An overview of quantum computing and in particular the Hidden Subgroup Problem are presented from a mathematical viewpoint. Detailed proofs are supplied for many important results from the literature, and nota- tion is unified, making it easier to absorb the background necessary to begin research on the Hidden Subgroup Problem. Proofs are provided which give very concrete algorithms and bounds for the finite abelian case with little out- side references, and future directions are provided for the nonabelian case. This summary is current as of October 2004.

The hidden subgroup problem - review and open problems

We study the complexity of performing Fourier analysis for the group SL2(F q ), where F q is the finite field of q elements. Direct computation of a complete set of Fourier transforms for a complex-valued function f onSL2(F q ) requires q 6 operations. A similar bound holds for performing Fourier inversion. Here we show that for both operations this naive upper bound may be reduced to O(q 4 log q), where the implied constant is universal, depending only on the complexity of the “classical” fast Fourier transform. The techniques we use depend strongly on explicit construct io ns of matrix representationsof the group. Additionally, the ability to construct the matrix representations permits certain numerical experiments. By quite general methods, recent work of others has shown that certain families of Cayley graphs on SL2 (F q ) are expanders. However, little is known about their exact spectra. Computation of the relevant Four ier transform permits extensive numerical investigations of the spectra of these...

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Fast Fourier analysis for {${\rm SL}\sb 2$} over a finite field and related numerical experiments

/pdf/separation-of-variables-and-the-computation-of-fourier-32t97tpp1i.pdf

Separation of variables and the computation of Fourier transforms of finite groups, I

The linear complexityL
K(A) of a matrixA over a fieldK is defined as the minimal number of additions, subtractions and scalar multiplications sufficient to evaluateA at a generic input vector. IfG is a finite group andK a field containing a primitive exp(G)-th root of unity,L
K(G):= min{L
K(A)|A a Fourier transform forKG} is called theK-linear complexity ofG. We show that every supersolvable groupG has amonomial Fourier Transform adapted to a chief series ofG. The proof is constructive and gives rise to an efficient algorithm with running timeO(|G|2log|G|). Moreover, we prove that these Fourier transforms are efficient to evaluate:L
K(G)≤8.5|G|log|G| for any supersolvable groupG andL
K(G)≤1.5|G|log|G| for any 2-groupG.

Existence and efficient construction of fast Fourier transforms on supersolvable groups

Variations on computing reciprocals of power series

The linear complexityL2(G) of a finite groupG is the minimal number of additions, subtractions and multiplications by complex constants of absolute value ≦2 sufficient to evaluate a suitable Fourier transform of ℂG. Combining and modifying several classical FFT-algorithms, we show thatL2(G)≦8|G|log2|G| for any finite metabelian groupG.

Benno Tietz

Papers

Improved Upper Complexity Bounds for the Discrete Fourier Transform