One of the current trends in quantum computing is the application of algebraic topological methods in the design of new algorithms and quantum computers, giving rise to topological quantum computing. Two-spectral Yang-Baxter operators in topological quantum computation Consequently, it opens up a way to further investigate entanglements in a Yang-Baxter equation. Contrasting to conventional methods in which the two hand sides of the equation are simulated separately, they are simulated simultaneously in this proposal. In this work, we propose a duality quantum simulation algorithm of the Yang-Baxter equation, which contains the Yang-Baxter system and an ancillary qubit. It is desirable to investigate the quantum simulation of the Yang-Baxter equation itself, exploring the connections between quantum integrability and quantum information processing, in which the unity of both the Yang-Baxter equation system and its quantum entanglement should be kept as a whole. The Yang-Baxter equation has become a significant theoretical tool in a variety of areas of physics. In this paper we present detailed considerations of the above scheme on the example of the algebra Uq (sl (2)) leading to discrete Liouville equations, however the approach is rather general and can be applied to any quantized Lie algebra.Дuality Quantum Simulation of the Yang-Baxter Equation As a result one obtains an integrable discrete Hamiltonian evolution system, where the local equation of motion are determined by a classical Yang-Baxter map and the action functional is determined by the quasi-classical asymptotics of the universal R-matrix of the underlying quantum algebra. All elements of the above construction have a meaningful quasi-classical limit. The commuting Integrals of Motion are defined in the standard way via the Quantum Inverse Problem Method, utilizing Baxter's famous commuting transfer matrix approach.

The corresponding equations of motion admit the zero curvature representation. This map allows one to define an integrable discrete quantum evolution system on quadrilateral lattices, where local degrees of freedom (dynamical variables) take values in a tensor power of the quantized Lie algebra. Sergeev, Sergey M.ДЇor every quantized Lie algebra there exists a map from the tensor square of the algebra to itself, which by construction satisfies the set-theoretic Yang-Baxter equation. Yang-Baxter maps, discrete integrable equations and quantum groupsД«azhanov, Vladimir V. The quasi-determinant expression of the quantum Yang-Baxter map reduces to ratios of determinants, which give a new expression of a classical Yang-Baxter map. We also consider the quasi-classical limit, where the underlying quantum algebra reduces to a Poisson algebra.

Moreover, the map is identified with products of quasi-PlГѓВјcker coordinates over a matrix composed of the L-operators. Thereby obtained a quasi-determinant expression of the quantum Yang-Baxter map associated with the quantum algebra Uq (gl (n)). We find that the zero curvature representation can be solved by the Gauss decomposition of a product of L-operators.

The map has a zero curvature representation among L-operators defined as images of the universal R-matrix.

It is known that the adjoint action of the universal R-matrix on the elements of the tensor square of the algebra constitutes a quantum Yang-Baxter map, which satisfies the set-theoretic Yang-Baxter equation. Quantum groups, Yang-Baxter maps and quasi-determinantsДЇor any quasi-triangular Hopf algebra, there exists the universal R-matrix, which satisfies the Yang-Baxter equation.

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