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arXiv:2104.14436 · Quantum Information and Computation

Deterministic algorithms for the hidden subgroup problem

Ashwin Nayak

Published May 30, 2022• 3 citations
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Abstract

We present deterministic algorithms for the Hidden Subgroup Problem. The first algorithm, for abelian groups, achieves the same asymptotic worst-case query complexity as the optimal randomized algorithm, namely~$ \Order(\sqrt{ n}\, )$, where~$n$ is the order of the group. The analogous algorithm for non-abelian groups comes within a~$\sqrt{ \log n}$ factor of the optimal randomized query complexity. The best known randomized algorithm for the Hidden Subgroup Problem has \emph{expected\/} query complexity that is sensitive to the input, namely~$ \Order(\sqrt{ n/m}\, )$, where~$m$ is the order of the hidden subgroup. In the first version of this article~\cite[Sec.~5]{Nayak21-hsp-classical}, we asked if there is a deterministic algorithm whose query complexity has a similar dependence on the order of the hidden subgroup. Prompted by this question, Ye and Li~\cite{YL21-hsp-classical} present deterministic algorithms for \emph{abelian\/} groups which solve the problem with~$ \Order(\sqrt{ n/m }\, )$ queries, and find the hidden subgroup with~$ \Order( \sqrt{ n (\log m) / m} + \log m ) $ queries. Moreover, they exhibit instances which show that in general, the deterministic query complexity of the problem may be~$\order(\sqrt{ n/m } \,)$, and that of \emph{finding\/} the entire subgroup may also be~$\order(\sqrt{ n/m } \,)$ or even~$\upomega(\sqrt{ n/m } \,) $.}We present a different deterministic algorithm for the Hidden Subgroup Problem that also has query complexity~$ \Order(\sqrt{ n/m }\, )$ for abelian groups. The algorithm is arguably simpler. Moreover, it works for non-abelian groups, and has query complexity~$ \Order(\sqrt{ (n/m) \log (n/m) }\,) $ for a large class of instances, such as those over supersolvable groups. We build on this to design deterministic algorithms to find the hidden subgroup for all abelian and some non-abelian instances, at the cost of a~$\log m$ multiplicative factor increase in the query complexity.

DOI
10.26421/qic22.9-10-3

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We present deterministic algorithms for the Hidden Subgroup Problem. The first algorithm, for abelian groups, achieves the same asymptotic worst-case query complexity as the optimal randomized algorithm, namely~\Order(n ) \Order(\sqrt{ n}\, )\Order(n​), where~nnn is the order of the group. The analogous algorithm for non-abelian groups comes within a~log⁡n\sqrt{ \log n}logn​ factor of the optimal randomized query complexity. The best known randomized algorithm for the Hidden Subgroup Problem has \emph{expected/} query complexity that is sensitive to the input, namely~\Order(n/m ) \Order(\sqrt{ n/m}\, )\Order(n/m​), where~mmm is the order of the hidden subgroup. In the first version of this article~\cite[Sec.5]{Nayak21-hsp-classical}, we asked if there is a deterministic algorithm whose query complexity has a similar dependence on the order of the hidden subgroup. Prompted by this question, Ye and Li\cite{YL21-hsp-classical} present deterministic algorithms for \emph{abelian/} groups which solve the problem with~\Order(n/m ) \Order(\sqrt{ n/m }\, )\Order(n/m​) queries, and find the hidden subgroup with~\Order(n(log⁡m)/m+log⁡m)\Order( \sqrt{ n (\log m) / m} + \log m )\Order(n(logm)/m​+logm) queries. Moreover, they exhibit instances which show that in general, the deterministic query complexity of the problem may be~\order(n/m )\order(\sqrt{ n/m } \,)\order(n/m​), and that of \emph{finding/} the entire subgroup may also be~\order(n/m )\order(\sqrt{ n/m } \,)\order(n/m​) or even~\upomega(n/m )\upomega(\sqrt{ n/m } \,) \upomega(n/m​).}We present a different deterministic algorithm for the Hidden Subgroup Problem that also has query complexity~\Order(n/m ) \Order(\sqrt{ n/m }\, )\Order(n/m​) for abelian groups. The algorithm is arguably simpler. Moreover, it works for non-abelian groups, and has query complexity~\Order((n/m)log⁡(n/m) )\Order(\sqrt{ (n/m) \log (n/m) }\,)\Order((n/m)log(n/m)​) for a large class of instances, such as those over supersolvable groups. We build on this to design deterministic algorithms to find the hidden subgroup for all abelian and some non-abelian instances, at the cost of a~log⁡m\log mlogm multiplicative factor increase in the query complexity.
Several early quantum algorithms, including Simon's algorithm and Shor's period-finding are instances of the hidden subgroup problem (HSP) over finite abelian groups. No polynomial-time quantum algorithm is known for the HSP over arbitrary non-abelian finite groups. The non-Abelian case is of particular interest because some instances, such as the dihedral and symmetric group HSPs, are connected to lattice problems and graph isomorphism, respectively. In this work, we give polynomial-time quantum algorithms for two further families containing non-Abelian groups. First, we consider groups of the form G=A⋊φZpkG=A\rtimes_{\varphi} \mathbb{Z}_{p^k}G=A⋊φ​Zpk​, with AAA finite Abelian, ppp prime, k∈Nk\in \mathbb{N}k∈N and the action of Zpk\mathbb Z_{p^k}Zpk​ is generated by the scalar automorphism a↦μaa\mapstoμaa↦μa, for some μ∈ZExp⁡(A)×μ\in\mathbb Z_{\operatorname{Exp}(A)}^\timesμ∈ZExp(A)×​, where Exp(A)\mathrm{Exp}(A)Exp(A) is the exponent of AAA. Our algorithm is efficient when AAA has bounded generator rank and Exp(A)/p=polylog(∣G∣)\mathrm{Exp}(A)/p=\mathrm{polylog}(|G|)Exp(A)/p=polylog(∣G∣). This includes the case A=ZNA=\mathbb{Z}_NA=ZN​ for N∈NN\in\mathbb{N}N∈N and k=1k=1k=1, studied by Bacon, Childs and van Dam (FOCS 2005), and A=ZqrA=\mathbb{Z}_{q^r}A=Zqr​ with qqq prime and r∈Nr\in \mathbb{N}r∈N studied by van Dam and Dey (TQC 2014). Second, we give a polynomial-time quantum algorithm for finite quasi-Hamiltonian groups under a mild assumption on the input structure. Quasi-Hamiltonian groups are finite nilpotent groups with modular subgroup lattice, or equivalently the finite groups in which every subgroup is permutable. As far as we know, this is the first quantum algorithm to exploit the modularity of the subgroup lattice for solving the HSP. This extends, under the aforementioned structured input assumption, the quantum algorithm for Dedekind groups given by Hallgren, Russell, and Ta-Shma (SIAM J. Comput. 32, 2003).
Following the example of Shor's algorithm for period-finding in the integers, we explore the hidden subgroup problem (HSP) for discrete infinite groups. On the hardness side, we show that HSP is NP-hard for the additive group of rational numbers, and for normal subgroups of non-abelian free groups. We also indirectly reduce a version of the short vector problem to HSP in Zk\mathbb{Z}^kZk with pseudo-polynomial query cost. On the algorithm side, we generalize the Shor-Kitaev algorithm for HSP in Zk\mathbb{Z}^kZk (with standard polynomial query cost) to the case where the hidden subgroup has deficient rank or equivalently infinite index. Finally, we outline a stretched exponential time algorithm for the abelian hidden shift problem (AHShP), extending prior work of the author as well as Regev and Peikert. It follows that HSP in any finitely generated, virtually abelian group also has a stretched exponential time algorithm.
We provide a survey on the Hidden Subgroup Problem (HSP), which plays an important role in studying the security of public-key cryptosystems. We first review the abelian case, where Kitaev's algorithm yields an efficient quantum solution to the HSP, recalling how classical problems (such as order finding, integer factorization, and discrete logarithm) can be formulated as abelian HSP instances. We then examine the current state of the art for non-abelian HSP, where no general efficient quantum solution is known, focusing on some relevant groups including dihedral group (connected to the shortest vector problem), symmetric groups (connected to the graph isomorphism problem), and semidirect product constructions (connected, in a special case, to the code equivalence problem). We also describe the main techniques for addressing the HSP in non-abelian cases, namely Fourier sampling and the black-box approach. Throughout the paper, we highlight the mathematical notions required and exploited in this context, providing a cryptography-oriented perspective.
We revisit the finite Abelian hidden subgroup problem (AHSP) from a mathematical perspective and make the following contributions. First, by employing amplitude amplification, we present an exact quantum algorithm for the finite AHSP, our algorithm is more concise than the previous exact algorithm and applies to any finite Abelian group. Second, utilizing the Chinese Remainder Theorem, we propose a distributed exact quantum algorithm for finite AHSP, which requires fewer qudits, lower quantum query complexity, and no quantum communication. We further show that our distributed approach can be extended to certain classes of non-Abelian groups. Finally, we develop a parallel exact classical algorithm for finite AHSP with reduced query complexity; even without parallel execution, the total number of queries across all nodes does not exceed that of the original centralized algorithm under mild conditions.
The hidden subgroup problem (HSP) plays an important role in quantum computation, because many quantum algorithms that are exponentially faster than classical algorithms are special cases of the HSP. In this paper we show that there exist a new efficient quantum algorithm for the HSP on groups ZN⋊Zqs\Z_{N}\rtimes\Z_{q^s}ZN​⋊Zqs​ where NNN is an integer with a special prime factorization, qqq prime number and sss any positive integer.
In the quest to identify further quantum algorithms exhibiting superpolynomial speed-ups, a recurring theme is that the complexity of a problem is largely shaped by the input access model. Here, we study this phenomenon for the state hidden subgroup problem (StateHSP), a quantum generalization of the hidden subgroup problem in which the goal is to identify the symmetries of an unknown quantum state. For finite abelian groups, existing Fourier-sampling algorithms use O(log⁡(∣G∣)/ε)O(\log(|G|)/ε)O(log(∣G∣)/ε) copies of the state, but whether this scaling is optimal has remained open. We settle the complexity of the abelian StateHSP in both the previously studied sample model and a new query model, which is a stronger and operationally natural generalization that provides access to the state-preparation unitary and its inverse. In the query model, we give a time-efficient quantum algorithm using O(log⁡(∣G/H∣)/ε)O(\log(|G/H|)/\sqrtε)O(log(∣G/H∣)/ε​) forward and inverse queries, and prove a matching Ω(log⁡(∣G/H∣)/ε)Ω(\log(|G/H|)/\sqrtε)Ω(log(∣G/H∣)/ε​) lower bound which holds even in the stronger conjugate-query and controlled-query settings. By contrast, we show that in the sample model, Θ(log⁡(∣G/H∣)/ε)Θ(\log(|G/H|)/ε)Θ(log(∣G/H∣)/ε) copies are both sufficient and information-theoretically necessary, even if one allows for arbitrary collective measurements. Thus, the quadratic improvement in εεε genuinely arises from coherent access to the preparation circuit. As applications, we obtain faster algorithms for learning stabilizer groups, locating unentanglement, and identifying hidden translation symmetries.