Many-Qudit representation for the Travelling Salesman Problem Optimisation

TL;DR

This paper introduces a many-qudit approach to map the TSP onto a quantum system with a smaller Hilbert space, enabling faster quantum and classical solutions, validated through simulations up to 100 cities.

Contribution

It proposes a novel many-qudit representation for TSP that reduces the Hilbert space size compared to traditional QUBO mappings.

Findings

Reduced Hilbert space dimension from 2^{N^2} to 2^{N log_2 N}

Successful simulation of TSP with nearly 100 cities

Potential for significant speedup in quantum and classical algorithms

Abstract

We present a map from the travelling salesman problem (TSP), a prototypical NP-complete combinatorial optimisation task, to the ground state associated with a system of many-qudits. Conventionally, the TSP is cast into a quadratic unconstrained binary optimisation (QUBO) problem, that can be solved on an Ising machine. The size of the corresponding physical system's Hilbert space is $2^{N^2}$, where $N$ is the number of cities considered in the TSP. Our proposal provides a many-qudit system with a Hilbert space of dimension $2^{N\log_2N}$, which is considerably smaller than the dimension of the Hilbert space of the system resulting from the usual QUBO map. This reduction can yield a significant speedup in quantum and classical computers. We simulate and validate our proposal using variational Monte Carlo with a neural quantum state, solving the TSP in a linear layout for up to almost 100 cities.