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Toward a quantum Fourier transform on SU(2).

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Almost all known superpolynomial quantum speedups over classical algorithms have used the quantum Fourier transform (QFT). Most known applications of the QFT make use of the QFT over abelian groups, including Shor’s well known factoring algorithm [1]. However, the QFT can be generalised to act on non-abelian groups allowing different applications. For example, Kuperberg solves the dihedral hidden subgroup problem in subexponential time using the QFT on the dihedral group. The aim of this research is to construct an efficient QFT on SU(2). Most of the progress in constructing QFTs has come from applying ideas from classical algorithms such as subgroup adapted bases. For example, Moore et al.  have applied classical ideas from e.g.  to build non-abelian QFTs. Applying these ideas to infinite groups such as SU(2) requires new tools. The function must be sampled or discretised in a way so as to minimise the error. I will present some ideas based on classical algorithms which may lead to a QFT over the group SU(2) for band limited functions. There are problems with making these algorithms unitary that must be addressed and efficient methods for calculating coefficients (cf. the controlled phase gates in the abelian case) must be found.