Solving Constrained Optimization Problems Using Hybrid Qubit-Qumode Quantum Devices

TL;DR

This work addresses constrained optimization by recasting problems as QUBOs and solving them with a hybrid qubit–qumode quantum device using Echoed Conditional Displacement VQE (ECD-VQE). By encoding QUBO Hamiltonians across one qubit and two qumodes and reading out via photon-number measurements, the approach achieves high-quality solutions with substantially lower circuit depth than qubit-only implementations. In benchmarks on the Binary Knapsack Problem and multi-constraint instances, ECD-VQE delivers exact or near-exact solutions with dramatic resource advantages over QAOA and transpiled qubit circuits. The framework extends to chemically motivated tasks, notably active-space selection for multireference electronic structure, demonstrating broad applicability to NP-hard and chemistry-related optimization problems and suggesting strong potential for early fault-tolerant quantum computing with qubit–qumode platforms.

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.

Figures (14)

• Figure 1: A constrained optimization problem is reformulated as a QUBO instance and solved using a qubit--qumode device. The problem is encoded into a qubit--qumode Hamiltonian whose ground state corresponds to the optimal solution. This ground state is approximated via variational optimization of the circuit parameters, and the solution is read out through photon-number measurements on the qumodes.
• Figure 2: Schematic to implement quantum nondemolition (QND) approach for photon number measurements for two qumodes with computational measurements for the coupled qubit. Two microwave cavities are dispersively coupled to a coupler transmon qubit with a readout resonator for computational basis measurement of the coupler qubit. Each of the cavities are also coupled to an ancillary transmon qubit with a readout resonator which allows for photon number detection followed by the approach discussed in Ref. Wang2020vibronic.
• Figure 3: Hybrid one-qubit two-qumode circuit followed by measurements for computation of expectation values of a diagonal Hamiltonian as defined in Eq. (\ref{['eq: diag_ham_qubit']}). The circuit consists of echoed conditional displacement (ECD) qubit-qumode gates with one-qubit rotations, as discussed in Section \ref{['sec: vqe']}.
• Figure 4: Trial energy values defined in Eq. (\ref{['eq: vqe']}) at different ECD-VQE iterations while finding the ground state of $H_Q$ defined in Eq. (\ref{['eq: bkp_q7_example_ham']}). The horizontal black line represents the exact ground state energy. The circuit depth for the trial state is $N_d = 5$.
• Figure 5: Probabilities $S_{q, n, m} = | \braket{q, n, m | \psi} |^2$ at different numbers of iterations of the ECD-VQE method for the one-qubit two-qumode Hamiltonian $H_Q$ defined in Eq. (\ref{['eq: bkp_q7_example_ham']}). The histograms are split into two parts for better readability of the basis states. The circuit depth for the trial state is $N_d = 5$. The corresponding trial energy values are shown in Figure \ref{['fig: bkp_energies_lval2_nd5']}.