Improved Quantum Algorithms for Fidelity Estimation

TL;DR

This paper introduces new quantum algorithms for fidelity estimation that are efficient when at least one state is low-rank, providing performance guarantees and highlighting the problem's inherent complexity in general.

Contribution

The work develops novel quantum algorithms for fidelity estimation with provable guarantees, improving efficiency over prior methods especially for low-rank states.

Findings

Algorithms achieve efficient fidelity estimation for low-rank states.

Proves fidelity estimation is hard in the general case with high-dimensional states.

Establishes lower bounds on sample complexity depending on dimension.

Abstract

Fidelity is a fundamental measure for the closeness of two quantum states, which is important both from a theoretical and a practical point of view. Yet, in general, it is difficult to give good estimates of fidelity, especially when one works with mixed states over Hilbert spaces of very high dimension. Although, there has been some progress on fidelity estimation, all prior work either requires a large number of identical copies of the relevant states, or relies on unproven heuristics. In this work, we improve on both of these aspects by developing new and efficient quantum algorithms for fidelity estimation with provable performance guarantees in case at least one of the states is approximately low-rank. Our algorithms use advanced quantum linear algebra techniques, such as the quantum singular value transformation, as well as density matrix exponentiation and quantum spectral sampling. As a complementary result, we prove that fidelity estimation to any non-trivial constant additive accuracy is hard in general, by giving a sample complexity lower bound that depends polynomially on the dimension. Moreover, if circuit descriptions for the relevant states are provided, we show that the task is hard for the complexity class called (honest verifier) quantum statistical zero knowledge via a reduction to a closely related result by Watrous.