Grover's search with faults on some marked elements

TL;DR

This paper investigates Grover's quantum search algorithm's robustness when some marked elements are faulty, demonstrating it can still find non-faulty marked items efficiently and analyzing its limiting behavior under such conditions.

Contribution

It introduces a model with faulty and non-faulty marked elements and shows that Grover's algorithm can find non-faulty items in optimal query complexity, analyzing its limiting state.

Findings

Grover's algorithm finds non-faulty marked items in O(√N) queries despite faults.

The paper characterizes the limiting state of the algorithm under fault conditions.

Faulty elements do not prevent the algorithm from achieving quantum speed-up.

Abstract

Grover's algorithm is a quantum query algorithm solving the unstructured search problem of size $N$ using $O(\sqrt{N})$ queries. It provides a significant speed-up over any classical algorithm \cite{Gro96}. The running time of the algorithm, however, is very sensitive to errors in queries. It is known that if query may fail (report all marked elements as unmarked) the algorithm needs $\Omega(N)$ queries to find a marked element \cite{RS08}. \cite{AB+13} have proved the same result for the model where each marked element has its own probability to be reported as unmarked. We study the behavior of Grover's algorithm in the model where the search space contains both faulty and non-faulty marked elements. We show that in this setting it is indeed possible to find one of non-faulty marked items in $O(\sqrt{N})$ queries. We also analyze the limiting behavior of the algorithm for a large number of steps and show the existence and the structure of limiting state $\rho_{lim}$.