scholarly journals Polyadic Braid Operators and Higher Braiding Gates

Universe ◽  
2021 ◽  
Vol 7 (8) ◽  
pp. 301
Author(s):  
Steven Duplij ◽  
Raimund Vogl
Keyword(s):  
Algebraic Structure ◽  
Braid Group ◽  
Quantum Gates ◽  
Baxter Equation ◽  
Bounded Operators ◽  
Cartan Decomposition ◽  
Classes Of Matrices ◽  
Speed Up ◽  
Qubit Loss ◽  
Qubit States

A new kind of quantum gates, higher braiding gates, as matrix solutions of the polyadic braid equations (different from the generalized Yang--Baxter equations) is introduced. Such gates lead to another special multiqubit entanglement that can speed up key distribution and accelerate algorithms. Ternary braiding gates acting on three qubit states are studied in detail. We also consider exotic non-invertible gates, which can be related with qubit loss, and define partial identities (which can be orthogonal), partial unitarity, and partially bounded operators (which can be non-invertible). We define two classes of matrices, star and circle ones, such that the magic matrices (connected with the Cartan decomposition) belong to the star class. The general algebraic structure of the introduced classes is described in terms of semigroups, ternary and $5$-ary groups and modules. The higher braid group and its representation by the higher braid operators are given. Finally, we show, that for each multiqubit state, there exist higher braiding gates that are not entangling, and the concrete conditions to be non-entangling are given for the obtained binary and ternary gates.Yang--Baxter equation; braid group; qubit; ternary; polyadic; braiding quantum gate.

scholarly journals GENERALIZED YANG–BAXTER EQUATIONS AND BRAIDING QUANTUM GATES

2012 ◽  
Vol 21 (09) ◽  
pp. 1250087 ◽  
Author(s):  
REBECCA S. CHEN
Keyword(s):  
Braid Group ◽  
Knot Theory ◽  
Information Science ◽  
Quantum Gates ◽  
Quantum Computers ◽  
Baxter Equation ◽  
Group Representations ◽  
Ongoing Research ◽  
Topological Quantum ◽  
Mathematics And Physics

Solutions to the Yang–Baxter equation — an important equation in mathematics and physics — and their afforded braid group representations have applications in fields such as knot theory, statistical mechanics, and, most recently, quantum information science. In particular, unitary representations of the braid group are desired because they generate braiding quantum gates. These are actively studied in the ongoing research into topological quantum computing. A generalized Yang–Baxter equation was proposed a few years ago by Eric Rowell et al. By finding solutions to the generalized Yang–Baxter equation, we obtain new unitary braid group representations. Our representations give rise to braiding quantum gates and thus have the potential to aid in the construction of useful quantum computers.

scholarly journals Braiding quantum gates from partition algebras

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 311
Author(s):  
Pramod Padmanabhan ◽  
Fumihiko Sugino ◽  
Diego Trancanelli
Keyword(s):  
Statistical Mechanics ◽  
Braid Group ◽  
Quantum Gates ◽  
Entangled States ◽  
Baxter Equation ◽  
Classical Communication ◽  
Qubit System ◽  
Partition Algebras

Unitary braiding operators can be used as robust entangling quantum gates. We introduce a solution-generating technique to solve the (d,m,l)-generalized Yang-Baxter equation, for m/2≤l≤m, which allows to systematically construct such braiding operators. This is achieved by using partition algebras, a generalization of the Temperley-Lieb algebra encountered in statistical mechanics. We obtain families of unitary and non-unitary braiding operators that generate the full braid group. Explicit examples are given for a 2-, 3-, and 4-qubit system, including the classification of the entangled states generated by these operators based on Stochastic Local Operations and Classical Communication.

scholarly journals Measuring the Tangle of Three-Qubit States

Entropy ◽  
2020 ◽  
Vol 22 (4) ◽  
pp. 436 ◽  
Author(s):  
Adrián Pérez-Salinas ◽  
Diego García-Martín ◽  
Carlos Bravo-Prieto ◽  
José Latorre
Keyword(s):  
Direct Measurement ◽  
Canonical Form ◽  
Quantum Circuit ◽  
Quantum Gates ◽  
Tripartite Entanglement ◽  
Single Qubit ◽  
Optimal Values ◽  
Random States ◽  
Qubit States ◽  
Computational Basis

We present a quantum circuit that transforms an unknown three-qubit state into its canonical form, up to relative phases, given many copies of the original state. The circuit is made of three single-qubit parametrized quantum gates, and the optimal values for the parameters are learned in a variational fashion. Once this transformation is achieved, direct measurement of outcome probabilities in the computational basis provides an estimate of the tangle, which quantifies genuine tripartite entanglement. We perform simulations on a set of random states under different noise conditions to asses the validity of the method.

FERMIONS AND LINK INVARIANTS

1992 ◽  
Vol 07 (supp01a) ◽  
pp. 493-532 ◽  
Author(s):  
L. Kauffman ◽  
H. Saleur
Keyword(s):  
Braid Group ◽  
Degrees Of Freedom ◽  
Knot Theory ◽  
Alexander Polynomial ◽  
Baxter Equation ◽  
Group Representations ◽  
R Matrix ◽  
Link Invariants ◽  
Linear Representations ◽  
State Models

This paper deals with various aspects of knot theory when fermionic degrees of freedom are taken into account in the braid group representations and in the state models. We discuss how the Ř matrix for the Alexander polynomial arises from the Fox differential calculus, and how it is related to the quantum group Uqgl(1,1). We investigate new families of solutions of the Yang Baxter equation obtained from "linear" representations of the braid group and exterior algebra. We study state models associated with Uqsl(n,m), and in the case n=m=1 a state model for the multivariable Alexander polynomial. We consider invariants of links in solid handlebodies and show how the non trivial topology lifts the boson fermion degeneracy that is present in S3. We use "gauge like" changes of basis to obtain invariants in thickened surfaces Σ×[0,1].

REALIZATION OF FRT ALGEBRA RELATED TO A COLORED BRAID GROUP REPRESENTATION

1994 ◽  
Vol 09 (29) ◽  
pp. 2733-2743 ◽  
Author(s):  
B. BASU-MALLICK
Keyword(s):  
Braid Group ◽  
Group Representation ◽  
Toda Chain ◽  
Baxter Equation ◽  
Color Parameter ◽  
Triangular Matrices ◽  
Explicit Realization ◽  
Lower Triangular ◽  
Parameter Dependent ◽  
Lower Triangular Matrices

A colored braid group representation (CBGR) is constructed by using some modified universal ℛ-matrix associated with U q( gl (2)) quantized algebra. Explicit realization of Faddeev–Reshetikhin–Takhtajan (FRT) algebra, involving color parameter dependent upper and lower triangular matrices, is built up for this CBGR and subsequently applied to generate nonadditive type solutions of quantum Yang–Baxter equation. Rational limit of such solutions interestingly yields 'colored' extension of known Lax operators associated with lattice nonlinear Schrödinger model and Toda chain.

scholarly journals BRAID GROUP REPRESENTATIONS ARISING FROM THE YANG–BAXTER EQUATION

2010 ◽  
Vol 19 (04) ◽  
pp. 525-538 ◽  
Author(s):  
JENNIFER M. FRANKO
Keyword(s):  
Group Algebra ◽  
Braid Group ◽  
Roots Of Unity ◽  
Baxter Equation ◽  
Group Representations ◽  
Link Invariant

This paper aims to determine the images of the braid group under representations afforded by the Yang–Baxter equation when the solution is a non-trivial 4 × 4 matrix. Making the assumption that all the eigenvalues of the Yang–Baxter solution are roots of unity, leads to the conclusion that all the images are finite. Using results of Turaev, we have also identified cases in which one would get a link invariant. Finally, by observing the group algebra generated by the image of the braid group sometimes factor through known algebras, in certain instances we can identify the invariant as particular specializations of a known invariant.

Graphical solutions of the Yang–Baxter Equation with cut strands

2017 ◽  
Vol 26 (04) ◽  
pp. 1750012
Author(s):  
Péter Varga
Keyword(s):  
Braid Group ◽  
Baxter Equation

We study the representations of the braid group in a generalization of the Temperley–Lieb algebra. This algebra was introduced by Kloster and contains the diagrams of the TL algebra with cut strands. We address the question of existence of Markov traces on these representations and make some observations on the possibility of their Baxterizations.

scholarly journals Quantum Spectral Curve of γ-Twisted 𝒩 = 4 SYM Theory and Fishnet CFT

2018 ◽  
Vol 30 (07) ◽  
pp. 1840010 ◽  
Author(s):  
Vladimir Kazakov
Keyword(s):  
Algebraic Structure ◽  
Spectral Curve ◽  
Scaling Limit ◽  
Integrable Model ◽  
Quantization Condition ◽  
Spin Chains ◽  
Hasse Diagram ◽  
Baxter Equation ◽  
Heisenberg Spin Chain ◽  
Wheel Graphs

We review the quantum spectral curve (QSC) formalism for the spectrum of anomalous dimensions of [Formula: see text] SYM, including its [Formula: see text]-deformation. Leaving aside its derivation, we concentrate on the formulation of the “final product” in its most general form: a minimal set of assumptions about the algebraic structure and the analyticity of the [Formula: see text]-system — the full system of Baxter [Formula: see text]-functions of the underlying integrable model. The algebraic structure of the [Formula: see text]-system is entirely based on (super)symmetry of the model and is efficiently described by Wronskian formulas for [Formula: see text]-functions organized into the Hasse diagram. When supplemented with analyticity conditions on [Formula: see text]-functions, it fixes completely the set of physical solutions for the spectrum of an integrable model. First, we demonstrate the spectral equations on the example of [Formula: see text] and [Formula: see text] Heisenberg (super)spin chains. Supersymmetry [Formula: see text] occurs as a simple “rotation” of the Hasse diagram for a [Formula: see text] system. Then we apply this method to the spectral problem of [Formula: see text]/CFT4-duality, describing the QSC formalism. The main difference with the spin chains consists in more complicated analyticity constraints on [Formula: see text]-functions which involve an infinitely branching Riemann surface and a set of Riemann–Hilbert conditions. As an example of application of QSC, we consider a special double scaling limit of [Formula: see text]-twisted [Formula: see text] SYM, combining weak coupling and strong imaginary twist. This leads to a new type of non-unitary CFT dominated by particular integrable, and often computable, 4D fishnet Feynman graphs. For the simplest of such models — the bi-scalar theory — the QSC degenerates into the [Formula: see text]-system for integrable non-compact Heisenberg spin chain with conformal, [Formula: see text] symmetry. We describe the QSC derivation of Baxter equation and the quantization condition for particular fishnet graphs — wheel graphs, and review numerical and analytic results for them.

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