Generalized Yang-Baxter Equation Research Articles

New series of multi-parametric solutions to GYBE: Quantum gates and integrability

We obtain two series of spectral parameter dependent solutions to the generalized Yang-Baxter equations (GYBE), for definite types of N12×N22 matrices with general dimensions N1 and N2. Appropriate extensions are presented for the inhomogeneous GYBEs. The first series of the solutions includes as particular cases the X-shaped trigonometric braiding matrices. For construction of the second series the colored and graded permutation operators are defined, and multi-spectral parameter Yang-Baxterization is performed. For some examples the corresponding integrable models are discussed. The unitary solutions existing in these two series can be considered as generalizations of the multipartite Bell matrices in the quantum information theory.

Polyadic Braid Operators and Higher Braiding Gates

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.

Braiding quantum gates from 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.

Quantum entanglement, supersymmetry, and the generalized Yang-Baxter equation

Entangled states, such as the Bell and GHZ states, are generated from separable states using matrices known to satisfy the Yang-Baxter equation and its generalization. This remarkable fact hints at the possibility of using braiding operators as quantum entanglers, and is part of a larger speculated connection between topological and quantum entanglement. We push the analysis of this connection forward, by showing that supersymmetry algebras can be used to construct large families of solutions of the spectral parameter-dependent generalized Yang-Baxter equation. We present a number of explicit examples and outline a general algorithm for arbitrary numbers of qubits. The operators we obtain produce, in turn, all the entangled states in a multi-qubit system classified by the Stochastic Local Operations and Classical Communication protocol introduced in quantum information theory.

New type of solutions of Yang-Baxter equations, quantum entanglement and related physical models

Starting from the Kauffman-Lomonaco braiding matrix transforming the natural basis to Bell states, the spectral parameter describing the entanglement is introduced through Yang-Baxterization. It gives rise to a new type of solutions for Yang-Baxter equation, called the type-II that differs from the familiar solution called type-I of YBE associated with the usual chain models. The Majorana fermionic version of type-II yields the Kitaev Hamiltonian. The introduced ℓ1-norm leads to the maximum of the entanglement by taking the extreme value and shows that it is related to the Wigner’s D-function. Based on the Yang-Baxter equation the 3-body S-Matrix for type-II is explicitly given. Different from the type-I solution, the type-II solution of YBE should be considered in describing quantum information. The idea is further extended to ℤ3 parafermion model based on SU(3) principal representation. The type-II is in difference from the familiar type-I in many respects. For example, the quantities corresponding to velocity in the chain models obey the Lorentzian additivity rather than Galilean rule (u + v). Most possibly, for the type-II solutions of YBE there may not exist RTT relation. Further more, for ℤ3 parafermion model we only need the rational Yang-Baxterization, which seems like trigonometric. Similar discussions are also made in terms of generalized Yang-Baxter equation with three spin spaces .

Gaussian (N, z)-generalized Yang-Baxter operators

We find unitary matrix solutions R˜(a) to the (multiplicative parameter-dependent) (N, z)-generalized Yang-Baxter equation that carry the standard measurement basis to m-level N-partite entangled states that generalize the 2-level bipartite entangled Bell states. This is achieved by a careful study of solutions to the Yang-Baxter equation discovered by Fateev and Zamolodchikov in 1982.

Extraspecial two-Groups, generalized Yang-Baxter equations and braiding quantum gates

In this paper we describe connections among extraspecial 2-groups, unitary representations of the braid group and multi-qubit braiding quantum gates. We first construct new representations of extraspecial 2-groups. Extending the latter by the symmetric group, we construct new unitary braid representations, which are solutions to generalized Yang-Baxter equations and use them to realize new braiding quantum gates. These gates generate the GHZ (Greenberger-Horne-Zeilinger) states, for an arbitrary (particularly an \emph{odd}) number of qubits, from the product basis. We also discuss the Yang-Baxterization of the new braid group representations, which describes unitary evolution of the GHZ states. Our study suggests that through their connection with braiding gates, extraspecial 2-groups and the GHZ states may play an important role in quantum error correction and topological quantum computing.

Extraspecial two-groups, generalized yang-baxter equations and braiding quantum gates

In this paper we describe connections among extraspecial 2-groups, unitary representa-tions of the braid group and multi-qubit braiding quantum gates. We first construct newrepresentations of extraspecial 2-groups. Extending the latter by the symmetric group,we construct new unitary braid representations, which are solutions to generalized Yang-Baxter equations and use them to realize new braiding quantum gates. These gates gen-erate the GHZ (Greenberger-Horne-Zeilinger) states, for an arbitrary (particularly anodd) number of qubits, from the product basis. We also discuss the Yang-Baxterizationof the new braid group representations, which describes unitary evolution of the GHZstates. Our study suggests that through their connection with braiding gates, extraspe-cial 2-groups and the GHZ states may play an important role in quantum error correctionand topological quantum computing.

On the integrability of N = 2 supersymmetric massive theories

In this paper we propose a criterion to establish the integrability of N = 2 supersymmetric massive theories. The basic data required are the vacua and the spectrum of Bogomolnyi solitons, which can be neatly encoded in a graph (nodes=vacua and links=Bogomolnyi solitons). Integrability is then equivalent to the existence of solutions of a generalized Yang-Baxter equation which is built up from the graph (graph-Yang-Baxter equation). We solve this equation for two general types of graphs: circular and daisy, proving, in particular, the integrability of the following Landau-Ginzburg superpotentials: A n ( t 1), A n ( t 2), D n ( τ), E 6( t 7) and E 8( t 16). For circular graphs the solution are intertwiners of the affine Hopf algebra U q( A 1 (1)) , while for daisy graphs the solution corresponds to a SUSY generalization of the Boltzmann weights of the chiral Potts model in the trigonometric regime. A chiral-Potts-like solution is conjectured for the more tricky case D n ( t 2). The scattering theory of circular models, for instance A n ( t 1) or D n ( τ), is Toda-like. The physical spectrum of daisy models, as A n ( t 2), E 6( t 7) or E 8( t 16), is given by confined states of radial solitons. The scattering theory of the confined states is again Toda-like. Bootstrap factors for the confined solitons are given by fusing the SUSY chiral Potts S-matrices of the elementary constituents; i.e the radial solitons of the daisy graph.

EQUIVALENCE OF S3-INVARIANT LATTICE MODEL AND MODEL OF VERTEX-FACE TYPE

We show that the S3-invariant model proposed in Ref. 2 is equivalent to vertex-face model with two kinds of Boltzmann weights, which appeared to satisfy the generalized Yang-Baxter equation. The more general solution of this equation is constructed in the framework of some appropriate ansatz. This solution depends on the continuous parameter q. The initial model is a particular case of q=±ie∓2π/3i. The integrability of this model is discussed.

Generalized Yang-Baxter equations, Koszul operators and Poisson Lie groups

Generalized Yang-Baxter equations, Koszul operators and Poisson Lie groups

Bethe Ansatz and the generalized Yang-Baxter equations

Bethe Ansatz and the generalized Yang-Baxter equations