Qumode-Based Variational Quantum Eigensolver for Molecular Excited States

TL;DR

This work introduces QSS-VQE, a bosonic-qumode–based variational algorithm for computing molecular excited states by embedding the electronic Hamiltonian into a qumode Fock space and using a universal SNAP–displacement ansatz. Energies are evaluated via photon-number measurements and a shared variational unitary across an orthonormal set of input states, following the SSQVE/S[SVQE] framework. The authors demonstrate competitive accuracy on dihydrogen and a conical intersection in cytosine, and show that simple qumode gates can outperform deeper qubit-based circuits on model Hamiltonians, highlighting a potential resource advantage of bosonic quantum computation. These results underscore the potential of bosonic degrees of freedom to enable efficient excited-state simulations and motivate future multi-qumode, hardware-native implementations on cQED platforms.

Abstract

We introduce the Qumode Subspace Variational Quantum Eigensolver (QSS-VQE), a hybrid quantum-classical algorithm for computing molecular excited states using the Fock basis of bosonic qumodes in circuit quantum electrodynamics (cQED) devices. This approach harnesses the native universal gate sets of qubit-qumode architectures to construct highly expressive variational ansatze, offering potential advantages over conventional qubit-based methods. In QSS-VQE, the electronic structure Hamiltonian is first mapped to a qubit representation and subsequently embedded into the Fock space of bosonic qumodes, enabling efficient state preparation and reduced quantum resource requirements. We demonstrate the performance of QSS-VQE through simulations of molecular excited states, including dihydrogen and a conical intersection in cytosine. Additionally, we explore a bosonic model Hamiltonian to assess the expressivity of qumode gates, identifying regimes where qumode-based implementations outperform purely qubit-based approaches. These results highlight the promise of leveraging bosonic degrees of freedom for enhanced quantum simulation of complex molecular systems.

Figures (6)

• Figure 1: Full circuit for computing the expectation value of a Pauli word with respect to a qumode state. After state preparation of Fock basis state $\ket{n}$, the expectation value is computed by applying a set of SNAP-displacement gates followed by photon number measurements as discussed in Section \ref{['sec: exp_val']}.
• Figure 2: Comparison between the energies from exact diagonalization (dashed lines) and subspace VQE (circles) as defined in Section \ref{['sec: subspace_vqe_qumode']} for the ground and first two excited states of the dihydrogen molecule in STO-3G basis. The circuit depth for the subspace VQE SNAP-displacement ansatz is $D = 4$ with the weight parameters $\textbf{w} = ( 1.0, 0.9, 0.8 )$.
• Figure 3: Comparison between different qubit-based excited state methods and the subspace VQE method as defined in Section \ref{['sec: subspace_vqe_qumode']} for the ground and first two excited states of dihydrogen molecules in STO-3G basis. The vertical axis plots the average error function defined in Eq. (\ref{['eq: error_sum_function']}). The circuit depth for the subspace VQE with SNAP-displacement ansatz for the trial states is $D = 4$ with the weight parameters $\textbf{w} = ( 1.0, 0.9, 0.8 )$.
• Figure 4: Comparison of exact energies with those obtained from qubit-based and qumode-based SSVQE methods. The qubit-based subspace VQE uses a TwoLocal ansatz with circuit depth $D = 10$, while the qumode-based QSS-VQE employs a SNAP-displacement ansatz with circuit depths up to $D = 3$. Both methods use uniform weight parameters $\mathbf{w} = (1.0, 1.0, 1.0)$ in the cost function.
• Figure 5: Relative errors of the trial state energies, $\braket{\psi_n | H_Q | \psi_n}$, compared to the exact eigenvalues of the mapped Hamiltonian $H_Q$ (defined in Section \ref{['sec: model_qubit_hams']}), as a function of $\alpha$. Trial states are defined as $\ket{\psi_n} = D(\alpha) \ket{n}$, where $D(\alpha)$ is a displacement operator and $\ket{n}$ denotes the $n$-th Fock state. Results are shown for the ground state and the first two excited states using $N_Q = 3$ qubits.