Related papers: A Hardware-Efficient Approach to Computing the Rot…
We use a representability theorem of G. L. Watson to examine sums of squares in Quaternion rings with integer coefficients. This allows us to determine a large family of such rings where every element expressible as the sum of squares can…
In the past few years, successive improvements of the asymptotic complexity of square matrix multiplication have been obtained by developing novel methods to analyze the powers of the Coppersmith-Winograd tensor, a basic construction…
We exploit the truncated singular value decomposition and the recently proposed circulant decomposition for an efficient first-order approximation of the multiplication of large dense matrices. A decomposition of each matrix into a sum of a…
Variational formulations of reconstruction in computed tomography have the notable drawback of requiring repeated evaluations of both the forward Radon transform and either its adjoint or an approximate inverse transform which are…
The purpose of this effort is to investigate if the use of quaternion mathematics can be used to better model and simulate the electromagnetic fields that occur from moving electromagnetic charges. One observed deficiency with the commonly…
We show how one can use non-prime-power, composite moduli for computing representations of the product of two $n\times n$ matrices using only $n^{2+o(1)}$ multiplications.
Matrix square roots and their inverses arise frequently in machine learning, e.g., when sampling from high-dimensional Gaussians $\mathcal{N}(\mathbf 0, \mathbf K)$ or whitening a vector $\mathbf b$ against covariance matrix $\mathbf K$.…
In this paper we present two different variants of method for symmetric matrix inversion, based on modified Gaussian elimination. Both methods avoid computation of square roots and have a reduced machine time's spending. Further, both of…
We propose a quadrature-based formula for computing the exponential function of matrices with a non-oscillatory integral on an infinite interval and an oscillatory integral on a finite interval. In the literature, existing quadrature-based…
In this paper we introduce the algorithm and the fixed point hardware to calculate the normalized singular value decomposition of a non-symmetric matrices using Givens fast (approximate) rotations. This algorithm only uses the basic…
In this work we represent the $1/2$ Spin particles with complex quaternions using a transformation to 2x2 matrices in order to obtain the Pauli matrices. With this representation we determine the states, rotation operators and the total…
Recent results for rotations expressed as polynomials of spin matrices are derived here by elementary differential equation methods. Structural features of the results are then examined in the framework of biorthogonal systems, to obtain an…
We address a linear fractional differential equation and develop effective solution methods using algorithms for inversion of triangular Toeplitz matrices and the recently proposed QTT format. The inverses of such matrices can be computed…
We propose a class of randomized quantum algorithms for the task of sampling from matrix functions, without the use of quantum block encodings or any other coherent oracle access to the matrix elements. As such, our use of qubits is purely…
Quaternions provide a unified algebraic and geometric framework for representing three-dimensional rotations without the singularities that afflict Euler-angle parametrisations. This article develops a pedagogical and conceptual analysis of…
We present a novel set of reversible modular multipliers applicable to quantum computing, derived from three classical techniques: 1) traditional integer division, 2) Montgomery residue arithmetic, and 3) Barrett reduction. Each multiplier…
An efficient algorithm for computing eigenvectors of a matrix of integers by exact computation is proposed. The components of calculated eigenvectors are expressed as polynomials in the eigenvalue to which the eigenvector is associated, as…
We provide a new approach for compiling quantum simulation circuits that appear in Trotter, qDRIFT and multi-product formulas to Clifford and non-Clifford operations that can reduce the number of non-Clifford operations by a factor of up to…
Quaternion symmetry is ubiquitous in the physical sciences. As such, much work has been afforded over the years to the development of efficient schemes to exploit this symmetry using real and complex linear algebra. Recent years have also…
We present two deterministic algorithms to approximate single-qutrit gates. These algorithms utilize the Clifford + $\mathbf{R}$ group to find the best approximation of diagonal rotations. The first algorithm exhaustively searches over the…