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Register a new userBerry phase and quantum criticality in Yang--Baxter systems
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Spin interaction Hamiltonians are obtained from the unitary Yang--Baxter $breve{R}$-matrix. Based on which, we study Berry phase and quantum criticality in the Yang--Baxter systems.
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In this paper, we mainly present some new solutions of the Hom-Yang-Baxter equation from Hom-algebras, Hom-coalgebras and Hom-Lie algebras, respectively. Also, we prove that these solutions are all self-inverse and give some examples. Finally, we introduce the notion of Hom-Yang-Baxter systems and obtain two kinds of Hom-Yang-Baxter systems.
Any solution to the Yang-Baxter equation yields a family of representations of braid groups. Under certain conditions, identified by Turaev, the appropriately normalized trace of these representations yields a link invariant. Any Yang-Baxter solution can be interpreted as a two-qudit quantum gate. Here we show that if this gate is non-entangling, then the resulting invariant of knots is trivial. We thus obtain a general connection between topological entanglement and quantum entanglement, as suggested by Kauffman et al.
In this paper, several proposals of optically simulating Yang-Baxter equations have been presented. Motivated by the recent development of anyon theory, we apply Temperley-Lieb algebra as a bridge to recast four-dimentional Yang-Baxter equation into its two-dimensional counterpart. In accordance with both representations, we find the corresponding linear-optical simulations, based on the highly efficient optical elements. Both the freedom degrees of photon polarization and location are utilized as the qubit basis, in which the unitary Yang-Baxter matrices are decomposed into combination of actions of basic optical elements.
A unitary operator that satisfies the constant Yang-Baxter equation immediately yields a unitary representation of the braid group B n for every $n ge 2$. If we view such an operator as a quantum-computational gate, then topological braiding corresponds to a quantum circuit. A basic question is when such a representation affords universal quantum computation. In this work, we show how to classically simulate these circuits when the gate in question belongs to certain families of solutions to the Yang-Baxter equation. These include all of the qubit (i.e., $d = 2$) solutions, and some simple families that include solutions for arbitrary $d ge 2$. Our main tool is a probabilistic classical algorithm for efficient simulation of a more general class of quantum circuits. This algorithm may be of use outside the present setting.
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