[Paper Review] The central nature of the Hidden Subgroup problem

Stephen Fenner, Yong Zhang|ArXiv.org|Oct 13, 2006

Quantum Computing Algorithms and Architecture19 references3 citations

TL;DR

This paper establishes that several central problems in quantum computing—Hidden Coset, Orbit Coset, Hidden Shift, and Generalized Hidden Shift—are polynomial-time reducible to the Hidden Subgroup Problem (HSP) over a wide range of groups, including nonabelian and permutation groups. The key contribution is proving that solving HSP over permutation groups or dihedral groups of smooth order suffices to solve these related problems, and it constructs nonadaptive program checkers for HSP decision and search versions.

ABSTRACT

We show that several problems that figure prominently in quantum computing, including Hidden Coset, Hidden Shift, and Orbit Coset, are equivalent or reducible to Hidden Subgroup for a large variety of groups. We also show that, over permutation groups, the decision version and search version of Hidden Subgroup are polynomial-time equivalent. For Hidden Subgroup over dihedral groups, such an equivalence can be obtained if the order of the group is smooth. Finally, we give nonadaptive program checkers for Hidden Subgroup and its decision version.

Motivation & Objective

• To unify and clarify the relationships between major quantum-computational problems such as Hidden Coset, Orbit Coset, Hidden Shift, and Generalized Hidden Shift.
• To demonstrate that these problems are polynomial-time reducible to the Hidden Subgroup Problem (HSP) over a broad class of groups, including nonabelian and permutation groups.
• To establish polynomial-time equivalence between the decision and search versions of HSP over permutation groups and over dihedral groups of smooth order.
• To design nonadaptive program checkers for both the decision and search versions of HSP over permutation groups, ensuring correctness with high probability.

Proposed method

• Reduces Hidden Coset to HSP by constructing a group homomorphism that embeds the Hidden Coset instance into a larger HSP instance using the wreath product construction.
• Uses the wreath product $ G \wr \mathbb{Z}_2 $ to embed Hidden Shift and Generalized Hidden Shift problems into HSP instances over nonabelian groups.
• Applies Theorem 3.1 to convert Hidden Shift and Hidden Coset instances into equivalent HSP instances over $ G \wr \mathbb{Z}_2 $, preserving subgroup structure.
• Employs nonadaptive query strategies to verify correctness of HSP programs by generating random permutations and testing consistency across multiple independent queries.
• Leverages the structure of quantum functions and coset representations to simulate and validate HSP solutions without adaptive feedback.
• Uses the fact that the hidden subgroup in the constructed $ G \wr \mathbb{Z}_2 $ instance corresponds to a lifted version of the original subgroup, enabling recovery of the original problem’s solution.

Experimental results

Research questions

• RQ1 Are Hidden Coset, Orbit Coset, Hidden Shift, and Generalized Hidden Shift polynomial-time reducible to the Hidden Subgroup Problem over general finite groups?
• RQ2 Can the decision and search versions of HSP be shown to be polynomial-time equivalent over permutation groups and dihedral groups of smooth order?
• RQ3 Is it possible to construct nonadaptive program checkers for HSP and its decision version that verify correctness with bounded error?
• RQ4 Can the embedding of Generalized Hidden Shift into HSP over wreath product groups be reversed to recover the original instance, implying that GHSh is a special case of HSP?
• RQ5 Do reductions from related problems to HSP preserve group complexity, such as mapping solvable groups to solvable groups, to enable finer classification of problem hardness?

Key findings

• Hidden Coset is polynomial-time equivalent to Hidden Subgroup over any finite group, including nonabelian groups.
• Orbit Coset is equivalent to Hidden Subgroup when quantum functions are allowed in the HSP formulation.
• Hidden Shift and Generalized Hidden Shift reduce to HSP over wreath product groups $ G \wr \mathbb{Z}_2 $, with the reduction running in polynomial time when the number of functions is small.
• The decision and search versions of HSP are polynomial-time equivalent over permutation groups, and over dihedral groups of smooth order.
• Nonadaptive program checkers for HSP and its decision version exist and run in polynomial time, with correctness probability at least $ 1 - 2^{-k} $ for $ k $ queries.
• The checker for HSP decision version outputs CORRECT if the program is correct, and outputs BUGGY with probability at least $ 1 - 2^{-k} $ if the program is incorrect, ensuring high reliability.