An Optimum Algorithm for Quantum Search

TL;DR

This paper presents an improved quantum search algorithm tailored for specific target states, significantly enhancing efficiency when the target state's Hamming weight is imbalanced, with applications in Dicke state preparation.

Contribution

It introduces a modified Grover's algorithm that exploits Hamming weight properties to achieve exponential speedups in certain cases, especially when the number of 0's or 1's is small.

Findings

Efficiency depends on smaller number of 0's or 1's, not total length.

Can achieve exponential speedup when one Hamming weight is very small.

Poly-efficient Dicke state preparation in all cases.

Abstract

This paper discusses an improvement to Grover's algorithm for searches where target states are Hamming weight eigenstates and search space is not ordered. It is shown that under these conditions search efficiency depends on the smaller number of 0's and 1's, not the total length, of binary string of target state, and that Grover's algorithm can be improved whenever number of 0's and number 1's are not equal. In particular, improvement can be exponential when number of 0's or number of 1's is very small relative to binary string length. One interesting application is that Dicke state preparation, which in Grover's algorithm is P on average, can be made poly-efficient in all cases. For decision making process, this improvement won't improve computation efficiency, but can make implementation much simpler.