Inequality constraints in variational quantum circuits with qudits

TL;DR

This paper introduces a direct method using qudit-SUM gates to implement inequality constraints in variational quantum circuits, outperforming traditional slack variable methods, and enabling more efficient quantum optimization for complex, real-world problems.

Contribution

The paper proposes a novel direct implementation of inequality constraints in QAOA using qudit-SUM gates, reducing resource requirements and improving performance over slack variable methods.

Findings

Direct implementation outperforms slack variable approach.

Linear energy penalty is most effective.

Method enables tackling large-scale constrained problems.

Abstract

Quantum optimization is emerging as a prominent candidate for exploiting the capabilities of near-term quantum devices. Many application-relevant optimization tasks require the inclusion of inequality constraints, usually handled by enlarging the Hilbert space through the addition of slack variables. This approach, however, requires significant additional resources especially when considering multiple constraints. Here, we study an alternative direct implementation of these constraints within the QAOA algorithm, achieved using qudit-SUM gates, and compare it to the slack variable method generalized to qudits. We benchmark these approaches on three paradigmatic optimization problems. We find that the direct implementation of the inequality penalties vastly outperforms the slack variables method, especially when studying real-world inspired problems with many constraints. Within the direct penalty implementation, a linear energy penalty for unfeasible states outperforms other investigated functional forms, such as the canonical quadratic penalty. The proposed approach may thus be an enabling step for approaching realistic industry-scale and fundamental science problems with large numbers of inequality constraints.