Subspace Selected Variational Quantum Configuration Interaction with a Partial Walsh Series

Estimating the ground-state energy of a quantum system is one of the most promising applications for quantum algorithms. Here we propose a variational quantum eigensolver (VQE) \emph{Ansatz} for finding ground state configuration interaction (CI) wavefunctions. We map CI for fermions to a quantum circuit using a subspace superposition, then apply diagonal Walsh operators to encode the wavefunction. The algorithm can be used to solve both full CI and selected CI wavefunctions, resuling in exact and near-exact solutions for electronic ground states. Both the subspace selection and wavefunction \emph{Ansatz} can be applied to any Hamiltonian that can be written in a qubit basis. The algorithm bypasses costly classical matrix diagonalizations, which is advantageous for large-scale applications. We demonstrate results for several molecules using quantum simulators and hardware.

💡 Research Summary

The paper introduces a novel variational quantum eigensolver (VQE) ansatz designed to prepare configuration‑interaction (CI) wavefunctions directly on a quantum processor, thereby avoiding costly classical diagonalization of the Hamiltonian. The method consists of two main stages. First, a superposition is created over a selected subspace of Slater determinants that are physically relevant (e.g., those conserving particle number and spin). This subspace can be prepared either with a Dicke‑state circuit that scales as O(r N) CNOT gates (r = number of qubits, N = number of electrons) or with a quantum‑walk based circuit that scales as O(r D) CNOT gates (D = number of selected determinants). Second, a diagonal operator encoding the CI coefficients is applied. Because a purely diagonal operator is generally non‑unitary, the authors introduce a single ancilla qubit and construct a dilated unitary U = Y ∏_{j odd} exp(i a_j ŵ_j), where ŵ_j are Walsh operators and a_j are Walsh coefficients derived from the CI amplitudes c_k via a Walsh‑Fourier transform: a_j = 2^{-r} Σ_k c_k (−1)^{⟨k·j⟩}. The circuit proceeds as follows: (i) prepare the subspace state |S⟩, (ii) apply a Hadamard to the ancilla, (iii) apply U on the combined system, (iv) apply a second Hadamard to the ancilla and measure it. If the ancilla collapses to |0⟩, the desired CI state |Ψ⟩ = Σ_k c_k|ψ_k⟩ is obtained; if it collapses to |1⟩, the sign‑flipped counterpart appears. Both outcomes occur with exactly 50 % probability, making the algorithm intrinsically probabilistic but with a fixed success rate.

The Walsh ansatz requires only CNOT and single‑qubit R_z rotations; its gate count scales linearly with the number of determinants D, while the number of variational parameters scales as O(D log D). To keep the parameter count manageable, the authors oversample the Walsh basis by selecting O(D log D) functions at random from the full set, then use QR decomposition to obtain a full‑rank Walsh‑Fourier transform (WFT). Numerical experiments show that selecting roughly half of the oversampled set already yields a full‑rank transform with >90 % probability, confirming the theoretical O(1 − D^{−1}) success scaling.

The authors benchmark the approach on several molecular systems. For H₂ and a linear H₆ chain in minimal STO‑6G basis, they prepare the particle‑ and spin‑conserving subspace via quantum walk and achieve dissociation curves that match exact full CI (FCI) results within chemical accuracy (1.6 mHa) on both a noisy IBM Torino device and a noiseless simulator. For H₂O in a 6‑31G basis, they select 16 determinants (44 circuit parameters) and demonstrate that the oversampled Walsh ansatz reproduces the FCI energy to within 10⁻⁴ Ha. They also explore the trade‑off between oversampling rate and rank‑deficiency, presenting statistical data for molecules ranging from LiH to H₈, confirming that the required oversampling fraction decreases as system size grows.

Resource analysis reveals that subspace preparation costs O(r N) or O(r D) CNOTs, while the Walsh block adds O(D) CNOTs and D R_z rotations. Compared with recent ADAPT‑VQE circuits, the overall gate count and parameter count are comparable or lower, yet the ansatz avoids over‑parameterization of the chosen subspace, mitigating barren‑plateau issues. Moreover, because the Walsh coefficients are directly linked to the CI amplitudes, the optimization landscape is physically motivated and less prone to spurious local minima.

In summary, the work combines (1) efficient quantum preparation of a physically motivated determinant subspace, (2) a diagonal Walsh‑based encoding of CI coefficients via a dilated unitary, and (3) a probabilistic but fixed‑success ancilla measurement scheme. This hybrid approach eliminates the need for classical Hamiltonian diagonalization, scales favorably with system size, and demonstrates practical accuracy on current NISQ hardware, offering a promising pathway toward scalable quantum chemistry simulations.