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Theoretical and Mathematical Physics

Solutions of the Yang-Baxter equation associated with a topological basis and applications in quantum information

Mo‐Lin Ge, Li-Wei Yu, Kang Xue, Qing Bo Zhao

Published Oct 1, 2014• 5 citations
Publisher
DOI
10.1007/s11232-014-0205-7

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The solutions of Yang–Baxter equation (YBE) associated with quantum information have been reviewed with new progress. The additivity rule for two particles is shown to obey Lorentzian type other than the Gallilean. Acting the braiding operations on the topological basis, it turns out that the [Formula: see text]-dimensional representations obeying YBE are nothing but the Wigner’s [Formula: see text]-functions. The connection between the extremization of [Formula: see text]-norm as well as of von Neumann entropy and the reduction of YBE to braid relation is also discussed.
The Yang-Baxter equation and it's various forms have applications in many fields, including statistical mechanics, knot theory, and quantum information. Unitary solutions of the braided Yang-Baxter equation are of particular interest as quantum gates for topological quantum computers. We demonstrate a simple construction for solutions in any dimension, which are both unitary and universal for quantum computation. We also fully classify a family of solutions to certain generalized Yang-Baxter equations and prove that certain instances of the equation only have solutions that are scalar multiples of the identity.
The main goal of this paper is to apply the Yang-Baxter-Rowell equation in topological quantum computation, using the symbolic computational software Maple. Initially, using Maple, I reproduced the Dye classification of the 4x4 unitary solutions of the Yang-Baxter equation. Then, also using Maple, I reproduced the solutions given by Rebecca Chen to the Yang-Baxter-Rowell equation. Finally, the obtained solutions were applied to the problem of topological quantum categorification of the chromatic polynomial for graphs. It was claimed that the Yang-Baxter-Rowell operators, considered as quantum gates, can be incorporated in quantum circuit models and quantum games, such as the Grover algorithm and the game Battle of the Sexes. It is expected that a future investigation can provide a more complete classification of the solution to the Yang-Baxter-Rowell equation and its possible applications on topological quantum computation.