Iterative quantum-phase-estimation protocol for shallow circuits

Abstract

Given ${N}_{\text{tot}}$ applications of a unitary operation, parametrized by an unknown phase, a phase-estimation protocol on a large-scale fault-tolerant quantum system can reduce the standard deviation of an estimate of the phase from scaling as $O[1/\sqrt{{N}_{\text{tot}}}]$ to scaling as $O[1/{N}_{\text{tot}}]$. Owing to the limited resources available to near-term quantum devices, protocols that do not entangle probes have been developed. Their mean absolute error scales as $O[log({N}_{\text{tot}})/{N}_{\text{tot}}]$. Here, we propose a two-step protocol for near-term phase estimation, with an improved error scaling. Our protocol's first step produces several low-standard-deviation estimates of $\ensuremath{\theta}$, within $\ensuremath{\theta}$'s parameter range. The second step iteratively homes in on one of these estimates. Our protocol achieves a mean-absolute-error scaling of $O[\sqrt{log(log{N}_{\text{tot}})}/{N}_{\text{tot}}]$ and a root-mean-square-error scaling of $O[\sqrt{log{N}_{\text{tot}}}/{N}_{\text{tot}}]$. Furthermore, we demonstrate a reduction in the constant scaling factor and the required circuit depths. This allows our protocol to outperform the asymptotically optimal quantum-phase-estimation algorithm for realistic values of ${N}_{\text{tot}}$.