Distinguishing symmetric quantum oracles and quantum group multiplication

TL;DR

This paper develops quantum algorithms for group-related oracle problems, including a generalization of Deutsch's algorithm and solutions for the hidden conjugating element problem, demonstrating optimality and high success probabilities.

Contribution

It introduces a quantum algorithm for the GROUP MULTIPLICATION problem for any finite group and proves its optimality, also solving the hidden conjugating element problem efficiently.

Findings

Single-query algorithm for group multiplication with success probability 2/|G|

Optimality proof for the group multiplication algorithm

High-probability solution for the hidden conjugating element problem

Abstract

Given a unitary representation of a finite group on a finite-dimensional Hilbert space, we show how to find a state whose translates under the group are distinguishable with the highest probability. We apply this to several quantum oracle problems, including the GROUP MULTIPLICATION problem, in which the product of an ordered $n$-tuple of group elements is to be determined by querying elements of the tuple. For any finite group $G$, we give an algorithm to find the product of two elements of $G$ with a single quantum query with probability $2/|G|$. This generalizes Deutsch's Algorithm from $Z_2$ to an arbitrary finite group. We further prove that this algorithm is optimal. We also introduce the HIDDEN CONJUGATING ELEMENT PROBLEM, in which the oracle acts by conjugating by an unknown element of the group. We show that for many groups, including dihedral and symmetric groups, the unknown element can be determined with probability $1$ using a single quantum query.