Extending the Promise of the Deutsch--Jozsa--Hoyer Algorithm for Finite Groups

Abstract

Hoyer has given a generalisation of the Deutsch--Jozsa algorithm which uses the Fourier transform on a group G which is (in general) non-Abelian. His algorithm distinguishes between functions which are either perfectly balanced (m-to-one) or constant, with certainty, and using a single quantum query. Here, we show that this algorithm (which we call the Deutsch--Jozsa--Hoyer algorithm) can in fact deal with a broader range of promises, which we define in terms of the irreducible representations of G.

24 pages, 5 figures, to appear in LMS JCM Updated on 9th August 2005 following the referees comments. Added: Overview of questions surrounding the Fourier transform; Appendix on group representations. Corrected typos and improved notation