Qumode Subspace VQE: Hybrid Quantum Simulation

• Qumode Subspace VQE (QSS-VQE) is a hybrid algorithm that embeds qubit Hamiltonians into the infinite-dimensional Fock space of bosonic modes for efficient molecular eigenstate simulations.
• It leverages displacement and SNAP gates to construct variational ansätze, achieving high expressivity with lower circuit depth and reduced quantum resource requirements.
• The method demonstrates superior performance in excited-state and bosonic model simulations, paving the way for advanced quantum simulations in chemistry and materials science.

The Qumode Subspace Variational Quantum Eigensolver (QSS-VQE) is a hybrid quantum–classical algorithm for molecular ground and excited state computations that employs bosonic qumodes to realize a subspace-search variational scheme. QSS-VQE exploits the infinite-dimensional Fock space of bosonic modes to embed qubit-encoded Hamiltonians, constructing hardware-native variational ansätze using universal bosonic gate sets (such as displacement and SNAP gates) and optimizing energy expectations of orthonormal Fock-basis–labeled trial states. This approach extends and generalizes the subspace-search paradigms from qubit-based SSVQE to qubit–qumode architectures, delivering resource-efficient, highly expressive state preparation for challenging excited-state quantum chemistry and bosonic model Hamiltonians (Dutta et al., 5 Sep 2025).

1. Theoretical Foundation and Subspace Strategy

QSS-VQE formalizes the search for multiple eigenstates of a given Hamiltonian $H_Q$ by constructing an orthonormal subspace in the Hilbert space of a bosonic mode, truncated to accommodate $L=2^{N_Q}$ Fock states for $N_Q$ qubits. The Fock states $|n\rangle$, $n=0,1,...,L-1$, provide a natural computational basis, with trial states prepared as

$$|\psi_n(\theta)\rangle = \mathcal{U}(\alpha^{\psi_n}, \theta^{\psi_n})|n\rangle$$

where $\mathcal{U}(\alpha, \theta)$ denotes a parameterized sequence of hardware-native gates (typically composed of SNAP and displacement operations).

The cost function to be minimized is a weighted sum over state energies: $$F(\theta) = \sum_{n} w_n E_n, \quad E_n = \langle \psi_n(\theta) | H_Q | \psi_n(\theta)\rangle.$$ The variational optimization aligns the subspace $\mathrm{span}\{\mathcal{U}(\cdot)|n\rangle\}$ with the low-lying eigenspace of $H_Q$. For excited-state targeting, weighting schemes ($w_0 \gg w_1 \gg ...$) ensure that all levels up to a chosen $k$ are found simultaneously, generalizing the weighted SSVQE principle (Nakanishi et al., 2018, Dutta et al., 5 Sep 2025). Owing to the unitarity of $\mathcal{U}$, orthogonality is preserved without ancilla–based overlap measurements.

2. Mapping Qubit Hamiltonians to Qumode Fock Space

After transforming the molecular Hamiltonian (typically in fermionic second quantized form) to a qubit Hamiltonian via Jordan–Wigner or Bravyi–Kitaev mapping, QSS-VQE embeds it into the bosonic Fock space through a binary–to–integer mapping: $$|q_1,\ldots, q_{N_Q}\rangle \mapsto |n\rangle,\quad n = 2^{N_Q-1}q_1 + \cdots + 2^0 q_{N_Q}.$$ Each Pauli term $P_j$ in $H_Q$ is associated with a photon–number–indexed operator in the Fock representation. Expectation values $\langle \psi_n | P_j | \psi_n \rangle$ are extracted via photon number–resolved measurements, natively supported on cQED hardware (Dutta et al., 5 Sep 2025).

For bosonic models such as displaced harmonic oscillators,

$$H_B = \frac{1}{2}p^2 + \frac{1}{2}(x - \sqrt{2}\alpha)^2,$$

the eigenstates $D(\alpha)|n\rangle$ (where $D(\alpha)$ is the displacement operator) are prepared directly by qumode hardware (Dutta et al., 5 Sep 2025). The qubit embedding formalism applies to either fermionic (molecular) or bosonic Hamiltonians, depending on the application domain.

3. Variational Ansätze and Hardware-Native Gate Sets

The central variational building blocks are the displacement,

$$D(\alpha) = \exp\left[\alpha a^\dagger - \alpha^*a\right],$$

and SNAP (Selective Number-dependent Arbitrary Phase),

$$S(\theta) = \exp\left[i\sum_n \theta_n |n\rangle\langle n|\right],$$

gates. The ansatz consists of repeating $d$ layers of these gates: $$\mathcal{U}_{SD}(\alpha, \theta) = S(\theta)D(\alpha),$$ and the full variational circuit is

$$U(\alpha^\psi, \theta^\psi) = \prod_{l=1}^d S(\theta_l) D(\alpha_l).$$

Each trial state $|\psi_n(\theta)\rangle$ is initialized in Fock state $|n\rangle$ and then evolved by the same $\mathcal{U}$ sequence. These gates are implemented natively in cQED architectures, yielding low-depth, high-expressivity circuits—especially beneficial for excited-state manifolds that exhibit non-trivial structure in Hilbert space (Dutta et al., 5 Sep 2025).

4. Numerical Performance Benchmarks

Benchmarks on electronic structure problems and bosonic models show the following:

• Molecules (dihydrogen, cytosine): For H$_2$, using three Fock-basis states and depth $D=4$ with weights $w=(1.0,\,0.9,\,0.8)$, QSS-VQE matches exact diagonalization within chemical accuracy across bond distances. At a conical intersection in cytosine (after symmetry reduction to $N_Q=4$), QSS-VQE (D=1) resolves near-degenerate excited states at accuracy surpassing that of qubit-based SSVQE (D=10) (Dutta et al., 5 Sep 2025).
• Bosonic Model Hamiltonian: For a weakly displaced oscillator $(\alpha \leq 0.4)$, using only a single displacement gate, QSS-VQE achieves significantly lower energy error than fixed-depth qubit-based variational circuits, leveraging the compactness of bosonic state transformations (Dutta et al., 5 Sep 2025).

These results affirm that for certain regimes and systems, the bosonic expressivity and hardware-native gate access in QSS-VQE provide superior performance at lower circuit depths and reduced quantum resource requirements.

5. Quantum Resource Efficiency and Scalability

QSS-VQE benefits from the exponentially large Hilbert space of a single bosonic mode: representing $N_Q$ qubits requires only a single qumode truncated at $L=2^{N_Q}$, rather than $N_Q$ physical qubits. This enables compact Hilbert-space representations and reduces total qubit count. The native implementation of elementary bosonic operations (displacement, SNAP, number measurement) further reduces circuit depth and aggregate two-qubit gate count relative to conventional all-qubit-based VQE/SSVQE circuits of comparable expressivity (Dutta et al., 5 Sep 2025).

Measurement of state energies is achieved by photon number–resolved readout, with no need for tomographic ancillae or swap tests for subspace orthogonality. The mapping enables reuse of established Pauli measurement groupings and adaptive grouping strategies.

Additionally, the algorithm demonstrates flexibility for extension to multi-qumode devices, enabling further compactification and parallelization for larger molecular or many-body systems.

6. Positioning within the Family of Subspace and Hybrid VQE Algorithms

QSS-VQE extends and generalizes the SSVQE framework (Nakanishi et al., 2018) by exploiting the Fock subspace in continuous-variable hardware. Its design is closely related to weighted SSVQE and non-orthogonal subspace VQE methods (Nakanishi et al., 2018, Hong et al., 2023, Huggins et al., 2019), in which trial states spanning a low-energy subspace are simultaneously prepared and optimized to target ground and excited states via unitary mappings that preserve orthogonality. The algorithm integrates naturally with hybrid quantum–classical post-processing (e.g., diagonalization of projected Hamiltonians), similar to other subspace expansion or Krylov-based approaches (Huggins et al., 2019, Getelina et al., 14 Apr 2024, Patel et al., 1 Sep 2025).

The QSS-VQE architecture is also well suited to the inclusion of advanced optimization strategies (e.g., constraint/tabu search (Wakaura et al., 2021), block-wise optimization (Slattery et al., 2021), and circuit-depth–trading subspace schemes (Manrique et al., 2020)), and recent extensions support efficient parameter initialization and resource allocation consistent with generator-informed and subspace-optimized methods (Patra et al., 17 Apr 2025).

7. Implications and Outlook

Qumode Subspace VQE is highlighted as a promising strategy for quantum simulation of complex molecular excited-state features, ultrafast photochemistry, and bosonic many-body systems where Hilbert space size or strong correlation renders purely qubit-based methods challenging. Its hardware-compatibility with cQED and photonic platforms, together with observed gains in expressivity, accuracy, and resource efficiency (especially in low–circuit-depth and degenerate-state scenarios), suggest that QSS-VQE will be a critical component in the advancement of near- and mid-term quantum simulation of chemistry and materials science (Dutta et al., 5 Sep 2025). Future directions include generalization to multi-qumode registers, adaptive subspace selection, and integration with error-mitigated or fault-tolerant continuous-variable quantum computation architectures.