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Given $N_{\textrm{tot}}$ applications of a unitary operation with an unknownphase $\theta$, a large-scale fault-tolerant quantum system can {reduce} anestimate's {error} scaling from $\mathcal{O} \left[ 1 / \sqrt{N_{\textrm{tot}}}\right]$ to $\mathcal{O} \left[ 1 / {N_{\textrm{tot}}} \right]$. Owing to thelimited resources available to near-term quantum devices, entanglement-freeprotocols have been developed, which achieve a $\mathcal{O} \left[\log(N_{\textrm{tot}}) / N_{\textrm{tot}} \right]$ {mean-absolute-error}scaling. Here, we propose a new two-step protocol for near-term phaseestimation, with an improved {error} scaling. Our protocol's first stepproduces several low-{standard-deviation} estimates of $\theta $, within$\theta$'s parameter range. The second step iteratively hones in on one ofthese estimates. Our protocol's {mean absolute error} scales as $\mathcal{O}\left[ \sqrt{\log (\log N_{\textrm{tot}})} / N_{\textrm{tot}} \right]$.Furthermore, we demonstrate a reduction in the constant scaling factor and therequired circuit depths: our protocol can outperform the asymptotically optimalquantum-phase estimation algorithm for realistic values of $N_{\textrm{tot}}$.

An iterative quantum-phase-estimation protocol for near-term quantum hardware