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ZX-calculus is a strict mathematical formalism for graphical quantum computing which is based on the field of complex numbers. In this paper, we extend its power by generalising ZX-calculus to such an extent that it is universal both in an…
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 properties are a type of property of categories which includes the ones of being Mal'tsev, arithmetical, majority, unital, strongly unital and subtractive. Recently, an algorithm has been developed to determine implications…
In this paper, we investigate the conditions under which a diagonal quadratic form $\sum_{i=1}^{m}a_i X_i^2$ represents every $n \times n$ integral matrix, where $a_i$ ($1 \leq i \leq m$) are integers. For $n=2$, we give a necessary and…
A mixed basis approach based on density functional theory is employed for low dimensional systems. The basis functions are taken to be plane waves for the periodic direction multiplied by B-spline polynomials in the non-periodic direction.…
The minimax theory for estimating linear functionals is extended to the case of a finite union of convex parameter spaces. Upper and lower bounds for the minimax risk can still be described in terms of a modulus of continuity. However in…
As a new approach to efficiently describe correlation effects in the relativistic quantum world we propose to consider reduced density matrix functional theory, where the key quantity is the first-order reduced density matrix (1-RDM). In…
This paper introduces an algorithm for the nonnegative matrix factorization-and-completion problem, which aims to find nonnegative low-rank matrices X and Y so that the product XY approximates a nonnegative data matrix M whose elements are…
introduce {\sc Planar Disjoint Paths Completion}, a completion counterpart of the Disjoint Paths problem, and study its parameterized complexity. The problem can be stated as follows: given a, not necessarily connected, plane graph $G,$ $k$…
This paper is a continuation of our previous works where we study maps from $X_0(N)$, $N \ge 1$, into $\mathbb P^2$ constructed via modular forms of the same weight and criteria that such a map is birational (see [12]). In the present paper…
We consider the problem of finding a local minimum of a binary quadratic function, and show by an elementary construction that every descending local search algorithm takes exponential time in the worst case.
We generalize the concept of a field by allowing addition to be a partial operation. We show that elements of such a "partially additive field" share many similarities with physical quantities. In particular, they form subsets of mutually…
We construct six unitary trace invariants for 2 by 2 quaternionic matrices which separate the unitary similarity classes of such matrices, and show that this set is minimal. We prove two quaternionic versions of a well known…
Efficiently processing basic linear algebra subroutines is of great importance for a wide range of computational problems. In this paper, we consider techniques to implement matrix functions on a quantum computer, which are composed of…
Linear connections are introduced on a series of noncommutative geometries which have commutative limits. Quasicommutative corrections are calculated.
This paper studies convex quadratic minimization problems in which each continuous variable is coupled with a binary indicator variable. We focus on the structured setting where the Hessian matrix of the quadratic term is positive definite…
This survey highlights the recent advances in algorithms for numerical linear algebra that have come from the technique of linear sketching, whereby given a matrix, one first compresses it to a much smaller matrix by multiplying it by a…
An inexact Newton type method for numerical minimization of convex piecewise quadratic functions is considered and its convergence is analyzed. Earlier, a similar method was successfully applied to optimizaton problems arising in numerical…
The problem of fitting experimental data to a given model function $f(t; p_1,p_2,\dots,p_N)$ is conventionally solved numerically by methods such as that of Levenberg-Marquardt, which are based on approximating the Chi-squared measure of…
We discuss the hard-hexagon and hard-square problems, as well as the corresponding problem on the honeycomb lattice. The case when the activity is unity is of interest to combinatorialists, being the problem of counting binary matrices with…