Related papers: Computing with Classical Real Numbers

Surrogate models are ubiquitously used in industry and academia to efficiently approximate given black box functions. As state-of-the-art methods from classical machine learning frequently struggle to solve this problem accurately for the…

We reveal a natural algebraic problem whose complexity appears to interpolate between the well-known complexity classes BQP and NP: (*) Decide whether a univariate polynomial with exactly m monomial terms has a p-adic rational root. In…

Quantum Information Theory, the standard formalism used to represent information contained in quantum systems, is based on complex Hilbert spaces (CQT). It was recently shown that it predicts correlations in quantum networks which cannot be…

Clifford gates are a winsome class of quantum operations combining mathematical elegance with physical significance. The Gottesman-Knill theorem asserts that Clifford computations can be classically efficiently simulated but this is true…

interpreters are tools to compute approximations for behaviors of a program. These approximations can then be used for optimisation or for error detection. In this paper, we show how to describe an abstract interpreter using the type-theory…

Solving linear systems of equations is ubiquitous in all areas of science and engineering. With rapidly growing data sets, such a task can be intractable for classical computers, as the best known classical algorithms require a time…

Although classical mechanics and quantum mechanics are separate disciplines, we live in a world where Planck's constant \hbar>0, meaning that the classical and quantum world views must actually {\it coexist}. Traditionally, canonical…

This paper contains a discussion of a library of formalized mathematics for the proof assistant Coq which the author worked on in 2011-13.

Theorem provers are important tools for people working in formal verification. There are a myriad of interactive systems available today, with varying features and approaches motivating their development. These design choices impact their…

The realistic interpretation of classical theory assumes that every classical system has well-defined properties, which may be unknown to the observer but are nevertheless part of reality and can in principle be revealed by measurements.…

We present algorithmic and complexity results concerning computations with one and two real algebraic numbers, as well as real solving of univariate polynomials and bivariate polynomial systems with integer coefficients using Sturm-Habicht…

In recent years, strong expectations have been raised for the possible power of quantum computing for solving difficult optimization problems, based on theoretical, asymptotic worst-case bounds. Can we expect this to have consequences for…

We employ an algebraic procedure based on quantum mechanics to propose a `quantum number theory' (QNT) as a possible extension of the `classical number theory'. We built our QNT by defining pure quantum number operators ($q$-numbers) of a…

We consider quantum versions of two well-studied classical learning models: Angluin's model of exact learning from membership queries and Valiant's Probably Approximately Correct (PAC) model of learning from random examples. We give…

In this paper, we discuss questions related to the renormalizability of the classical statistical approximation, an approximation scheme that has been used recently in several studies of out-of-equilibrium problems in Quantum Field Theory.…

In this paper we give a preliminary formalization of the p-adic numbers, in the context of the second author's univalent foundations program. We also provide the corresponding code verifying the construction in the proof assistant Coq.…

Quantum neural networks (QNNs) are an analog of classical neural networks in the world of quantum computing, which are represented by a unitary matrix with trainable parameters. Inspired by the universal approximation property of classical…

There are many ways to construct the field R of real numbers. The most important and famous of these employ Cauchy sequences (Cantor) or cuts (Dedekind) in the field Q of rational numbers. These constructions sometimes overlook important…

Quantum computers can execute algorithms that dramatically outperform classical computation. As the best-known example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for…

Many interesting computational problems can be reformulated in terms of decision trees. A natural classical algorithm is to then run a random walk on the tree, starting at the root, to see if the tree contains a node n levels from the root.…