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Leibniz international proceedings in informatics

Another Subexponential-time Quantum Algorithm for the Dihedral Hidden Subgroup Problem

Greg Kuperberg

Published Jan 1, 2013• 50 citations
Publisher

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
10.4230/lipics.tqc.2013.20

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We study quantum algorithms for the hidden subgroup problem (HSP). The HSP provides a unified framework for many important problems, including factoring integers, graph isomorphism problem, etc. It has been known that the problem for abelian groups can be efficiently solved. However, for non-abelian groups, it remains widely open. In this dissertation we study the problem for some specific non-abelian groups, called q-hedral groups. We start with a quantum algorithm, proposed by Moore et al. (2003), for finding some hidden conjugate subgroups of affine groups, a special case of q-hedral groups for large q. We then adapt their algorithm to construct an algorithm for solving the hidden conjugate problem for q-hedral groups for large q and point out a difficulty in complexity analysis that involves certain exponential sums. We discuss certain bounds on these sums and present some computational data on their values to make a conjecture. For small q, Regev (2003) gave an efficient algorithm for solving the HSP for dihedral groups (q = 2) using an oracle that solves a certain subset sum problem (SSP). A crucial step of his algorithm relies on the probability that elements in a cyclic group ZN can be expressed as sums of random elements in the group. We generalize this problem to any finite group and prove the following theorem: for any finite group G and r ≥ 32 log2 |G| + 5, if r elements from G are chosen randomly, then the 2r possible sums (or products) of the r elements will cover all the elements in G with high probability. Also, by making a further generalization of the SSP, we show that Regev's algorithm can be generalized for q-hedral groups for any fixed q. We also study this generalized subset sum problem. In addition to the algorithm developing for small q, we give an efficient quantum algorithm for solving the coset problem and an efficient quantum algorithm for solving the hidden conjugate problem. Furthermore, we also give a survey of an algorithm for finding the largest normal subgroup in a hidden subgroup of any finite group, introduced by Hallgren et al. (2000) and provide a simplified and unified proof.

•Jan 1, 2005•0 citations
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 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 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.
Consider the following generalized hidden shift problem: given a function f on {0,...,M-1} x Z_N satisfying f(b,x)=f(b+1,x+s) for b=0,1,...,M-2, find the unknown shift s in Z_N. For M=N, this problem is an instance of the abelian hidden subgroup problem, which can be solved efficiently on a quantum computer, whereas for M=2, it is equivalent to the dihedral hidden subgroup problem, for which no efficient algorithm is known. For any fixed positive epsilon, we give an efficient (i.e., poly(log N)) quantum algorithm for this problem provided M > N^epsilon. The algorithm is based on the "pretty good measurement" and uses H. Lenstra's (classical) algorithm for integer programming as a subroutine.
We study quantum algorithms for the hidden subgroup problem (HSP). The HSP provides a unified framework for many important problems, including factoring integers, graph isomorphism problem, etc. It has been known that the problem for abelian groups can be efficiently solved. However, for non-abelian groups, it remains widely open. In this dissertation we study the problem for some specific non-abelian groups, called q-hedral groups. We start with a quantum algorithm, proposed by Moore et al. (2003), for finding some hidden conjugate subgroups of affine groups, a special case of q-hedral groups for large q. We then adapt their algorithm to construct an algorithm for solving the hidden conjugate problem for q-hedral groups for large q and point out a difficulty in complexity analysis that involves certain exponential sums. We discuss certain bounds on these sums and present some computational data on their values to make a conjecture. For small q, Regev (2003) gave an efficient algorithm for solving the HSP for dihedral groups (q = 2) using an oracle that solves a certain subset sum problem (SSP). A crucial step of his algorithm relies on the probability that elements in a cyclic group ZN can be expressed as sums of random elements in the group. We generalize this problem to any finite group and prove the following theorem: for any finite group G and r ≥ 32 log2 |G| + 5, if r elements from G are chosen randomly, then the 2r possible sums (or products) of the r elements will cover all the elements in G with high probability. Also, by making a further generalization of the SSP, we show that Regev's algorithm can be generalized for q-hedral groups for any fixed q. We also study this generalized subset sum problem. In addition to the algorithm developing for small q, we give an efficient quantum algorithm for solving the coset problem and an efficient quantum algorithm for solving the hidden conjugate problem. Furthermore, we also give a survey of an algorithm for finding the largest normal subgroup in a hidden subgroup of any finite group, introduced by Hallgren et al. (2000) and provide a simplified and unified proof.