Related papers: Quantum Computing and the Jones Polynomial
The quantum circuit model is the most widely used model of quantum computation. It provides both a framework for formulating quantum algorithms and an architecture for the physical construction of quantum computers. However, several other…
We introduce tensor network contraction algorithms for the evaluation of the Jones polynomial of arbitrary knots. The value of the Jones polynomial of a knot maps to the partition function of a $q$-state Potts model defined as a planar…
We present an end-to-end reconfigurable algorithmic pipeline for solving a famous problem in knot theory using a noisy digital quantum computer, namely computing the value of the Jones polynomial at the fifth root of unity within additive…
The repertoire of problems theoretically solvable by a quantum computer recently expanded to include the approximate evaluation of knot invariants, specifically the Jones polynomial. The experimental implementation of this evaluation,…
What is the computational power of a quantum computer? We show that determining the output of a quantum computation is equivalent to counting the number of solutions to an easily computed set of polynomials defined over the finite field…
In this paper we give a quantum statistical interpretation for the bracket polynomial state sum <K> and for the Jones polynomial. We use this quantum mechanical interpretation to give a new quantum algorithm for computing the Jones…
This paper will be an exposition of the Kauffman bracket polynomial model of the Jones polynomial, tangle methods for computing the Jones polynomial, and the use of these methods to produce non-trivial links that cannot be detected by the…
The fundamental group $\pi_1(L)$ of a knot or link $L$ may be used to generate magic states appropriate for performing universal quantum computation and simultaneously for retrieving complete information about the processed quantum states.…
This paper formulates a generalization of our work on quantum knots to explain how to make quantum versions of algebraic, combinatorial and topological structures. We include a description of previous work on the construction of Hilbert…
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…
We present two paradigms relating algebraic, topological and quantum computational statistics for the topological model for quantum computation. In particular we suggest correspondences between the computational power of topological quantum…
In this paper we study the complexity of quantum query algorithms computing the value of Boolean function and its relation to the degree of algebraic polynomial representing this function. We pay special attention to Boolean functions with…
In this paper, we give a description of a recent quantum algorithm created by Aharonov, Jones, and Landau for approximating the values of the Jones polynomial at roots of unity of the form exp(2$\pi$i/k). This description is given with two…
The Jones polynomial of a knot in 3-space is a Laurent polynomial in $q$, with integer coefficients. Many people have pondered why is this so, and what is a proper generalization of the Jones polynomial for knots in other closed…
The theory of quantum computation can be constructed from the abstract study of anyonic systems. In mathematical terms, these are unitary topological modular functors. They underlie the Jones polynomial and arise in Witten-Chern-Simons…
Knots, links and entangled filaments appear in many physical systems of interest in biology and engineering. Classifying knots and measuring entanglement is of interest both for advancing knot theory, as well as for analyzing large data…
In this paper, we explore algebraic structures and low dimensional topology underlying quantum information and computation. We revisit quantum teleportation from the perspective of the braid group, the symmetric group and the virtual braid…
In this paper we review our current results concerning the computational power of quantum read-once branching programs. First of all, based on the circuit presentation of quantum branching programs and our variant of quantum fingerprinting…
The relationship between quantum physics and discrete mathematics is reviewed in this article. The Boolean functions unitary representation is considered. The relationship between Zhegalkin polynomial, which defines the algebraic normal…
Let K be a 3-stranded knot (or link), and let L denote the number of crossings in K. Let $\epsilon_{1}$ and $\epsilon_{2}$ be two positive real numbers such that $\epsilon_{2}$ is less than or equal to 1. In this paper, we create two…