  Quantum phase estimation (QPE) is a key quantum algorithm, which has been
widely studied as a method to perform chemistry and solid-state calculations on
future fault-tolerant quantum computers. Recently, several authors have
proposed statistical alternatives to QPE that have benefits on early
fault-tolerant devices, including shorter circuits and better suitability for
error mitigation techniques. However, practical implementations of the
algorithm on real quantum processors are lacking. In this paper we practically
implement statistical phase estimation on Rigetti's superconducting processors.
We specifically use the method of Lin and Tong [PRX Quantum 3, 010318 (2022)]
using the improved Fourier approximation of Wan et al. [PRL 129, 030503
(2022)], and applying a variational compilation technique to reduce circuit
depth. We then incorporate error mitigation strategies including zero-noise
extrapolation and readout error mitigation with bit-flip averaging. We propose
a simple method to estimate energies from the statistical phase estimation
data, which is found to improve the accuracy in final energy estimates by one
to two orders of magnitude with respect to prior theoretical bounds, reducing
the cost to perform accurate phase estimation calculations. We apply these
methods to chemistry problems for active spaces up to 4 electrons in 4
orbitals, including the application of a quantum embedding method, and use them
to correctly estimate energies within chemical precision. Our work demonstrates
that statistical phase estimation has a natural resilience to noise,
particularly after mitigating coherent errors, and can achieve far higher
accuracy than suggested by previous analysis, demonstrating its potential as a
valuable quantum algorithm for early fault-tolerant devices.
