Quantum Algorithms and the Fourier Transform

The quantum algorithms of Deutsch, Simon and Shor are described in a way which highlights their dependence on the Fourier transform. The general construction of the Fourier transform on an Abelian group is outlined and this provides a unified way of understanding the efficacy of the algorithms. Finally we describe an efficient quantum factoring algorithm based on a general formalism of Kitaev and contrast its structure to the ingredients of Shor’s algorithm. Introduction The principal quantum algorithms which provide an exponential speedup over any known classical algorithms for the corresponding problems are Deutsch’s algorithm [2], Simon’s algorithm [4] and Shor’s algorithm [5]. Each of these rests essentially on the application of a suitable Fourier transform. In this paper we will outline the construction of the Fourier transform over a general (finite) Abelian group and highlight its origin and utility in the quantum algorithms. This provides a unified way of understanding the special efficacy of these algorithms. Indeed we have described elsewhere [8] how this efficacy may be explicitly seen as a property of quantum entanglement in the context of implementing the large unitary operation which is the Fourier transform. From our general group-theoretic viewpoint we will see that Simon’s and Shor’s algorithms are essentially identical in their basic formal structure differing only in the choice of underlying group. Both algorithms amount to the extraction of a periodicity relative to an Abelian group G using the Fourier transform of G in a uniform way. This general viewpoint may also be useful in developing new quantum algorithms by applying the formalism to other groups. Kitaev [7] has recently formulated a group–theoretic approach to quantum algorithms. We will describe below a special explicit case of his general formalism – an efficient quantum factoring algorithm which appears to be quite different from Shor’s. In particular, the Fourier transform as such, is not explicitly used. It is especially