Time complexity of a monitored quantum search with resetting

TL;DR

This paper analyzes a quantum search algorithm with feedback and resetting, showing how it affects search time complexity and under what conditions it can outperform traditional Grover's algorithm.

Contribution

It introduces a quantum search model with feedback and resetting, analyzing its time complexity and identifying conditions for potential speedup over Grover's algorithm.

Findings

Optimal parameters for search time identified

Quantum speedup possible within physical bounds

Resetting enhances convergence for finite databases

Abstract

Searching a database is a central task in computer science and is paradigmatic of transport and optimization problems in physics. For an unstructured search, Grover's algorithm predicts a quadratic speedup, with the search time $\tau(N)=\Theta(\sqrt{N})$ and $N$ the database size. Numerical studies suggest that the time complexity can change in the presence of feedback, injecting information during the search. Here, we determine the time complexity of the quantum analog of a randomized algorithm, which implements feedback in a simple form. The search is a continuous-time quantum walk on a complete graph, where the target is continuously monitored by a detector. Additionally, the quantum state is reset if the detector does not click within a specified time interval. This yields a non-unitary, non-Markovian dynamics. We optimize the search time as a function of the hopping amplitude, detection rate, and resetting rate, and identify the conditions under which time complexity could outperform Grover's scaling. The overall search time does not violate Grover's optimality bound when including the time budget of the physical implementation of the measurement. For databases of finite sizes monitoring can warrant rapid convergence and provides a promising avenue for fault-tolerant quantum searches.