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.