Quantum Computing, Ising Formulation, and the Traveling Salesman Problem

TL;DR

This paper explores the use of Ising formulation and quantum algorithms like VQE for solving the Traveling Salesman Problem, addressing issues and proposing novel approaches to improve quantum optimization methods.

Contribution

It introduces a detailed analysis of Ising models for TSP, discusses the relevance of VQE and qubit efficiency, and proposes a novel Discrete Quantum Exhaustive Search approach.

Findings

Identified issues with Ising model representation of TSP.

Clarified the role of VQE as a SAT-solver.

Proposed Discrete Quantum Exhaustive Search to enhance quantum optimization.

Abstract

Ising formulation is important for many NP problems (Lucas, 2014). This formulation enables implementing novel quantum computing methods including Quantum Approximate Optimization Algorithm and Variational Quantum Eigensolver (VQE). Here, we investigate closely the traveling salesman problem (TSP). First, we present some non-trivial issues related to Ising model view versus a realistic salesman. Then, focusing on VQE we discuss and clarify the use of: a.-- Conventional VQE and how it is relevant as a novel SAT-solver; b.-- Qubit efficiency and its importance in the Noisy Intermediate Scale Quantum-era; and c.-- the relevance and importance of a novel approach named Discrete Quantum Exhaustive Search (Alfassi, Meirom, and Mor, 2024), for enhancing VQE and other methods using mutually unbiased bases. The approach we present here in details can potentially be extended for analyzing approximating and solving various other NP complete problems. Our approach can also be extended beyond the Ising model and beyond the class NP, for example to the class Quantum Merlin Arthur (QMA) of problems, relevant for quantum chemistry and for general spin problems.