The ultimate objective of this paper is to create a stepping stone to the development of new quantum algorithms. The strategy chosen is to begin by focusing on the class of abelian quantum hidden subgroup algorithms, i.e., the class of abelian algorithms of the Shor/Simon genre. Our strategy is to make this class of algorithms as mathematically transparent as possible. By the phrase "mathematically transparent" we mean to expose, to bring to the surface, and to make explicit the concealed mathematical structures that are inherently and fundamentally a part of such algorithms. In so doing, we create symbolic abelian quantum hidden subgroup algorithms that are analogous to the those symbolic algorithms found within such software packages as Axiom, Cayley, Maple, Mathematica, and Magma. As a spin-off of this effort, we create three different generalizations of Shor's quantum factoring algorithm to free abelian groups of finite rank. We refer to these algorithms as wandering (or vintage Z_Q) Shor algorithms. They are essentially quantum algorithms on free abelian groups A of finite rank n which, with each iteration, first select a random cyclic direct summand Z of the group A and then apply one iteration of the standard Shor algorithm to produce a random character of the "approximating" finite group A=Z_Q, called the group probe. These characters are then in turn used to find either the order P of a maximal cyclic subgroup Z_P of the hidden quotient group H_phi, or the entire hidden quotient group H_phi. An integral part of these wandering quantum algorithms is the selection of a very special random transversal, which we refer to as a Shor transversal. The algorithmic time complexity of the first of these wandering Shor algorithms is found to be O(n^2(lgQ)^3(lglgQ)^(n+1)).
The ultimate objective of this paper is to create a stepping stone to the development of new quantum algorithms. The strategy chosen is to begin by focusing on the class of abelian quantum hidden subgroup algorithms, i.e., the class of abelian algorithms of the Shor/Simon genre. Our strategy is to make this class of algorithms as mathematically transparent as possible. By the phrase "mathematically transparent" we mean to expose, to bring to the surface, and to make explicit the concealed mathematical structures that are inherently and fundamentally a part of such algorithms. In so doing, we create symbolic abelian quantum hidden subgroup algorithms that are analogous to the those symbolic algorithms found within such software packages as Axiom, Cayley, Maple, Mathematica, and Magma. As a spin-off of this effort, we create three different generalizations of Shor's quantum factoring algorithm to free abelian groups of finite rank. We refer to these algorithms as wandering (or vintage Z_Q) Shor algorithms. They are essentially quantum algorithms on free abelian groups A of finite rank n which, with each iteration, first select a random cyclic direct summand Z of the group A and then apply one iteration of the standard Shor algorithm to produce a random character of the "approximating" finite group A=Z_Q, called the group probe. These characters are then in turn used to find either the order P of a maximal cyclic subgroup Z_P of the hidden quotient group H_phi, or the entire hidden quotient group H_phi. An integral part of these wandering quantum algorithms is the selection of a very special random transversal, which we refer to as a Shor transversal. The algorithmic time complexity of the first of these wandering Shor algorithms is found to be O(n^2(lgQ)^3(lglgQ)^(n+1)).
One of the most promising and versatile approaches to creating new quantum algorithms is based on the quantum hidden subgroup (QHS) paradigm, originally suggested by Alexei Kitaev. This class of quantum algorithms encompasses the Deutsch-Jozsa, Simon, Shor algorithms, and many more. In this paper, our strategy for finding new quantum algorithms is to decompose Shor's quantum factoring algorithm into its basic primitives, then to generalize these primitives, and finally to show how to reassemble them into new QHS algorithms. Taking an "alphabetic building blocks approach," we use these primitives to form an "algorithmic toolkit" for the creation of new quantum algorithms, such as wandering Shor algorithms, continuous Shor algorithms, the quantum circle algorithm, the dual Shor algorithm, a QHS algorithm for Feynman integrals, free QHS algorithms, and more. Toward the end of this paper, we show how Grover's algorithm is most surprisingly "almost" a QHS algorithm, and how this result suggests the possibility of an even more complete "algorithmic tookit" beyond the QHS algorithms.
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⋊φZpk, with A finite Abelian, p prime, k∈N and the action of Zpk is generated by the scalar automorphism a↦μa, for some μ∈ZExp(A)×, where Exp(A) is the exponent of A. Our algorithm is efficient when A has bounded generator rank and Exp(A)/p=polylog(∣G∣). This includes the case A=ZN for N∈N and k=1, studied by Bacon, Childs and van Dam (FOCS 2005), and A=Zqr with q prime and 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).
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.
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) 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...