Compactifying the Electronic Wavefunction II: Quantum Estimators for Spin-Coupled Generalized Valence Bond Wavefunctions Applied to H4

Valence-bond-based wavefunctions, such as the spin-coupled generalized valence bond (SCGVB) ansatz, provide compact and chemically interpretable descriptions of strong correlation. However, their non-orthogonal determinant structure poses a major challenge for quantum computing implementations. Although recent fermion-qubit mappings allow non-orthogonal orbitals to be encoded on qubit registers, the evaluation of overlap and Hamiltonian matrix elements remains a bottleneck on NISQ devices due to the need for ancilla qubits, controlled operations, and deep circuits. We present a measurement-driven quantum framework for evaluating these quantities in SCGVB wavefunctions. Instead of preparing the full wavefunction, we reformulate the problem in terms of vacuum expectation values of Pauli-string operators, enabling evaluation with shallow, ancilla-free circuits based on local Clifford rotations and computational-basis measurements. Unlike Hadamard-test-based approaches, this method avoids controlled operations by reducing the task to local Pauli measurements, yielding a low-depth strategy suitable for near-term devices. We demonstrate the framework on the H4 cluster along a dissociation pathway from square geometry to the separated-fragment limit, considering five nuclear configurations via quantum-circuit emulation. The overlap and Hamiltonian matrices agree well with classical Lowdin-based references, and Chirgwin-Coulson weights remain chemically consistent. These results highlight the robustness of the approach and its suitability as a NISQ-compatible building block for SCGVB-based quantum algorithms.