Exhaustive and feasible parametrisation with applications to the travelling salesperson problem

TL;DR

This paper presents a novel approach to quantum circuits for constrained optimization, ensuring all feasible solutions can be reached with certainty using a fixed set of parameters, demonstrated on the travelling salesperson problem.

Contribution

It introduces a group-theoretic pipeline for constructing exhaustively parametrised, feasibility-respecting quantum circuits, a significant advancement over conventional asymptotic methods.

Findings

Successfully applied to TSP with up to nine cities

Circuits can reach all feasible solutions with fixed parameters

Numerical comparisons show advantages over existing mixers

Abstract

This paper introduces the concept of exhaustively parametrised, feasibility-respecting quantum circuits for constrained combinatorial optimisation problems. Such circuits can reach, given the right parameter values, every feasible solution with certainty -- including the optimum -- with a fixed number of parameters, while avoiding infeasible solutions altogether. This is in sharp contrast to conventional quantum alternating operator ansatz schemes, which are merely guaranteed to reach the optimum asymptotically. We introduce an abstract pipeline for constructing exhaustively parametrised, feasibility-respecting circuits from a transitive group action on a problem's feasible set. Our constructions rely on the simple combination of the group action with group representation and the novel notion of generating sequences: group elements in fixed order, possibly with repetitions, that generate the entire group. That is, we trace expressivity of parametrised quantum circuits back to the most fundamental concepts of group theory. We apply this pipeline to two concrete examples for the travelling salesperson problem, thus showing that exhaustively parametrised, feasibility-respecting circuits are not an empty definition. Furthermore, we provide numerical proof-of-principles on instances with up to nine cities, comparing the suitability of our constructions for parameter optimisation purposes against established mixers.