Whiteboard talk: Matrix perturbation theory for resource estimates of an early-fault-tolerant quantum algorithm for quantum chemistry

Non-orthogonal quantum eigensolver [U. Baek et al., PRX Quantum 4, 030307 (2023)] is a promising algorithm for solving the electronic structure problem of quantum chemistry using early-fault-tolerant quantum computers. In this algorithm, each element of the Hamiltonian and the overlap matrices, with respect to a basis of non-orthogonal states cleverly designed to capture both weak and strong electronic correlations, is evaluated from repetitions of a shallow quantum circuit. An important task, relevant to the economic cost of the algorithm, is to estimate the number of circuit repetitions required to reach the target accuracy in the eigenvalues, which translates to a problem in matrix perturbation theory. I will introduce the algorithm and discuss its relation to results from Stewart (1979) and Mathias and Li (2004) as well as my anticipations for achieving tighter bounds.