A Quantum Algorithm for Finding the Minimum

TL;DR

The paper addresses finding the index $y$ of the minimum value in an unsorted table $T[0..N-1]$, a task requiring $\Theta(N)$ probes classically. It introduces a quantum algorithm that finds the minimum with $O(\sqrt N)$ probes by iteratively applying the quantum exponential searching subroutine of BBHT96 to tighten a threshold $T[y]$, leveraging Grover-like speedups. The authors bound the expected time by $m_0 = \frac{45}{4} \sqrt N + \frac{7}{10} \lg^2 N$ and show that after at most $2 m_0$ iterations the minimum is found with probability at least $\frac{1}{2}$, yielding an overall $O(\sqrt N)$ time, near the quantum lower bound implied by BBBV95. They further discuss probability amplification through repetition, robustness to non-distinct values, and a time-out variant, highlighting the algorithm's potential for speedups in unstructured search problems.

Abstract

We give a quantum algorithm to find the index y in a table T of size N such that in time O(c sqrt N), T[y] is minimum with probability at least 1-1/2^c.

Theorems & Definitions (3)

• Theorem 1
• Lemma 1
• Lemma 2