Beyond QUBO and HOBO formulations, solving the Travelling Salesman Problem on a quantum boson sampler

TL;DR

This paper introduces a novel penalty-free formulation for the Travelling Salesman Problem that leverages quantum boson sampling, enabling larger problem sizes to be solved more efficiently than traditional penalty-based methods.

Contribution

A new formulation for TSP that reduces variables and eliminates penalties, demonstrated through simulations on a quantum boson sampler and hardware implementation.

Findings

Larger TSP networks can be solved with the new formulation.

Penalty-free approach improves problem size scalability.

Successful hardware translation on an experimental boson sampler.

Abstract

The Travelling Salesman Problem (TSP) is an important combinatorial optimisation problem, and is usually solved on a quantum computer using a Quadratic Unconstrained Binary Optimisation (QUBO) formulation or a Higher Order Binary Optimisation(HOBO) formulation. In these formulations, penalty terms are added to the objective function for outputs that don't map to valid routes. We present a novel formulation which needs fewer binary variables, and where, by design, there are no penalty terms because all outputs from the quantum device are mapped to valid routes. Simulations of a quantum boson sampler were carried out which demonstrate that larger networks can be solved with this penalty-free formulation than with formulations with penalties. Simulations were successfully translated to hardware by running a non-QUBO formulation with penalties on an early experimental prototype ORCA PT-1 boson sampler. Although we worked with a boson sampler, we believe that this novel formulation is relevant to other quantum devices. This work shows that a good embedding for combinatorial optimisation problems can solve larger problems with the same quantum computing resource. The flexibility of boson sampling quantum devices is a powerful asset in solving combinatorial optimisation problem, because it enables formulations where the output string is always mapped to a valid solution, avoiding the need for penalties.