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Quantum phase estimation is the workhorse behind any quantum algorithm and apromising method for determining ground state energies of strongly correlatedquantum systems. Low-cost quantum phase estimation techniques make use ofcircuits which only use a single ancilla qubit, requiring classicalpost-processing to extract eigenvalue details of the system. We investigatechoices for phase estimation for a unitary matrix with low-depth noise-free ornoisy circuits, varying both the phase estimation circuits themselves as wellas the classical post-processing to determine the eigenvalue phases. We work inthe scenario when the input state is not an eigenstate of the unitary matrix.We develop a new post-processing technique to extract eigenvalues from phaseestimation data based on a classical time-series (or frequency) analysis andcontrast this to an analysis via Bayesian methods. We calculate the variance inestimating single eigenvalues via the time-series analysis analytically,finding that it scales to first order in the number of experiments performed,and to first or second order (depending on the experiment design) in thecircuit depth. Numerical simulations confirm this scaling for both estimators.We attempt to compensate for the noise with both classical post-processingtechniques, finding good results in the presence of depolarizing noise, butsmaller improvements in $9$-qubit circuit-level simulations of superconductingqubits aimed at resolving the electronic ground-state of a $H_4$-molecule.

Quantum phase estimation of multiple eigenvalues for small-scale (noisy) experiments