We give polynomial-time quantum algorithms for two problems from computational algebraic number theory. The first is Pell's equation. Given a positive non-square integer d, Pell's equation is x 2 − dy 2 = 1 and the goal is to find its integer solutions. Factoring integers reduces to finding integer solutions of Pell's equation, but a reduction in the other direction is not known and appears more difficult. The second problem is the principal ideal problem in real quadratic number fields. Solving this problem is at least as hard as solving Pell's equation, and is the basis of a cryptosystem which is broken by our algorithm.

Almost all of the most successful quantum algorithms discovered to date exploit the ability of the Fourier transform to recover subgroup structure of functions, especially periodicity. The fact that Fourier transforms can also be used to capture shift structure has received far less attention in the context of quantum computation.In this paper, we present three examples of "unknown shift" problems that can be solved efficiently on a quantum computer using the quantum Fourier transform. We also define the hidden coset problem, which generalizes the hidden shift problem and the hidden subgroup problem. This framework provides a unified way of viewing the ability of the Fourier transform to capture subgroup and shift structure.

The Hidden Subgroup Problem is the foundation of many quantum algorithms. An efficient solution is known for the problem over Abelian groups and this was used in Simon's algorithm and Shor's Factoring and Discrete Log algorithms. The non-Abelian case is open; an efficient solution would give rise to an efficient quantum algorithm for Graph Isomorphism. We fully analyze a natural generalization of the Abelian case solution to the non-Abelian case, and give an efficient solution to the problem for normal subgroups. We show, however, that this immediate generalization of the Abelian algorithm does not efficiently solve Graph Isomorphism.

Computing the group of units in a field of algebraic numbers is one of the central tasks of computational algebraic number theory. It is believed to be hard classically, which is of interest for cryptography. In the quantum setting, efficient algorithms were previously known for fields of constant degree. We give a quantum algorithm that is polynomial in the degree of the field and the logarithm of its discriminant. This is achieved by combining three new results. The first is a classical algorithm for computing a basis for certain ideal lattices with doubly exponentially large generators. The second shows that a Gaussian-weighted superposition of lattice points, with an appropriate encoding, can be used to provide a unique representation of a real-valued lattice. The third is an extension of the hidden subgroup problem to continuous groups and a quantum algorithm for solving the HSP over *

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