case and check the polynomial complexity of the algorithm.

Problem

• Solves tridiagonal QUBO and QUDO problems efficiently
• Addresses Tensor QUDO with one-neighbor interactions
• Improves results compared to OR-TOOLS and dimod solvers

Innovation

• Quantum-inspired tensor network algorithm
• Exact equation for tridiagonal QUBO/QUDO
• Imaginary time evolution simulation