A Critical Assessment of the Sample-Based Quantum Diagonalization for Heisenberg and Hubbard Models

Abstract: Sample-based quantum diagonalization (SQD) constructs subspaces from computational-basis configurations obtained via measurements of a quantum state, with the goal of approximating low-energy eigenspaces of many-body Hamiltonians. The effectiveness of this approach relies on the assumption that physically relevant states admit a compact representation in the computational basis. We investigate this assumption by analyzing SQD subspaces constructed directly from configurations of exact ground states of Heisenberg and Hubbard model lattices. By eliminating state-preparation and measurement inefficiencies, we isolate the intrinsic configuration-space structure of the wavefunction. We determine the minimal number of configurations required to reproduce the ground-state energy within fixed accuracy thresholds and find that this number grows exponentially with the system size. Notably, this scaling persists even under optimal inclusion of configurations in order of decreasing probability, demonstrating that it originates from intrinsic delocalization of the wavefunction rather than sampling inefficiencies. Our results indicate that SQD effectively probes the configuration-space entropy but faces fundamental scalability limitations for these models.