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arXiv:1112.3333

Another subexponential-time quantum algorithm for the dihedral hidden subgroup problem

Greg Kuperberg

Published Dec 14, 2011• 26 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.1112.3333

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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 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.
A new quantum algorithm for the hidden shift problem in Z2tn\mathbb{Z}_{2^t}^nZ2tn​
In this paper we make a step towards a time and space efficient algorithm for the hidden shift problem for groups of the form Zkn\mathbb{Z}_k^nZkn​. We give a solution to the case when kkk is a power of 2, which has polynomial running time in nnn, and only uses quadratic classical, and linear quantum space in nlog⁡(k)n\log (k)nlog(k). It can be a useful tool in the general case of the hidden shift and hidden subgroup problems too, since one of the main algorithms made to solve them can use this algorithm as a subroutine in its recursive steps, making it more efficient in some instances.
We examine the application of quantum algorithms to the non-Abelian hidden subgroup problem and focus on the dihedral hidden subgroup problem (DHSP). Ettinger and Høyer have recently discovered an algorithm which, although efficient in the number of operations of the quantum computation, requires classical post-processing which grows exponentially with the size of the input. We first show that the DHSP can be reduced to another problem: how to efficiently estimate an unknown shift k , in a one-to-one map f : M → M , given two oracles which provide x → f ( x ), and x → f ( x ⊕ k ), with k and f unknown. We devise an algorithm which uses amplitude amplification to obtain an estimate for the unknown shift k in a number (( M ) 1/2 ) oracle calls.
Abstract The hidden subgroup problem (HSP) is a cornerstone problem in quantum computing, which captures many problems of interest and provides a standard framework algorithm for their study based on Fourier sampling, one class of techniques known to provide quantum advantage, and which succeeds for some groups but not others. The quantum hardness of the HSP problem for the dihedral group is a critical question for post-quantum cryptosystems based on learning with errors and also appears in subexponential algorithms for constructing isogenies between elliptic curves over a finite field. In this article, we give an updated overview of the dihedral hidden subgroup problem as approached by the “standard” quantum algorithm for HSP on finite groups, detailing the obstructions for strong Fourier sampling to succeed and summarizing other known approaches and results. In our treatment, we “contrast and compare” as much as possible the cyclic and dihedral cases, with a view to determining bounds for the success probability of a quantum algorithm that uses m m coset samples to solve the HSP on these groups. In the last sections, we prove a number of no-go results for the dihedral coset problem (DCP), motivated by a connection between DCP and cloning of quantum states. The proofs of these no-go results are then adapted to give nontrivial upper bounds on the success probability of a quantum algorithm that uses m m coset samples to solve DCP.