# $XY$-mixers: analytical and numerical results for QAOA

## TL;DR

This paper investigates the use of $XY$-Hamiltonians as mixers in QAOA for constrained combinatorial problems, providing analytical and numerical results that show improved performance over traditional mixers, especially for graph coloring.

## Contribution

It introduces a method to implement $XY$-mixers in QAOA without Trotter error for certain problem encodings and validates their effectiveness through numerical experiments on challenging graph coloring problems.

## Key findings

- $XY$-mixers outperform $X$-mixers in numerical tests.
- Implementation strategies for all-to-all and linearly connected hardware are proposed.
- Generalized $W$-states improve QAOA performance with $XY$-mixers.

## Abstract

The Quantum Alternating Operator Ansatz (QAOA) is a promising gate-model meta-heuristic for combinatorial optimization. Applying the algorithm to problems with constraints presents an implementation challenge for near-term quantum resources. This work explores strategies for enforcing hard constraints by using $XY$-Hamiltonians as mixing operators (mixers). Despite the complexity of simulating the $XY$ model, we demonstrate that for problems represented through one-hot-encoding, certain classes of the mixer Hamiltonian can be implemented without Trotter error in depth $O(\kappa)$ where $\kappa$ is the number of assignable colors. We also specify general strategies for implementing QAOA circuits on all-to-all connected hardware graphs and linearly connected hardware graphs inspired by fermionic simulation techniques. Performance is validated on graph coloring problems that are known to be challenging for a given classical algorithm. The general strategy of using $XY$-mixers is borne out numerically, demonstrating a significant improvement over the general $X$-mixer, and moreover the generalized $W$-state yields better performance than easier-to-generate classical initial states when $XY$ mixers are used.