Quantum phase estimation with noisy qubits

We study how well small noisy quantum phase estimation circuits can estimate the ground-state energy of a Hamiltonian given an input 'starting' state with some overlap with the ground-state. We describe how one can estimate the ground-state energy from a series of quantum phase estimation experiments by an optimized and computationally feasible Bayesian analysis which includes a scalable modeling of the noise. We focus on quantum phase estimation experiments which consist of multiple rounds, each round coupling to a single ancilla qubit, which gradually project the input state onto a single eigenstate. To demonstrate the robustness and accuracy of our protocol, we numerically assess its performance for a few qubits, randomly picked Hamiltonians and starting states, on a simulation of superconducting quantum computing hardware. We show that the estimate of the ground-state energy and the classically-calculated variance of the posterior distribution of ground-state energies captures the uncertainty in learning the ground-state energy well.