# Quantum query complexity of symmetric oracle problems

## TL;DR

This paper analyzes the quantum query complexity of symmetric oracle problems, providing character-theoretic formulas for success probabilities and demonstrating quantum advantages in problems like permutation identification and coset learning.

## Contribution

It introduces a group character framework for quantum query problems and applies it to various symmetric oracle tasks, revealing quantum speedups.

## Key findings

- Quantum algorithms can identify permutations with fewer queries than classical methods.
- Character-theoretic formulas precisely determine success probabilities for symmetric oracle problems.
- Certain problems require only one quantum query compared to multiple classical queries.

## Abstract

We study the query complexity of quantum learning problems in which the oracles form a group $G$ of unitary matrices. In the simplest case, one wishes to identify the oracle, and we find a description of the optimal success probability of a $t$-query quantum algorithm in terms of group characters. As an application, we show that $\Omega(n)$ queries are required to identify a random permutation in $S_n$. More generally, suppose $H$ is a fixed subgroup of the group $G$ of oracles, and given access to an oracle sampled uniformly from $G$, we want to learn which coset of $H$ the oracle belongs to. We call this problem coset identification and it generalizes a number of well-known quantum algorithms including the Bernstein-Vazirani problem, the van Dam problem and finite field polynomial interpolation. We provide character-theoretic formulas for the optimal success probability achieved by a $t$-query algorithm for this problem. One application involves the Heisenberg group and provides a family of problems depending on $n$ which require $n+1$ queries classically and only $1$ query quantumly.