Variational quantum algorithms for permutation-based combinatorial problems: Optimal ansatz generation with applications to quadratic assignment problems and beyond

TL;DR

This paper introduces a quantum variational algorithm, QuPer, that efficiently generates permutation circuits for solving combinatorial problems like quadratic assignment, demonstrating near-term applicability and competitive performance with classical methods.

Contribution

The paper develops a novel permutation circuit construction based on group theory, enabling scalable quantum algorithms for permutation problems with applications to quadratic assignment and graph isomorphism.

Findings

Circuits scale logarithmically with permutation size.

QuPer performs competitively against classical heuristics.

Simulation up to 256 variables with 20 qubits is feasible.

Abstract

We present a quantum variational algorithm based on a novel circuit that generates all permutations that can be spanned by one- and two-qubits permutation gates. The construction of the circuits follows from group-theoretical results, most importantly the Bruhat decomposition of the group generated by the \(\mathtt{cx}\) gates. These circuits require a number of qubits that scale logarithmically with the permutation dimension, and are therefore employable in near-term applications. We further augment the circuits with ancilla qubits to enlarge their span, and with these we build ansatze to tackle permutation-based optimization problems such as quadratic assignment problems, and graph isomorphisms. The resulting quantum algorithm, \textsc{QuPer}, is competitive with respect to classical heuristics and we could simulate its behavior up to a problem with $256$ variables, requiring $20$ qubits.