Solving Constrained Optimization Problems Using Hybrid Qubit-Qumode Quantum Devices

Abstract: Variational Quantum Algorithms (VQAs) offer a promising approach for solving complex optimization problems on near-term quantum hardware. In this work, we show how hybrid qubit-qumode quantum devices can address Quadratic Unconstrained Binary Optimization (QUBO) problems using the Echoed Conditional Displacement Variational Quantum Eigensolver (ECD-VQE). Leveraging circuit quantum electrodynamics (cQED) architectures, we encode QUBO instances into multiple qumodes weakly coupled to a single qubit and extract solutions directly from photon-number measurements. As a demonstration, we apply ECD-VQE to the Binary Knapsack Problem and find that it outperforms the Quantum Approximate Optimization Algorithm (QAOA) implemented on conventional qubit circuits. These results highlight the potential of ECD-VQE for a broad class of NP-hard problems. We conclude that hybrid qubit-qumode platforms can efficiently realize variational ECD ansatze that would otherwise require deep circuits on standard qubit-based quantum computers, positioning them as promising candidates for constrained optimization in early fault-tolerant quantum computing.