Norm Inequality for Perturbed Quantum Evolutions and Its Application to Grover's Algorithm

TL;DR

This paper derives a general norm inequality to bound the effects of coherent control errors on quantum state evolution and applies it to analyze the robustness of Grover's algorithm against such errors.

Contribution

It introduces a broad, assumption-free inequality for quantifying deviations caused by coherent perturbations in quantum systems and applies it to improve understanding of quantum algorithm robustness.

Findings

Provides explicit bounds on state deviation due to coherent errors.

Characterizes how errors scale with algorithm runtime and error strength.

Offers guidelines for designing error-resilient quantum protocols.

Abstract

We investigate the impact of coherent control errors on quantum state evolution by deriving a general norm inequality based on Gronwall's lemma. This inequality provides an explicit upper bound on the deviation between an ideal quantum state and one subject to arbitrary coherent perturbations, including both time-dependent and time-independent cases. The framework is broadly applicable, requiring no assumptions about the detailed structure of the perturbation and full dynamics of the quantum system. We apply this approach to analyze the robustness of Grover's search algorithm in the presence of coherent errors. Our results characterizes quantitative scaling relations between the error strength, algorithm runtime, and success probability, offering practical guidelines for the design of resilient quantum protocols. We further compare the effects of time-dependent and time-independent perturbations, showing the distinctive ways in which coherent errors accumulate in quantum dynamics.