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arXiv:2004.07764 · Journal of Experimental and Theoretical Physics Letters

Quantum ℛ-Matrices as Universal Qubit Gates

Nikita Kolganov, An. Morozov

Published Apr 25, 2020• 17 citations
arXivHTMLPDF

Abstract

We study the Chern-Simons approach to the topological quantum computing. We use quantum ℛ-matrices as universal quantum gates and study the approximations of some one-qubit operations. We make some modifications to the known Solovay—Kitaev algorithm suitable for our particular problem.
DOI
10.1134/s0021364020090027

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Chern-Simons topological quantum computer is a device that can be effectively described by the Chern-Simons topological quantum field theory and used for quantum computations. Quantum qudit gates of this quantum computer are represented by sequences of quantum R-matrices. Its dimension and explicit form depend on the parameters of the Chern-Simons theory — level k, gauge group SU(N), and representation, which is chosen to be symmetric representation [r]. In this paper, we examine the universality of such a quantum computer. We prove that for sufficiently large k it is universal, and the minimum allowed value of k depends on the remaining parameters r and N.
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Topological quantum computation may provide a robust approach for encoding and manipulating information utilizing the topological properties of anyonic quasiparticle excitations. We develop an efficient means to map between dense and sparse representations of quantum information (qubits) and a simple construction of multiqubit gates, for all anyon models from Chern-Simons-Witten SU(2)k{}_{k}k​ theory that support universal quantum computation by braiding ($k\ensuremath{\geqslant}3,\phantom{\rule{4pt}{0ex}}k\ensuremath{\ne}4$). In the process, we show how the constructions of topological quantum memory and gates for k=2,4k=2,4k=2,4 connect naturally to those for $k\ensuremath{\geqslant}3,\phantom{\rule{4pt}{0ex}}k\ensuremath{\ne}4$, unifying these concepts in a simple framework. Furthermore, we illustrate potential extensions of these ideas to other anyon models outside of Chern-Simons-Witten field theory.