# Theoretical analysis of a nearly optimal analog quantum search

## TL;DR

This paper provides a theoretical analysis of a modified quantum search algorithm that achieves faster performance by allowing nearly optimal fidelity, with potential implications for practical quantum computing.

## Contribution

It introduces a nearly optimal fidelity-based modification to the Farhi-Gutmann Hamiltonian algorithm, demonstrating potential speed improvements over the original quantum search method.

## Key findings

- Modified algorithm can outperform original in speed
- Performance depends on energy ratio and state overlap
- Nearly optimal fidelity is achievable with minimal deviation

## Abstract

We analyze the possibility of modifying the original Farhi-Gutmann Hamiltonian algorithm in order to speed up the procedure for producing a suitably distributed unknown normalized quantum mechanical state. Such a modification is feasible provided only a nearly optimal fidelity is sought. We propose to select the lower bounds of the nearly optimal fidelity values such that their deviations from unit fidelity are less than the minimum error probability characterizing the optimum ambiguous discrimination scheme between the two nonorthogonal quantum states yielding the chosen nearly optimal fidelity values. Departing from the working assumptions of perfect state overlap and uniform distribution of the target state on the unit sphere in N-dimensional complex Hilbert space, we determine that the modified algorithm can indeed outperform the original analog counterpart of a quantum search algorithm. This performance enhancement occurs in terms of speed for a convenient choice of both the ratio E'/E between the energy eigenvalues E' and E of the modified search Hamiltonian and the quantum mechanical overlap x between the source and the target states. Finally, we briefly discuss possible analytical improvements of our investigation together with its potential relevance in practical quantum engineering applications.