From VQE To SQD: Modern Quantum Algorithms For The Electronic Structure Problem

Abstract: This thesis investigates sampling-based quantum algorithms for electronic ground state energy estimation, focusing on Quantum-Selected Configuration Interaction (QSCI) and Sample-Based Quantum Diagonalization (SQD) as near-term alternatives to the Variational Quantum Eigensolver (VQE). Unlike VQE, which suffers from barren plateaus and high measurement costs, these methods avoid variational optimization by sampling Slater determinants from quantum hardware and performing diagonalization classically. The central contribution is the first analytical expression for the sampling bottleneck: the determinant-discovery step is mapped to the classical coupon-collector problem, yielding both an exact formula and a scalable lower-bound estimator for the number of measurements required to recover all determinants contributing to the ground state. Previous work only relied on numerical sampling and estimation. The analysis is validated through simulations, hardware-calibrated noisy studies, and execution on IBM's 127-qubit IBM Brisbane processor, demonstrating the dominant role of sampling efficiency in the near-term feasibility of QSCI and SQD. This work was carried out under the supervision of Dr. Mohamed Taha Rouabah.