Statistical comparison of ensemble implementations of Grover's search algorithm to classical sequential searches

TL;DR

This paper compares ensemble quantum implementations of Grover's search with classical sequential searches, establishing conditions under which quantum methods outperform classical ones based on resource usage and success probabilities.

Contribution

It introduces a resource-based comparison criterion and derives bounds for polarization, showing when ensemble quantum search surpasses classical search in success probability.

Findings

Critical polarization scales as N^(-1/4).

Ensemble Grover's search outperforms classical search for N > 10^22.

Provides bounds for resource efficiency in quantum search implementations.

Abstract

We compare pseudopure state ensemble implementations, quantified by their initial polarization and ensemble size, of Grover's search algorithm to probabilistic classical sequential search algorithms in terms of their success and failure probabilities. We propose a criterion for quantifying the resources used by the ensemble implementation via the aggregate number of oracle invocations across the entire ensemble and use this as a basis for comparison with classical search algorithms. We determine bounds for a critical polarization such that the ensemble algorithm succeeds with a greater probability than the probabilistic classical sequential search. Our results indicate that the critical polarization scales as N^(-1/4) where N is the database size and that for typical room temperature solution state NMR, the polarization is such that the ensemble implementation of Grover's algorithm would be advantageous for N > 10^22