Almost all of the most successful quantum algorithms discovered to date exploit the ability of the Fourier transform to recover subgroup structures 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. For one of these problems, the shifted Legendre symbol problem, we give evidence that the problem is hard to solve classically, by showing a reduction from breaking algebraically homomorphic cryptosystems. 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.

We consider the problem of identifying a base k string given an oracle which returns information about the number of correct components in a query, specifically, the Hamming distance between the query and the solution, modulo r = max{2, 6 − k}. Classically this problem requires Ω(n log r k) queries. For k ∈ {2, 3, 4} we construct quantum algorithms requiring only a single quantum query. For k> 4, we show that O ( √ k) quantum queries suffice. In both case, the quantum algorithms are optimal.

Concept learning provides a natural framework in which to place the problems solved by the quantum algorithms of Bernstein-Vazirani and Grover. By combining the tools used in these algorithms---quantum fast transforms and amplitude amplification---with a novel (in this context) tool---a solution method for geometrical optimization problems--- we derive a general technique for quantum concept learning. We name this technique "Amplified Impatient Learning" and apply it to construct quantum algorithms solving two new problems: BATTLESHIP and MAJORITY, more e#ciently than is possible classically.

We consider the problem of locating a template as a subimage of a larger image. Computing the maxima of the correlation function solves this problem classically. Since the correlation can be calculated with the Fourier transform this problem is a good candidate for a superior quantum algorithmic solution. We outline how such an algorithm would work.

We consider the problem of recovering a hidden monic polynomial f(X) of degree d ≥ 1 over a finite field Fp of p elements given a black box which, for any x ∈ Fp, evaluates the quadratic character of f(x). We design a classical algorithm of complexity O(d 2 p d+ε) and also show that the quantum query complexity of this problem is O(d). Some of our results extend those of Wim van Dam, Sean Hallgren and Lawrence Ip obtained in the case of a linear polynomial f(X) = X +s (with unknown s); some are new even in this case. 1 1

In the past decade quantum algorithms have been found which outperform the best classical solutions known for certain classical problems as well as the best classical methods known for simulation of certain quantum systems. This suggests that they may also speed up the simulation of some classical systems. I describe one class of discrete quantum algorithms which do so---quantum lattice gas automata---and show how to implement them efficiently on standard quantum computers.

Concept learning provides a natural framework in which to place the problems solved by the quantum algorithms of Bernstein-Vazirani and Grover. By combining the tools used in these algorithms—quantum fast transforms and amplitude amplification—with a novel (in this context) tool—a solution method for geometrical optimization problems— we derive a general technique for quantum concept learning. We name this technique “Amplified Impatient Learning ” and apply it to construct quantum algorithms solving two new problems: BATTLESHIP and MAJORITY, more efficiently than is possible classically.

In the past decade quantum algorithms have been found which outperform the best classical solutions known for certain classical problems as well as the best classical methods known for simulation of certain quantum systems. This suggests that they may also speed up the simulation of some classical systems. I describe one class of discrete quantum algorithms which do so—quantum lattice gas automata—and show how to implement them efficiently on standard quantum computers.

New weighing matrices constructed