Related papers: How Fast Can We Multiply Large Integers on an Actu…
In ensemble (or bulk) quantum computation, measurements of qubits in an individual computer cannot be performed. Instead, only expectation values can be measured. As a result of this limitation on the model of computation, various important…
This is a draft of a book about algorithms for performing arithmetic, and their implementation on modern computers. We are concerned with software more than hardware - we do not cover computer architecture or the design of computer…
Almost asymptotically tight lower bounds are derived for the Input/Output (I/O) complexity $IO_\mathcal{A}\left(n,M\right)$ of a general class of hybrid algorithms computing the product of two integers, each represented with $n$ digits in a…
Suppose that we are given an infinite binary sequence which is random for a Bernoulli measure of parameter $p$. By the law of large numbers, the frequency of zeros in the sequence tends to~$p$, and thus we can get better and better…
Mixed integer sets have a strong modeling capacity to describe practical systems. Nevertheless, incorporating a mixed integer set often renders an optimization formulation drastically more challenging to compute. In this paper, we study how…
Similarity comparisons of the form "Is object a more similar to b than to c?" are useful for computer vision and machine learning applications. Unfortunately, an embedding of $n$ points is specified by $n^3$ triplets, making collecting…
In a typical Internet-of-Things setting that involves scientific applications, a target computation can be evaluated in many different ways depending on the split of computations among various devices. On the one hand, different…
The Kolmogorov complexity of a physical state is the minimal physical resources required to reproduce that state. We define a second quantized quantum Turing machine and use it to define second quantized Kolmogorov complexity. There are two…
What can humans compute in their heads? We are thinking of a variety of Crypto Protocols, games like Sudoku, Crossword Puzzles, Speed Chess, and so on. The intent of this paper is to apply the ideas and methods of theoretical computer…
This paper has a practical aim. For a long time, implementations of pseudorandom number generators in standard libraries of programming languages had poor quality. The situation started to improve only recently. Up to now, a large number of…
The fundamental principles of quantum mechanics, such as its probabilistic nature, allow for the theoretical ability of quantum computers to generate statistically random numbers, as opposed to classical computers which are only able to…
Computability on uncountable sets has no standard formalization, unlike that on countable sets, which is given by Turing machines. Some of the approaches to define computability in these sets rely on order-theoretic structures to translate…
Quantum measurement is universal for quantum computation. This universality allows alternative schemes to the traditional three-step organisation of quantum computation: initial state preparation, unitary transformation, measurement. In…
Numerous methods have been considered to create a fast integer factorization algorithm. Despite its apparent simplicity, the difficulty to find such an algorithm plays a crucial role in modern cryptography, notably, in the security of RSA…
We present an efficient algorithm for twirling a multi-qudit quantum state. The algorithm can be used for approximating the twirling operation in an ensemble of physical systems in which the systems cannot be individually accessed. It can…
The Turing Machine has two implicit properties that depend on its underlying notion of computing: the format is fully determinate and computations are information preserving. Distributed representations lack these properties and cannot be…
In this paper, new schemes for a squarer, multiplier and divider of complex numbers are proposed. Traditional structural solutions for each of these operations require the presence some number of general-purpose binary multipliers. The…
Quantum tomography is the main method used to assess the quality of quantum information processing devices, but its complexity presents a major obstacle for the characterization of even moderately large systems. The number of experimental…
Random number generators are widely used in practical algorithms. Examples include simulation, number theory (primality testing and integer factorization), fault tolerance, routing, cryptography, optimization by simulated annealing, and…
Performance of clustering algorithms is evaluated with the help of accuracy metrics. There is a great diversity of clustering algorithms, which are key components of many data analysis and exploration systems. However, there exist only few…