Solving Constrained Optimization Problems Using Hybrid Qubit-Qumode Quantum Devices

TL;DR

This paper introduces a hybrid qubit-qumode quantum approach using ECD-VQE to efficiently solve QUBO problems, outperforming traditional methods and extending to chemistry applications.

Contribution

It demonstrates a novel hybrid quantum algorithm with shallower circuits for constrained optimization, applicable to NP-hard and chemistry problems.

Findings

ECD-VQE outperforms QAOA on the Binary Knapsack Problem

Higher-quality solutions achieved with fewer resources

Method extends to active-space selection in electronic structure

Abstract

Variational Quantum Algorithms (VQAs) provide a promising framework for tackling complex optimization problems on near-term quantum hardware. Here, we demonstrate that hybrid qubit--qumode quantum devices offer an efficient route to solving 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 across multiple qumodes weakly coupled to a single qubit and extract binary solutions directly from photon-number measurements. We apply ECD-VQE to the Binary Knapsack Problem and show that it outperforms the Quantum Approximate Optimization Algorithm (QAOA) implemented on conventional qubit circuits, achieving higher-quality solutions with dramatically fewer resources. We also demonstrate that ECD-VQE can be extended to chemically motivated tasks such as active-space selection for multireference electronic structure methods. These results highlight the utility of hybrid qubit-qumode platforms for a broad class of NP-hard and chemistry-related optimization problems, and demonstrate that variational ECD ansatz can realize expressive state preparation with significantly shallower circuits than qubit-only architectures, positioning qubit-qumode gates as compelling candidates for constrained optimization in early fault-tolerant quantum computing.