Efficient algorithm for fidelity estimation of two quantum states

TL;DR

This paper introduces an efficient quantum algorithm for estimating the fidelity between two quantum states, applicable to mixed states and higher dimensions, with potential resource savings.

Contribution

It proposes a novel quantum algorithm based on density matrix exponentiation and interferometry for fidelity estimation of mixed states.

Findings

Algorithm has time complexity $O( appa^2 N^2 / epsilon^7)$.

Effective for mixed states with commuting density matrices.

Applicable to both pure and mixed quantum states.

Abstract

The fidelity estimation between two quantum states is crucial for quantum computation and information science. However, an efficacious method for this, especially for mixed states and higher-dimensional density matrices, remains elusive. While there are many existing algorithms on computing the fidelity between two pure states, there is not much work on how to obtain the fidelity between two mixed states. Here, an efficient quantum algorithm for the fidelity estimation is proposed, based primarily on the density matrix exponentiation and interferometeric scheme for mixed states, with a time complexity of $O(\kappa^2N^2/\epsilon^7)$, where $N$ is the system size, $\kappa$ is the condition number of the density matrices and $\epsilon$ is a precision error. This algorithm may serve as a resource-efficient technique to deduce fidelity of any two (pure or mixed) unknown or known quantum states, when the density matrices of the quantum states commute with each other.