Related papers: Uniform Approximation by (Quantum) Polynomials
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…
In this note we propose a version of the classical Stone-Weierstrass theorem in the context of quantum operations, by introducing a particular class of quantum operations, dubbed polynomial quantum operations. This result permits to…
Bernstein polynomials provide a constructive proof for the Weierstrass approximation theorem, which states that every continuous function on a closed bounded interval can be uniformly approximated by polynomials with arbitrary accuracy.…
Motivated by the result that an `approximate' evaluation of the Jones polynomial of a braid at a $5^{th}$ root of unity can be used to simulate the quantum part of any algorithm in the quantum complexity class BQP, and results relating BQP…
The celebrated and famous Weierstrass approximation theorem characterizes the set of continuous functions on a compact interval via uniform approximation by algebraic polynomials. This theorem is the first significant result in…
We strengthen the Weierstrass approximation theorem by proving that any real-valued continuous function on an interval $I \subset \mathbb{R}$ can be uniformly approximated by a real-valued polynomial whose only (possibly complex) critical…
We describe algorithms to obtain an approximate classical description of a $d$-dimensional quantum state when given access to a unitary (and its inverse) that prepares it. For pure states we characterize the query complexity for…
Motivated by algorithmic problems arising in quantum field theories whose dynamical variables are geometric in nature, we provide a quantum algorithm that efficiently approximates the colored Jones polynomial. The construction is based on…
In this paper we use the Vandermonde matrices and their properties to give a new proof of the classical result of Karl Weierstrass about the approximation of continuous functions $f$ on closed intervals, using a sequence of polynomials. The…
There is a natural relationship between Jones polynomials and quantum computation. We use this relationship to show that the complexity of evaluating relative-error approximations of Jones polynomials can be used to bound the classical…
It is a classical result in rational approximation theory that certain non-smooth or singular functions, such as $|x|$ and $x^{1/p}$, can be efficiently approximated using rational functions with root-exponential convergence in terms of…
Approximation algorithms for classical constraint satisfaction problems are one of the main research areas in theoretical computer science. Here we define a natural approximation version of the QMA-complete local Hamiltonian problem and…
Combinatorial optimization - a field of research addressing problems that feature strongly in a wealth of scientific and industrial contexts - has been identified as one of the core potential fields of applicability of quantum computers. It…
Universal approximation theorems are the foundations of classical neural networks, providing theoretical guarantees that the latter are able to approximate maps of interest. Recent results have shown that this can also be achieved in a…
We present a quantum algorithm for systems of (possibly inhomogeneous) linear ordinary differential equations with constant coefficients. The algorithm produces a quantum state that is proportional to the solution at a desired final time.…
We prove a version of the quantum de Finetti theorem: permutation-invariant quantum states are well approximated as a probabilistic mixture of multi-fold product states. The approximation is measured by distinguishability under fully…
We construct a quantum algorithm to approximate efficiently the colored Jones polynomial of the plat presentation of any oriented link L at a fixed root of unity q. Our construction is based on SU(2) Chern-Simons topological quantum field…
We show that, in spite of a rather common opinion, quantum mechanics can be represented as an approximation of classical statistical mechanics. The approximation under consideration is based on the ordinary Taylor expansion of physical…
We seek random versions of some classical theorems on complex approximation by polynomials and rational functions, as well as investigate properties of random compact sets in connection to complex approximation.
Quantum algorithm is an algorithm for solving mathematical problems using quantum systems encoded as information, which is found to outperform classical algorithms in some specific cases. The objective of this study is to develop a quantum…