PROCESSOR ARCHITECTURES FOR FAST COMPUTATION OF MULTI-DIMENSIONAL UNITARY TRANSFORMS.

This work presents the development of new algorithms and special purpose sequential processor architectures for the computation of a class of one-, two- and multi-dimensional unitary transforms. In particular, a technique is presented to factorize the transformation matrices of a class of multi-dimensional unitary transforms, having separable kernels, into products of sparse matrices. These sparse matrices consist of Kronecker products of factors of the one-dimensional transformation matrix. Such factorizations result in fast algorithms for the computation of a variety of multi-dimensional unitary transforms including Fourier, Walsh-Hadamard and generalized Walsh transforms. It is shown that the u-dimensional Fourier and generalized Walsh transforms can be implemented with a u-dimensional radix-r butterfly operation requiring considerably fewer complex multiplications than the conventional implementation using a one-dimensional radix-r butterfly operation. Residue number principles and techniques are applied to develop novel special purpose sequential processor architectures for the computation of one-dimensional discrete Fourier and Walsh-Hadamard transforms and convolutions in real-time. The residue number system (RNS) based implementations yield a significant improvement in processing speed over the conventional realizations using the binary number system. As an illustration of the factorization techniques developed in this work, novel sequential architectures of RNS-based fast Fourier, Walsh-Hadamard and generalized Walsh transform processors for real-time processing of two-dimensional signals are presented. These sequential processor architectures are capable of processing large bandwidth (\u3e 5 M.Hz) input sequences. The application of the proposed FFT processors for the real-time computation of two-dimensional convolutions is also investigated. A special memory structure to support two-dimensional convolution operations is presented and it is shown that the two-dimensional FFT processor architecture proposed in this work requires less hardware than the conventional implementations. The FFT algorithms and processor architectures are verified by computer simulation.Dept. of Electrical and Computer Engineering. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1981 .N246. Source: Dissertation Abstracts International, Volume: 42-08, Section: B, page: 3366. Thesis (Ph.D.)--University of Windsor (Canada), 1981

Efficient Scheme for Initializing a Quantum Register with an Arbitrary Superposed State

Preparation of a quantum register is an important step in quantum computation and quantum information processing. It is straightforward to build a simple quantum state such as |i_1 i_2 ... i_n\ket with $i_j$ being either 0 or 1, but is a non-trivial task to construct an {\it arbitrary} superposed quantum state. In this Paper, we present a scheme that can most generally initialize a quantum register with an arbitrary superposition of basis states. Implementation of this scheme requires $O(Nn^2)$ standard 1- and 2-bit gate operations, {\it without introducing additional quantum bits}. Application of the scheme in some special cases is discussed.Comment: 4 pages, 4 figures, accepted by Phys. Rev.

A Quantitative Measure of Interference

We introduce an interference measure which allows to quantify the amount of interference present in any physical process that maps an initial density matrix to a final density matrix. In particular, the interference measure enables one to monitor the amount of interference generated in each step of a quantum algorithm. We show that a Hadamard gate acting on a single qubit is a basic building block for interference generation and realizes one bit of interference, an ``i-bit''. We use the interference measure to quantify interference for various examples, including Grover's search algorithm and Shor's factorization algorithm. We distinguish between ``potentially available'' and ``actually used'' interference, and show that for both algorithms the potentially available interference is exponentially large. However, the amount of interference actually used in Grover's algorithm is only about 3 i-bits and asymptotically independent of the number of qubits, while Shor's algorithm indeed uses an exponential amount of interference.Comment: 13 pages of latex; research done at http://www.quantware.ups-tlse.fr

Quantum Mechanics helps in searching for a needle in a haystack

Quantum mechanics can speed up a range of search applications over unsorted data. For example imagine a phone directory containing N names arranged in completely random order. To find someone's phone number with a probability of 50%, any classical algorithm (whether deterministic or probabilistic) will need to access the database a minimum of O(N) times. Quantum mechanical systems can be in a superposition of states and simultaneously examine multiple names. By properly adjusting the phases of various operations, successful computations reinforce each other while others interfere randomly. As a result, the desired phone number can be obtained in only O(sqrt(N)) accesses to the database.Comment: Postscript, 4 pages. This is a modified version of the STOC paper (quant-ph/9605043) and is modified to make it more comprehensible to physicists. It appeared in Phys. Rev. Letters on July 14, 1997. (This paper was originally put out on quant-ph on June 13, 1997, the present version has some minor typographical changes