Quantum Seniority-based Subspace Expansion: Linear Combinations of Short-Circuit Unitary Transformations for the Electronic Structure Problem

TL;DR

The paper tackles the challenge of achieving chemical accuracy for electronic-structure problems on near-term quantum devices by reducing circuit depth and measurement overhead. It introduces Quantum SENiority-based Subspace Expansion (Q-SENSE), a hybrid quantum–classical framework that builds orthogonal basis states from seniority-symmetry considerations, enabling a tunable trade-off between quantum circuit complexity and the size of the classical subspace. By partitioning matrix-element measurements into efficiently computable classical fragments and a smaller quantum portion, and by using an extended swap-test formalism with constant-term optimization, Q-SENSE achieves substantial measurement savings while preserving accuracy. The authors demonstrate chemical accuracy for H$_2$O and N$_2$ across weak and strongly correlated regimes, comparing two operational modes (Variational Optimization and Perturbation Theory) that interpolate between VQE-like and CI-like behavior, and show favorable scaling of quantum resources and measurement costs. Overall, Q-SENSE provides a scalable, symmetry-exploiting pathway toward quantum advantage in electronic structure on current and near-future quantum hardware, with practical implications for handling strong correlation at reduced quantum costs.

Abstract

Quantum SENiority-based Subspace Expansion (Q-SENSE) is a hybrid quantum-classical algorithm that interpolates between the Variational Quantum Eigensolver (VQE) and Configuration Interaction (CI) methods. It constructs Hamiltonian matrix elements on a quantum device and solves the resulting eigenvalue problem classically. Unlike other expansion-based methods -- such as Quantum Subspace Expansion (QSE), Quantum Krylov Algorithms, and the Non-Orthogonal Quantum Eigensolver -- Q-SENSE introduces seniority operators as artificial symmetries to construct orthogonal basis states. This seniority-symmetry-based approach reduces one of the primary limitations of VQE on near-term quantum hardware -- circuit depth -- at the cost of measuring additional matrix elements. The artificial symmetries also reduce the number of Hamiltonian terms that must be measured, as only a small fraction of the terms couple basis states in different seniority subspaces. With all these merits, Q-SENSE offers a scalable and resource-efficient route to quantum advantage on near-term quantum devices and in the early fault-tolerant regime.

Figures (7)

• Figure 1: Illustration of a singlet CSF with two unpaired electrons ($\Omega=2$) and pair rotations.
• Figure 2: Orbital partitioning to virtual orbitals ($\mathcal{V}$), active orbitals (dashed box, $\mathcal{A} = \mathcal{A}_\text{occ} \cup \mathcal{A}_\text{virt}$), inactive orbitals ($\mathcal{I}$); blue and red arrows correspond to external and internal excitations.
• Figure 3: Quantum circuits (a) $\hat{S}_{ia}^{(1)}$, (b) $\hat{S}_{ijab}^{(2)}$, and (c) $\hat{S}_{ijab}^{(3)}$ applied to qubits corresponding to singly occupied orbitals to obtain seniority-$\Omega = 2$ and $\Omega = 4$ CSF states. The rotation gate $R_y$ in $\hat{S}_{ijab}^{(3)}$ is parameterized by $\theta_1 = -2\tan^{-1}(1/\sqrt{2})$.
• Figure 4: Quantum circuit implementing the rotation $\hat{U}_{ia}(\theta)$ generated by the compressed pair excitation in Eq. (\ref{['eqn:tap_pair_exc']}).
• Figure 5: Quantum circuit to prepare the states $\ket{\Phi_{\mu\nu}^{(\mathcal{Q})}} = (\ket{0}\ket{\phi_\mu^{(\mathcal{Q})}} + \ket{1}\ket{\phi_{\nu}^{(\mathcal{Q})}})/\sqrt{2}$, where $\hat{S}_\mu \in \{\hat{S}^{(0)}, \hat{S}_{ia}^{(1)}, \hat{S}_{ijab}^{(2)}, \hat{S}_{ijab}^{(3)}\}$ prepares the initial compressed CSF state and $n = |\mathcal{Q}|$. The operators $\hat{W}_\mu^{(c)}$ are compressed electron-pair rotations. A CSWAP network, consisting of $n$ CSWAP operations, is added to obtain the state $\ket{\Phi_{\mu\nu}^{(\mathcal{Q})}}$ on the first $n + 1$ qubits.