Constrained Quantum Optimization via Iterative Warm-Start XY-Mixers

TL;DR

This paper introduces a novel warm-started XY-mixer Hamiltonian for constrained quantum optimization, combined with an iterative classical heuristic, significantly improving solution quality and speed on quantum hardware.

Contribution

It formulates a new warm-started XY-mixer Hamiltonian with proven ground-state properties and develops an iterative heuristic that enhances QAOA performance on constrained problems.

Findings

IWS-QAOA increases the probability of sampling optimal solutions by orders of magnitude.

The approach accelerates convergence compared to standard XY-QAOA.

Successful implementation on ibm_boston QPU demonstrates practical viability.

Abstract

The Quantum Approximate Optimization Algorithm (QAOA) is a leading hybrid heuristic for combinatorial optimization, but efficiently handling hard constraints remains a significant challenge. XY-mixers successfully confine quantum state evolution to a feasible subspace, such as the Hamming-weight-1 sector for one-hot constraints. On the contrary, warm-starting biases the search toward promising regions based on preliminary solutions. Combining these two techniques requires maintaining the essential alignment between the initial state and the mixer Hamiltonian to preserve convergence guarantees. Previous work demonstrated warm-starting with XY-mixers via a biased initial state, but relying only on standard mixer Hamiltonians. Consequently, the initial state is no longer a ground state of the mixer. In this work, we overcome these limitations by formulating a warm-started XY-mixer Hamiltonian for one-hot constraints and proving its ground-state properties. Furthermore, we provide a shallow circuit implementation suitable for NISQ implementations. We embed the warm-starting into a classical heuristic that iteratively updates the bias based on previous samples, called Iterative Warm-Starting (IWS). Extensive numerical simulations on Max-$k$-Cut and Traveling Salesperson Problem instances demonstrate that IWS-QAOA significantly accelerates the solution-finding process, increasing the probability of sampling optimal solutions by orders of magnitude compared to standard XY-QAOA. Finally, we validate our approach on the ibm_boston QPU using hardware-tailored 144-qubit problem instances. By coupling IWS-QAOA with a greedy steepest-descent post-processing strategy to repair infeasible measurements caused by hardware noise, we successfully identify optimal solutions on actual quantum devices.