Direct phase encoding in QAOA: Describing combinatorial optimization problems through binary decision variables

TL;DR

This paper introduces a more qubit-efficient encoding method for the QAOA applied to the Traveling Salesperson Problem, reducing qubit count while maintaining approximation quality on small instances.

Contribution

It proposes a novel edge-based encoding for TSP in QAOA, decreasing qubit requirements by a linear factor compared to traditional node-based encoding.

Findings

Edge-based encoding reduces qubit count significantly.

Approximation quality remains comparable to traditional methods.

Classical optimizer iterations increase only slightly with the new encoding.

Abstract

The Quantum Approximate Optimization Algorithm (QAOA) and its derived variants are widely in use for approximating combinatorial optimization problem instances on gate-based Noisy Intermediate Scale Quantum (NISQ) computers. Commonly, circuits required for QAOA are constructed by first reformulating a given problem as a Quadratic Unconstrained Binary Optimization (QUBO) problem. It is then straightforward to synthesize a QAOA circuit from QUBO equations. In this work, we illustrate a more qubit-efficient circuit construction for combinatorial optimization problems by the example of the Traveling Salesperson Problem (TSP). Conventionally, the qubit encoding in QAOA for the TSP describes a tour using a sequence of nodes, where each node is written as a 1-hot binary vector. We propose to encode TSP tours by selecting edges included in the tour. Removing certain redundancies, the number of required qubits can be reduced by a linear factor compared to the aforementioned conventional encoding. We examined implementations of both QAOA encoding variants in terms of their approximation quality and runtime. Our experiments show that for small instances results are just as accurate using our proposed encoding, whereas the number of required classical optimizer iterations increases only slightly.