Constraint Preserving XY-Mixers under Trotterized Adiabatic Evolution

TL;DR

This paper analyzes the impact of Trotterization on constraint-preserving XY-mixers in quantum optimization, highlighting how constraint structure influences their effectiveness compared to standard mixers.

Contribution

It provides a theoretical and numerical investigation of Trotter errors in XY-mixers, emphasizing the importance of constraint locality for their performance.

Findings

Trotter errors depend mainly on individual constraint size and structure.

XY-mixers outperform X-mixers in problems with local, disjoint constraints.

Global constraints significantly impair XY-mixer performance under Trotterization.

Abstract

Constraint handling is a central challenge for quantum algorithms applied to combinatorial optimization. Standard penalty-based approaches increase problem size, distort energy landscapes, and often degrade performance. Constraint-preserving mixers, such as XY-mixers, restrict quantum evolution to feasible subspaces, but their implementation on gate-based hardware requires Trotterization, which introduces approximation errors. In this work, we systematically investigate the interplay between constraint-preserving XY-mixers and Trotterized Adiabatic Evolution (TAE). We present a theoretical analyses of the origin and scaling of Trotter errors in XY-mixers and show that the dominant contribution depends on the size and structure of individual constraints rather than on the total problem size. Our findings are validated through extensive numerical simulations on three representative problems: Portfolio Optimization, the Multi-Car Paint Shop problem, and a Multi-Commodity Flow problem. For problems with a single global equality constraint spanning all variables, Trotter errors significantly impair XY-mixer performance, making standard Pauli-X mixers more robust under realistic implementations. In contrast, for problems whose constraints decompose into multiple disjoint local blocks, XY-mixers outperform X-mixers by several orders of magnitude even under Trotterized evolution. These results identify constraint locality as the key criterion for the effective use of XY-mixers and demonstrate that TAE combined with structure-aware mixer design provides a robust and theoretically grounded alternative to variational quantum optimization methods. We further present a dedicated mixer Hamiltonian for TSP-like 2-way-1-hot constraints.