Quantum search algorithm by adiabatic evolution under a priori probability

TL;DR

This paper introduces a modified adiabatic quantum search algorithm that leverages a priori knowledge to significantly reduce search time, potentially to constant time, unlike traditional algorithms.

Contribution

It demonstrates that prior knowledge influences the adiabatic search's efficiency and proposes a method to arbitrarily reduce its running time, including to constant time.

Findings

The initial state's solution amplitude determines the algorithm's running time.

A priori probability knowledge can be used to accelerate the search.

The running time can be reduced to a constant, independent of database size.

Abstract

Grover's algorithm is one of the most important quantum algorithms, which performs the task of searching an unsorted database without a priori probability. Recently the adiabatic evolution has been used to design and reproduce quantum algorithms, including Grover's algorithm. In this paper, we show that quantum search algorithm by adiabatic evolution has two properties that conventional quantum search algorithm doesn't have. Firstly, we show that in the initial state of the algorithm only the amplitude of the basis state corresponding to the solution affects the running time of the algorithm, while other amplitudes do not. Using this property, if we know a priori probability about the location of the solution before search, we can modify the adiabatic evolution to make the algorithm faster. Secondly, we show that by a factor for the initial and finial Hamiltonians we can reduce the running time of the algorithm arbitrarily. Especially, we can reduce the running time of adiabatic search algorithm to a constant time independent of the size of the database. The second property can be extended to other adiabatic algorithms.