Compactifying the Electronic Wavefunction II: Quantum Estimators for Spin-Coupled Generalized Valence Bond Wavefunctions

Abstract

We present a measurement-driven quantum framework for evaluating overlap and Hamiltonian matrix elements in spin-coupled generalized valence bond (SCGVB) wavefunctions. The approach targets a central difficulty of nonorthogonal valence-bond methods: estimating matrix elements between distinct, generally nonorthogonal configuration state functions. Rather than preparing the full wavefunction on quantum hardware, we reformulate the required quantities as vacuum expectation values of Pauli-string operators that can be accessed using shallow, ancilla-free circuits composed of local Clifford rotations and computational-basis measurements. In contrast to Hadamard-test-based matrix-element estimation, this construction avoids ancilla qubits and controlled operations by reducing the problem to local Pauli measurements. This separates the algebraic construction of the SCGVB problem from the measurement task executed on the quantum register and yields a low-depth strategy compatible with near-term architectures. We demonstrate the framework on square and rectangular H4 using quantum-circuit emulation, where the resulting overlap and Hamiltonian matrices reproduce classical Lowdin-based references with good accuracy across the geometries considered, and where derived Coulson-Chirgwin weights remain chemically consistent. These results support the feasibility of measurement-based quantum assistance for nonorthogonal SCGVB expansions and provide a practical route for incorporating quantum measurements into valence-bond electronic-structure workflows.

Figures (4)

• Figure 1: Branching diagram of spin couplings for $N=1,\dots,14$ electrons. Each node at coordinates $(N,S)$ is labeled by $f_S^N = \dfrac{(2S+1)N!}{(\frac{N}{2}+S+1)!\,(\frac{N}{2}-S)!}$, which gives the number of linearly independent spin eigenfunctions $\Theta_{S,M;k}^N$ for that electron number and total spin McWeeny1992Pauncz1979cooper_rumer. Edges indicate the allowed couplings when a spin-$\tfrac{1}{2}$ electron is added, connecting $(N,S)$ to $(N+1,S\pm\tfrac{1}{2})$. Thus, the integer at each node equals the range of the index $k$ in Eq. (\ref{['eq1']}), and therefore the number of spin-coupled structures entering the SCGVB expansion for that $(N,S)$. The color gradient (from light pink to light green) indicates increasing total spin $S$. Adapted from R. McWeeny McWeeny1992 and R. Pauncz Pauncz1979.
• Figure 2: Quantum circuit used to evaluate a single Pauli-pair contribution $\langle 0^n| P_{j,b} Q_{i,a} |0^n\rangle$. Since each Pauli string is a tensor product of single-qubit Clifford gates. The circuit consists only of local operations followed by computational basis measurement.
• Figure 3: Quantum circuit used to measure all Pauli operators $P_k \in G_\ell$ belonging to a qubit-wise commuting group. A single local basis rotation $U_\ell$ simultaneously diagonalizes every operator in $G_\ell$, after which all qubits are measured in the computational basis. The resulting bitstring statistics are reused to obtain $\langle P_k\rangle_0$ for all $P_k\in G_\ell$, which are then assembled to recover $H_{ij}$.
• Figure 4: Chirgwin--Coulson weights $w_1$ and $w_2$ for the two SCGVB structures of H$_4$ along the H$_4 \rightarrow$ H$_2$ + H$_2$ dissociation coordinate.