[Paper Review] Improved Quantum Algorithms for Fidelity Estimation

András Gilyén, Alexander Poremba|arXiv (Cornell University)|Mar 30, 2022

Quantum Computing Algorithms and Architecture21 citations

TL;DR

The paper presents new quantum algorithms for fidelity estimation with provable guarantees in the low-rank regime, leveraging block-encodings, QSVT, and spectral sampling, and proves hardness and sample-complexity lower bounds for general cases.

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.

Motivation & Objective

• Motivate the need for reliable fidelity estimation between quantum states, especially for high-dimensional mixed states.
• Develop efficient quantum algorithms with provable performance guarantees under a low-rank approximation assumption.
• Compare new algorithms to prior approaches and establish hardness and lower bounds for the general problem.

Proposed method

• Block-encoding based fidelity estimation using quantum singular value transformation (QSVT) to construct a block-encoding of a matrix related to sqrt(rho) sqrt(sigma).
• Use Hadamard tests and purified access to implement and combine block-encodings for rho, sigma, and their transformations.
• Introduce a soft-thresholded version rho_theta of rho to bound conditioning and enable efficient estimation.
• Apply quantum spectral sampling to sample eigenvalues/eigenvectors of rho and estimate off-diagonal matrix elements via Hadamard tests.
• Develop a non-uniform coupon-collector analysis to bound the number of samples needed to collect eigenvalues for spectral estimation.
• Provide a purified-access and a sampling-access workflow, with corresponding time/sample complexities.

Experimental results

Research questions

• RQ1 Can fidelity F(rho, sigma) be estimated efficiently when at least one state is approximately low-rank?
• RQ2 What are the time/sample complexities for purified-access versus sampling-access models for fidelity estimation?
• RQ3 Is fidelity estimation hard in general, and if so, under what frameworks or hardness classes?
• RQ4 How does truncation (soft-thresholding) of rho affect estimation accuracy and resource requirements?
• RQ5 What are realistic lower bounds on sample complexity for constant-precision fidelity estimation?

Key findings

• A block-encoding based fidelity estimation algorithm achieves poly(r, 1/ε) time and sample complexity in the purified-access model; complexity scales as Õ(r^{5/2}/ε^{5})(T_ρ+T_σ) (with further refinements in θ/delta parameters).
• A spectral sampling algorithm yields an ε-approximate fidelity estimate with Õ((T_ρ+T_σ)/(θ^{10.5} ε^4 Δ) + T_ρ/(θ^3 min{θ^3 ε, Δ}^3)) time in the purified-access model, with favorable low-rank regime behavior under truncation.
• The paper proves Fidelity estimation to any non-trivial constant additive accuracy is QSZK_HV-hard in general.
• Corollaries and propositions establish polynomial copy complexity lower bounds: at least Ω(r/δ) copies are needed for constant-precision estimation, and stronger lower bounds under specific settings.
• Compared to prior purified-access approaches, the block-encoding method improves dependence on rank and error, while spectral sampling offers complementary robustness in low-rank scenarios.
• The authors discuss concurrent work and emphasize that their spectral-sampling method, while slower in the worst case, benefits significantly from truncation in practical low-rank cases.