Quantum Computing, Ising Formulation, and the Traveling Salesman Problem
For researchers in quantum computing and optimization, this work clarifies limitations of current Ising-based approaches and introduces a method that may improve quantum solvers for NP-complete problems.
The paper identifies non-trivial issues in applying Ising formulations to the traveling salesman problem and proposes a novel approach, Discrete Quantum Exhaustive Search, to enhance Variational Quantum Eigensolver for solving NP-complete problems, with potential extensions to QMA problems.
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.