## Abstract

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 - ϵ, the required number of qubits is t≥n+log2+δ22 ϵ ΔE2 . The parameter δ quantifies the error associated with the inexact eigenvector and/or the unitary operator, and ΔE 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 δ < ΔE, 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.

Original language | English (US) |
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Article number | 325303 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 55 |

Issue number | 32 |

DOIs | |

State | Published - Aug 12 2022 |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Statistics and Probability
- Modeling and Simulation
- Mathematical Physics
- General Physics and Astronomy