Some error analysis for the quantum phase estimation algorithms

This paper is concerned with the phase estimation algorithm in quantum computing, especially the scenarios where (1) the input vector is not an eigenvector; (2) the unitary operator is approximated by Trotter or Taylor expansion methods; (3) random approximations are used for the unitary operator. We characterize the probability of computing the phase values in terms of the consistency error, including the residual error, Trotter splitting error, or statistical mean-square error. In the first two cases, we show that in order to obtain the phase value with error less or equal to 2(-n) and probability at least 1 - epsilon, the required number of qubits is t >= n + log (2 +delta(2)/2 epsilon Delta E-2). The parameter delta quantifies the error associated with the inexact eigenvector and/or the unitary operator, and Delta E characterizes the spectral gap, i.e., the separation from the rest of the phase values. This analysis generalizes the standard result (Cleve et al 1998 Phys. Rev X 11 011020; Nielsen and Chuang 2002 Quantum Computation and Quantum Information) by including these effects. More importantly, it shows that when delta < Delta E, the complexity remains the same. For the third case, we found a similar estimate, but the number of random steps has to be sufficiently large.