arXiv:2105.03980 · Nuclear Physics B

Large k topological quantum computer

Nikita Kolganov, Sergey Mironov, Andrey Morozov

Published Jan 9, 2023• 15 citations

Abstract

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.

DOI: 10.1016/j.nuclphysb.2023.116072

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arXiv:2105.03980 · Nuclear Physics B

Large k topological quantum computer

Nikita Kolganov, Sergey Mironov, Andrey Morozov

Published Jan 9, 2023• 15 citations

arXiv HTML PDF

Abstract

DOI: 10.1016/j.nuclphysb.2023.116072

Related papers

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Abstract Chern–Simons theory in application to the quantum computing is actively developing at the present. However, most discussed are the questions of using materials with known parameters and building corresponding quantum gates and algorithms. In this paper we discuss opposite problem of finding Chern–Simons level k in the unknown material. For this purpose, we use the previously derived braiding rules for Chern–Simons $SU(2)_k$ S U ( 2 ) k anyons. Using certain operations (turnarounds) on three anyons, one can measure probabilities of annihilation of pairs of anyons, which depend on the parameter of the theory. Therefore, Chern–Simons level k can be found from such an experiment. It is implied that anyons additionally possess certain properties which are required for topological quantum computations.Abstract Chern–Simons theory in application to the quantum computing is actively developing at the present. However, most discussed are the questions of using materials with known parameters and building corresponding quantum gates and algorithms. In this paper we discuss opposite problem of finding Chern–Simons level k in the unknown material. For this purpose, we use the previously derived braiding rules for Chern–Simons $$SU(2)_k$$ S U ( 2 ) k anyons. Using certain operations (turnarounds) on three anyons, one can measure probabilities of annihilation of pairs of anyons, which depend on the parameter of the theory. Therefore, Chern–Simons level k can be found from such an experiment. It is implied that anyons additionally possess certain properties which are required for topological quantum computations.

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{}_{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=2,4

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.

It is shown, using level-rank duality that a universal topological quantum computer based on Chern-Simons theory for SU(2)

_3

also implies an analogous universal quantum computer based on SU(3)

_2

. Suggestions are made for the possible role of level-rank duality in entanglement from topology.

Measuring Chern–Simons level k by braiding

SU(2)_k

anyons

SU(2)_k

S U ( 2 ) k anyons. Using certain operations (turnarounds) on three anyons, one can measure probabilities of annihilation of pairs of anyons, which depend on the parameter of the theory. Therefore, Chern–Simons level k can be found from such an experiment. It is implied that anyons additionally possess certain properties which are required for topological quantum computations.