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The hidden subgroup problem for generalized quaternions

Julia Tumasova Upton

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

The hidden subgroup problem is a pivotal problem in quantum computation since it reflects the structure of tasks for which quantum algorithms significantly outperform classical algorithms. In this dissertation, a quantum algorithm that solves the hidden subgroup problem over the generalized quaternion group is developed. The algorithm employs the abelian quantum Fourier transform and Kuperberg sieve to reveal the hidden subgroup. (Published By University of Alabama Libraries)

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The hidden subgroup problem is a pivotal problem in quantum computation since it reflects the structure of tasks for which quantum algorithms significantly outperform classical algorithms. In this paper, a quantum algorithm that solves the hidden subgroup problem over the generalized quaternion group is developed. The algorithm employs the abelian quantum Fourier transform and Kuperberg sieve to reveal the hidden subgroup.

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The hidden subgroup problem is a pivotal problem in quantum computation since it reflects the structure of tasks for which quantum algorithms significantly outperform classical algorithms. In this dissertation, a quantum algorithm that solves the hidden subgroup problem over the generalized quaternion group is developed. The algorithm employs the abelian quantum Fourier transform and Kuperberg sieve to reveal the hidden subgroup. (Published By University of Alabama Libraries)
The Hidden Subgroup Problem (HSP) provides the fundamental framework for most quantum algorithms. Until very recently, all known problems where quantum computation provides a super-polynomial speedup over classical algorithms have been variants of the HSP for particular abelian groups. Examples include factoring integers and computing the discrete log, for which Shor found efficient quantum algorithms. The algorithm for these problems may be viewed as consisting of the main HSP solution plus an ad hoc method of dealing with the particular variant. In this dissertation, we give a systematic way of dealing with all these variants. The key component of our solution is a new theorem about the robustness of Fourier sampling. The purpose of this theorem is to better understand the structure underlying existing algorithms as well as to provide an algorithmic tool for the construction of future quantum algorithms. In addition, we also derive a new algorithm for computing the quantum Fourier transform which is asymptotically faster than any previously known algorithm. By contrast, the nonabelian HSP is wide open. It includes as a special case the longstanding open question of graph isomorphism, where the group is the symmetric group, S n . It is natural to carry over the abelian HSP algorithm to the nonabelian case. We show that in the case that the hidden subgroup is normal, this algorithm succeeds in reconstructing the subgroup in polynomial time. On the other hand, we give evidence that this algorithm is inadequate to even distinguish a trivial subgroup from an involution in t...
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.
The hidden subgroup problem is the foundation of many quantum algorithms. An efficient solution is known for the problem over abelian groups, employed by both Simon's algorithm and Shor's factoring and discrete log algorithms. The nonabelian case, however, remains open; an efficient solution would give rise to an efficient quantum algorithm for graph isomorphism. We fully analyze a natural generalization of the algorithm for the abelian case to the nonabelian case and show that the algorithm determines the normal core of a hidden subgroup: in particular, normal subgroups can be determined. We show, however, that this immediate generalization of the abelian algorithm does not efficiently solve graph isomorphism.
The hidden subgroup problem (HSP) plays an important role in quantum computation, because many quantum algorithms that are exponentially faster than classical algorithms can be casted in the HSP structure. In this paper, we present a new polynomial-time quantum algorithm that solves the HSP over the group Zpr⋊Zqs\Z_{p^r} \rtimes \Z_{q^s}Zpr​⋊Zqs​, when $p^r/q= \up{poly}(\log p^r)$, where ppp, qqq are any odd prime numbers and r,sr, sr,s are any positive integers. To find the hidden subgroup, our algorithm uses the abelian quantum Fourier transform and a reduction procedure that simplifies the problem to find cyclic subgroups.
The quantum Fourier transform (QFT) has emerged as the primary tool in quantum algorithms which achieve exponential advantage over classical computation and lies at the heart of the solution to the abelian hidden subgroup problem, of which Shor's celebrated factoring and discrete log algorithms are a special case. We begin by addressing various computational issues surrounding the QFT and give improved parallel circuits for both the QFT over a power of 2 and the QFT over an arbitrary cyclic group. These circuits are based on new insight into the relationship between the discrete Fourier transform over different cyclic groups. We then exploit this insight to extend the class of hidden subgroup problems with efficient quantum solutions. First we relax the condition that the underlying hidden subgroup function be distinct on distinct cosets of the subgroup in question and show that this relaxation can be solved whenever G is a finitely-generated abelian group. We then extend this reasoning to the hidden cyclic subgroup problem over the reals, showing how to efficiently generate the bits of the period of any sufficiently piecewise-continuous function on R. Finally, we show that this problem of period-finding over R, viewed as an oracle promise problem, is strictly harder than its integral counterpart. In particular, period-finding over R lies outside the complexity class MA, a class which contains period-finding over the integers.