Schmidt decompositions of the quantum Fourier transform on C1 ⊗C2 are computed for all N1,N2 ≥ 2. The Schmidt decomposition is shown to be completely degenerate when N1 is a factor of N2 and when N1 > N2. This generalizes previous decompositions, found for N1 = N2 = 2 n [M. A. Nielsen, PhD Thesis, University of New Mexico (1998), Chapter 6; arXiv:quant-ph/0011036], and, more generally, for N1 =...

Operator-Schmidt decompositions of the quantum Fourier transform on C N1 ⊗ C N2 are computed for all N1,N2 ≥ 2. The decomposition is shown to be completely degenerate when N1 is a factor of N2 and when N1 > N2. The first known special case, N1 = N2 = 2 , was computed by Nielsen in his study of the communication cost of computing the quantum Fourier transform of a collection of qubits equally di...

-The vibrational modes of clay minerals are uniquely accessible to FT Raman spectroscopy, but this potentially powerful technique has found limited application to the study of clay mineral structure. Raman spectra in the 50 to 3800 cm-t region were obtained for a number of kandite clays. The kandite clay minerals are characterised by relatively intense bands centred at 142.7 cm -I for kaolinite...

The ability to implement the Quantum Fourier Transform (QFT) efficiently on a quantum computer enables the advantages offered by a variety of fundamental quantum algorithms, such as those for integer factoring, computing discrete logarithm over Abelian groups, and phase estimation. The standard fault-tolerant implementation of an n-qubit QFT approximates the desired transformation by removing s...

The Fourier Transform is one of the most important linear transformations used in science and engineering. Cooley and Tukey’s Fast Fourier Transform (FFT) from 1964 is a method for computing this transformation in time O(n logn). From a lower bound perspective, Morgenstern’s result from 1974 provides an Ω(n logn) lower bound for the unnormalized Fourier Transform (of determinant n), assuming th...

This paper derives a simple form for the group representation based Fourier transform of permutations acting on a n-dimensional vector space, and shows that the simplified transform requires less than 2n multiplications and 2n additions. The transform is based on the wellknown Young orthogonal form for constructing irreducible representations of the symmetric group.

For a skip-free Markov process on Z + with generator matrix Q, we evaluate the joint Laplace transform of the occupation times before hitting the state n (starting at 0). This Laplace transform has a very straightforward and familiar expression. We investigate the properties of this Laplace transform, especially the conditions under which the occupation times form a Markov chain.

In this paper, a new scheme employing the time-recursive approach to compute the discrete cosine transform (DCT) and discrete sine transform (DST) is presented. Using such approach, parallel lattice structure that can dually generate the DCT and DST simultaneously is developed. The resulting architecture is regular, module, and without global communication and can be applied to any transform si...