Exponential convergence dynamics in Grover's search algorithm

TL;DR

This paper introduces a modified version of Grover's search algorithm that eliminates oscillatory dynamics, enabling an exponential decay process into solutions while maintaining quadratic speedup, addressing the souffle problem.

Contribution

The authors propose a novel modification to Grover's algorithm that converts oscillatory search dynamics into exponential decay, improving efficiency and robustness.

Findings

Achieves exponential convergence into solution states

Maintains quadratic quantum speedup

Eliminates the souffle problem in Grover's algorithm

Abstract

Grover's search algorithm is the cornerstone of many applications of quantum computing, providing a quadratic speed-up over classical methods. One limitation of the algorithm is that it requires knowledge of the number of solutions to obtain an optimal success probability, due to the oscillatory dynamics between the initial and solutions states (the ``souffl\'e problem''). While various methods have been proposed to solve this problem, each has their drawbacks in terms of inefficiency or sensitivity to control errors. Here, we modify Grover's algorithm to eliminate the oscillatory dynamics, such that the search proceeds as an exponential decay into the solution states. The basic idea is to convert the solution states into a reservoir by using ancilla qubits such that the initial state is nonreflectively absorbed. Trotterizing the continuous algorithm yields a quantum circuit that gives equivalent performance, which has the same quadratic quantum speedup as the original algorithm.