One-Query Quantum Algorithms for the Index-$q$ Hidden Subgroup Problem

TL;DR

This work investigates whether the quantum Fourier transform (QFT) is essential in canonical quantum query problems by separating algebraic structure from circuit implementation. It introduces the index-$q$ hidden subgroup problem (HSP), proving a one-query algorithm that distinguishes $[G:H]=1$ from $q$ for any abelian output structure and, under explicit minimal conditions (namely a cyclic quotient $G/H$ of order $q$ and a faithful compatible $\mathbb{Z}_q$ structure on the outputs), exactly recovers $H$ from a single query; this unconditional recovery holds for $q\in\{2,3\}$. The Bernstein--Vazirani problem is shown to be a special case of the index-$q$ HSP with $q=2$, tying BV directly to the one-query framework, while the Shor--Kitaev sampling approach is shown unable to guarantee exact single-sample recovery in general. Collectively, these results sharpen the understanding of when one-query quantum algorithms can solve abelian HSPs and clarify the precise role of the QFT across foundational quantum algorithms.

Abstract

The quantum Fourier transform (QFT) is central to many quantum algorithms, yet its necessity is not always well understood. We re-examine its role in canonical query problems. The Deutsch-Jozsa algorithm requires neither a QFT nor a domain group structure. In contrast, the Bernstein-Vazirani problem is an instance of the hidden subgroup problem (HSP), where the hidden subgroup has either index $1$ or $2$; and the Bernstein-Vazirani algorithm exploits this promise to solve the problem with a single query. Motivated by these insights, we introduce the index-$q$ HSP: determine whether a hidden subgroup $H \le G$ has index $1$ or $q$, and, when possible, identify $H$. We present a single-query algorithm that always distinguishes index $1$ from $q$, for any choice of abelian structure on the oracle's codomain. Moreover, with suitable pre- and post-oracle unitaries (inverse-QFT/QFT over $G$), the same query exactly identifies $H$ under explicit minimal conditions: $G/H$ is cyclic of order $q$, and the output alphabet admits a faithful, compatible group structure. These conditions hold automatically for $q \in \left\{ 2,3 \right\}$, giving unconditional single-query identification in these cases. In contrast, the Shor-Kitaev sampling approach cannot guarantee exact recovery from a single sample. Our results sharpen the landscape of one-query quantum solvability for abelian HSPs.