Related papers: How Fast Can We Multiply Large Integers on an Actu…
We define a Graphics Turing Test to measure graphics performance in a similar manner to the definition of the traditional Turing Test. To pass the test one needs to reach a computational scale, the Graphics Turing Scale, for which Computer…
Projective measurements with high quantum efficiency is often assumed to be required for efficient circuit based quantum computing. We argue that this is not the case and show that this fact has actually be known previously though not…
We investigate the time T a quantum computer requires to factorize a given number dependent on the number of bits L required to represent this number. We stress the fact that in most cases one has to take into account that the execution…
Complex networks are ubiquitous to several Computer Science domains. Centrality measures are an important analysis mechanism to uncover vital elements of complex networks. However, these metrics have high computational costs and…
This is yet another version of the course notes in chao-dyn/9407003. Here we change the universal Turing machine that is used to measure program-size complexity so that the constants in our information-theoretic incompleteness theorems are…
Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject…
We show that, for all reasonable functions $T(n)=o(n\log n)$, we can algorithmically verify whether a given one-tape Turing machine runs in time at most $T(n)$. This is a tight bound on the order of growth for the function $T$ because we…
An algorithm is given to factor an integer with $N$ digits in $\ln^m N$ steps, with $m$ approximately 4 or 5. Textbook quadratic sieve methods are exponentially slower. An improvement with the aid of an a particular function would provide a…
Permutation resemblance measures the distance of a function from being a permutation. Here we show how to determine the permutation resemblance through linear integer programming techniques. We also present an algorithm for constructing…
Almost all public secure communication relies on the inability to factor large numbers. There is no known analytic or classical numeric method to rapidly factor large numbers. Shor[1] has shown that a quantum computer can factor numbers in…
We found that the actual computational time-cost of the QFT is O(n 2^n) for large n in a quantum computer using nuclear spins. The computational cost of a quantum algorithm has usually been estimated as the sum of the universal gates…
Most calculations performed by the average R user are unremarkable in the sense that nowadays, any computer can crush the related code in a matter of seconds. But more and more often, heavy calculations are also performed using R, something…
Computational problems are classified into computable and uncomputable problems. If there exists an effective procedure (algorithm) to compute a problem then the problem is computable otherwise it is uncomputable. Turing machines can…
Ising machines are a form of quantum-inspired processing-in-memory computer which has shown great promise for overcoming the limitations of traditional computing paradigms while operating at a fraction of the energy use. The process of…
We give an $O(N\cdot \log N\cdot 2^{O(\log^*N)})$ algorithm for multiplying two $N$-bit integers that improves the $O(N\cdot \log N\cdot \log\log N)$ algorithm by Sch\"{o}nhage-Strassen. Both these algorithms use modular arithmetic.…
How important is fast measurement for fault-tolerant quantum computation? Using a combination of existing and new ideas, we argue that measurement times as long as even 1,000 gate times or more have a very minimal effect on the quantum…
The ultimate limits of computation are not just logical, but physical. We investigate the physical resources -- time, energy, entropy, and free energy -- required to perform computational work. We apply the resulting measures of physical…
Fast algorithms for matrix multiplication, namely those that perform asymptotically fewer scalar operations than the classical algorithm, have been considered primarily of theoretical interest. Apart from Strassen's original algorithm, few…
Most continuous mathematical formulations arising in science and engineering can only be solved numerically and therefore approximately. We shall always assume that we're dealing with a numerical approximation to the solution. There are two…
Residue arithmetic is an elegant and convenient way of computing with integers that exceed the natural word size of a computer. The algorithms are highly parallel and hence naturally adapted to quantum computation. The process differs from…