Constrained Quantum Optimization at Utility Scale: Application to the Knapsack Problem

TL;DR

This paper demonstrates the application of copula-QAOA, a quantum algorithm, to solve large-scale constrained optimization problems like the knapsack problem on IBM Quantum hardware, showing promising results close to classical solvers.

Contribution

It introduces and implements copula-QAOA for constrained optimization at scale, achieving the largest quantum demonstration of such problems with shallow mixers.

Findings

Cop-QAOA often outperforms greedy baselines.

Solutions are comparable or better than Gurobi in some instances.

Largest quantum demonstration of constrained optimization with 150 qubits.

Abstract

Constrained combinatorial optimization problems are challenging for quantum computing, particularly at utility-relevant scales and on near-term hardware. At the same time, these problems are of practical significance in industry; for example, the Unit Commitment (UC) problem in energy systems involves complex operational constraints. To address this challenge, we apply copula-QAOA (cop-QAOA), a hardware-efficient approach for constrained optimization to a single-period UC that can be reduced to a one-dimensional knapsack. Cop-QAOA biases the quantum state toward feasible solutions using constant-depth mixers and appropriately biased initial states. We implement our benchmark on problem instances that are confirmed to be hard for classical solvers such as Gurobi. Our results show that cop-QAOA often finds solutions better than a lazy greedy baseline and very close to, and in some instances surpasses, those obtained by Gurobi, with only a few QAOA rounds. This work presents the largest successful demonstration of the knapsack problem on IBM Quantum hardware using up to 150 qubits, and more generally, the largest demonstration of constrained combinatorial optimization where constraints are enforced via shallow mixers.