Quantum Algorithms for Identifying Hidden Strings with Applications to Matroid Problems

TL;DR

This paper demonstrates quantum algorithms that significantly outperform classical algorithms in identifying specific binary string pairs and finding matroid bases, showcasing quantum speedups in these combinatorial problems.

Contribution

The paper introduces quantum algorithms with exponential and quadratic speedups for string identification and matroid basis problems, respectively, compared to classical methods.

Findings

Quantum algorithm identifies string pairs with O(1) queries

Classical lower bound for string pair identification is Ω(n/log n) queries

Quantum algorithm finds all matroid bases with fewer queries than classical methods

Abstract

In this paper, we explore quantum speedups for the problem, inspired by matroid theory, of identifying a pair of $n$-bit binary strings that are promised to have the same number of 1s and differ in exactly two bits, by using the max inner product oracle and the sub-set oracle. More specifically, given two string $s, s'\in\{0, 1\}^n$ satisfying the above constraints, for any $x\in\{0, 1\}^n$ the max inner product oracle $O_{max}(x)$ returns the max value between $s\cdot x$ and $s'\cdot x$, and the sub-set oracle $O_{sub}(x)$ indicates whether the index set of the 1s in $x$ is a subset of that in $s$ or $s'$. We present a quantum algorithm consuming $O(1)$ queries to the max inner product oracle for identifying the pair $\{s, s'\}$, and prove that any classical algorithm requires $\Omega(n/\log_{2}n)$ queries. Also, we present a quantum algorithm consuming $\frac{n}{2}+O(\sqrt{n})$ queries to the subset oracle, and prove that any classical algorithm requires at least $n+\Omega(1)$ queries. Therefore, quantum speedups are revealed in the two oracle models. Furthermore, the above results are applied to the problem in matroid theory of finding all the bases of a 2-bases matroid, where a matroid is called $k$-bases if it has $k$ bases.