Improved Algorithm and Lower Bound for Variable Time Quantum Search

TL;DR

This paper introduces a new quantum algorithm for variable time search with improved complexity and establishes a matching lower bound, advancing the understanding of quantum search efficiency for items with differing check times.

Contribution

It presents a simpler quantum algorithm for variable time search with better complexity and proves a tighter lower bound, improving upon previous results.

Findings

Quantum algorithm with $O(\sqrt{T}\log n)$ complexity

Lower bound of $\Omega(\sqrt{T ext{log} T})$ established

Both results improve previous bounds by a factor of $\sqrt{ ext{log} T}$

Abstract

We study variable time search, a form of quantum search where queries to different items take different time. Our first result is a new quantum algorithm that performs variable time search with complexity $O(\sqrt{T}\log n)$ where $T=\sum_{i=1}^n t_i^2$ with $t_i$ denoting the time to check the $i$-th item. Our second result is a quantum lower bound of $\Omega(\sqrt{T\log T})$. Both the algorithm and the lower bound improve over previously known results by a factor of $\sqrt{\log T}$ but the algorithm is also substantially simpler than the previously known quantum algorithms.