Abstract

We develop a framework that enables direct and meaningful comparison of two early fault-tolerant methods for the computation of eigen energies, namely quantum Krylov subspace diagonalization (QKSD) and statistical phase estimation (SPE), within which both methods use expectation values of Chebyshev polynomials of the Hamiltonian as input. For QKSD we analyze methods for distributing shots and the importance of ensuring sufficient non-linearity of states spanning the Krylov space. For SPE we improve rigorous error-bounds, achieving a factor 1.5 reduction of circuit depth. We provide insights into the scalability of these methods by computing the maximum circuit depth, linearly related to the Chebyshev degree, and the respective number of repetitions required for the computation of molecular energies by means of density matrix renormalization group-enabled computations for active spaces with up to 54 electrons in 36 orbitals.