CANOE: Classically Assisted Non-Orthogonal Eigensolver

Current, noisy quantum hardware limits circuit width and depth, so a clear advantage in quantum chemistry —specifically for ground-state energies—has not yet been demonstrated. However, as hardware steadily improves, the ability to prepare and sample from complex quantum states creates opportunities for quantum advantage. We propose and analyze CANOE, a Classically Assisted Non-Orthogonal Eigensolver: a generalized-eigenvalue method that hybridizes a large number of carefully chosen simple, classical states with a small number of complex quantum states which are feasible on current devices. In simulation within a truncated Hilbert space for a chromium with 14 electrons in 38 orbitals, fewer than around ten quantum Krylov states reach chemical accuracy with the assistance of 1.4 M classical determinants. We thoroughly analyze the cost of obtaining the necessary overlap data using multiple methods, including Hadamard tests, classical shadows, and a histogram method. We also implemented a classical non-orthogonal eigensolver that stabilizes near-singular overlaps between classical and quantum bases via the use of deflation in a Schur-complement space. These results indicate that hybrid classical–quantum ansatzs can substantially improve electronic-structure accuracy within near-term resources.