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arXiv:quant-ph/0406151

A Subexponential Time Algorithm for the Dihedral Hidden Subgroup Problem with Polynomial Space

Oded Regev

Published Jun 21, 2004• 67 citations
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Abstract

In a recent paper, Kuperberg described the first subexponential time algorithm for solving the dihedral hidden subgroup problem. The space requirement of his algorithm is super-polynomial. We describe a modified algorithm whose running time is still subexponential and whose space requirement is only polynomial.
DOI
10.48550/arxiv.quant-ph/0406151

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We give an algorithm for the hidden subgroup problem for the dihedral group DND_NDN​, or equivalently the cyclic hidden shift problem, that supersedes our first algorithm and is suggested by Regev's algorithm. It runs in exp⁡(O(log⁡N))\exp(O(\sqrt{\log N}))exp(O(logN​)) quantum time and uses exp⁡(O(log⁡N))\exp(O(\sqrt{\log N}))exp(O(logN​)) classical space, but only O(log⁡N)O(\log N)O(logN) quantum space. The algorithm also runs faster with quantumly addressable classical space than with fully classical space. In the hidden shift form, which is more natural for this algorithm regardless, it can also make use of multiple hidden shifts. It can also be extended with two parameters that trade classical space and classical time for quantum time. At the extreme space-saving end, the algorithm becomes Regev's algorithm. At the other end, if the algorithm is allowed classical memory with quantum random access, then many trade-offs between classical and quantum time are possible.
We give an algorithm for the hidden subgroup problem for the dihedral group D_N, or equivalently the cyclic hidden shift problem, that supersedes our first algorithm and is suggested by Regev's algorithm. It runs in exp(O(sqrt(log N))) quantum time and uses exp(O(sqrt(log N))) classical space, but only O(log N) quantum space. The algorithm also runs faster with quantumly addressable classical space than with fully classical space. In the hidden shift form, which is more natural for this algorithm regardless, it can also make use of multiple hidden shifts. It can also be extended with two parameters that trade classical space and classical time for quantum time. At the extreme space-saving end, the algorithm becomes Regev's algorithm. At the other end, if the algorithm is allowed classical memory with quantum random access, then many trade-offs between classical and quantum time are possible.
We present a polynomial-time quantum algorithm for the Hidden Subgroup Problem over D2n\mathbb{D}_{2^n}D2n​. The usual approach to the Hidden Subgroup Problem relies on harmonic analysis in the domain of the problem, and the best known algorithm using this approach has time complexity in 2O(n)2^{\mathcal{O}(\sqrt{n})}2O(n​). By focusing on structure encoded in the codomain of the problem, we develop a polynomial-time algorithm which uses this structure to direct a "walk" down the subgroup lattice of D2n\mathbb{D}_{2^n}D2n​ terminating at the hidden subgroup.
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).
We present a quantum algorithm for the dihedral hidden subgroup problem (DHSP) with time and query complexity 2O(log⁡ N)2^{O(\sqrt{\log\ N})}2O(log N​). In this problem an oracle computes a function f on the dihedral group DND_NDN​ which is invariant under a hidden reflection in DND_NDN​. By contrast, the classical query complexity of DHSP is O(N)O(\sqrt{N})O(N​). The algorithm also applies to the hidden shift problem for an arbitrary finitely generated abelian group. The algorithm begins as usual with a quantum character transform, which in the case of DND_NDN​ is essentially the abelian quantum Fourier transform. This yields the name of a group representation of DND_NDN​, which is not by itself useful, and a state in the representation, which is a valuable but indecipherable qubit. The algorithm proceeds by repeatedly pairing two unfavorable qubits to make a new qubit in a more favorable representation of DND_NDN​. Once the algorithm obtains certain target representations, direct measurements reveal the hidden subgroup.
After Shor's discovery of an efficient quantum algorithm for integer factoring, hidden subgroup problems play a central role in developing efficient quantum algorithms. In spite of many intensive studies, no efficient quantum algorithms are known for hidden subgroup problems for many non-Abelian groups. Of particular interest are the hidden subgroup problems for the symmetric group and for the dihedral group, because an efficient algorithm for the former implies an efficient solution to the graph isomorphism problem, and that for the latter essentially solves a certain lattice-related problem whose hardness is assumed in cryptography. This paper focuses on the latter case and gives a comprehensive survey of known facts related to the dihedral hidden subgroup problem.