Benchmarking and Resource Analysis for Augmented-Lagrangian Quantum Hamiltonian Descent

TL;DR

This paper evaluates AL-QHD, a hybrid quantum-classical optimization framework combining Quantum Hamiltonian Descent with Augmented Lagrangian Method, analyzing its performance and resource requirements for constrained nonconvex problems.

Contribution

It introduces AL-QHD for constrained optimization, benchmarks its performance on test functions, and estimates quantum resource needs for practical power system applications.

Findings

AL-QHD effectively solves nonconvex constrained problems in tests.

Iterative refinement improves solution accuracy at fixed qubit cost.

Resource analysis indicates large-scale fault-tolerant quantum hardware is needed for practical ACOPF applications.

Abstract

Quantum Hamiltonian Descent (QHD) is a continuous optimization algorithm based on simulating a time-dependent quantum Hamiltonian whose potential energy encodes the objective function and whose kinetic energy promotes exploration through quantum interference and tunneling. While QHD is formulated for unconstrained optimization, many real-world optimization problems are constrained and highly nonconvex. In this paper, we benchmark AL-QHD, a hybrid framework that embeds QHD within the Augmented Lagrangian Method (ALM), thereby solving a sequence of unconstrained subproblems while using ALM to enforce constraints. We evaluate AL-QHD on standard nonconvex test functions and use iterative refinement to improve solution accuracy at fixed per-run qubit cost. We also perform a gate-based resource analysis on ACOPF-derived power system subproblems constructed from power-network data to estimate the quantum-computer scale required for practical applications. Resource estimates on Texas7k-derived ACOPF instances show steep hard-gate scaling, reaching $\sim 4.46 \times 10^7$ entangling gates in a NISQ-oriented model and $\sim 9.42 \times 10^8$ T gates in a fault-tolerant model at $\sim 5.3 \times 10^2$ active variables. These results suggest that AL-QHD is a useful framework for studying constrained nonconvex optimization with QHD, but that practical ACOPF-scale applications would likely require large-scale fault-tolerant quantum hardware.