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

• URL: http://arxiv.org/abs/2510.10538v1
• Date: Sun, 12 Oct 2025 10:42:50 GMT
• Title: One-Query Quantum Algorithms for the Index-$q$ Hidden Subgroup Problem
• Authors: Amit Te'eni, Yaron Oz, Eliahu Cohen,
• Abstract summary: The Bernstein-Vazirani problem is an instance of the hidden subgroup problem (HSP)<n>We present the index-$q$ HSP: determine whether a hidden subgroup $H le G$ has index $1$ or $q$, and, when possible, identify $H$.
• Score: 1.4146420810689422
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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