Speedup of iterated quantum search by parallel performance

TL;DR

This paper demonstrates a quantum algorithm that speeds up iterated search processes by parallel evaluation of functions, achieving a performance improvement over sequential Grover searches.

Contribution

It introduces a method to find specific points in a sequence of Boolean functions using parallel quantum evaluations, surpassing traditional sequential Grover search efficiency.

Findings

Achieves a speedup factor of  over sequential Grover search.

Uses  simultaneous evaluations of functions.

Models amplitude evolution with linear differential equations.

Abstract

Given a sequence $f_1 (x_1), f_2 (x_1, x_2), ..., f_k (x_1, ..., x_k)$ of Boolean functions, each of which $f_i$ takes the value 1 in a single point of the form $x_1^0, x_2^0, ..., x_i^0, i=1,2,..., k$. A length of all $x_i^0$ is $n, N=2^n$. It is shown how to find $x_k^0 (k\geq 2)$ using \frac{k\pi\sqrt{N}}{4\sqrt{2}}$ simultaneous evaluations of functions of the form $f_i, f_{i+1}$ with an error probability of order $k/\sqrt{N}$ which is $\sqrt{2}$ times as fast as by the $k$ sequential applications of Grover algorithm for the quantum search. Evolutions of amplitudes in parallel quantum computations are approximated by systems of linear differential equations. Some advantage of simultaneous evaluations of all $f_1 ,... f_k$ are discussed.