Improved Quantum Algorithms for Fidelity Estimation
András Gilyén, Alexander Poremba
TL;DR
The paper tackles fidelity estimation F(ρ,σ) for mixed quantum states in high dimensions by introducing two provably efficient quantum approaches: a block-encoding/QSVT framework and a quantum spectral-sampling scheme. The block-encoding method leverages truncated, soft-thresholded versions of ρ and polynomial transforms to achieve ε-accurate fidelity with poly(r,1/ε) complexity in the purified-access model, while the spectral-sampling approach exploits eigenvalue sampling and Hadamard tests to reconstruct Λ= √ρ σ √ρ and compute F via Tr[√Λ_+], with explicit bounds that depend on the eigenvalue gap Δ and rank. The work also proves QSZK_HV-hardness for constant-precision fidelity estimation and establishes a polynomial lower bound on sample requirements, highlighting fundamental limitations. Together, these results advance fidelity estimation beyond tomography and heuristic methods, providing concrete quantum-algorithmic pathways for practical state comparison in noisy, high-dimensional quantum systems.
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.