arXiv:2408.14188 · The European Physical Journal C

Measuring Chern–Simons level k by braiding $$SU(2)_k$$ anyons

A. A. Belov, Andrey Morozov

Published Jan 27, 2025• 0 citations

Abstract

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.

DOI: 10.1140/epjc/s10052-024-13734-1

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arXiv:2408.14188 · The European Physical Journal C

Measuring Chern–Simons level k by braiding $SU(2)_k$ anyonsMeasuring Chern–Simons level k by braiding $$SU(2)_k$$ anyons

A. A. Belov, Andrey Morozov

Published Jan 27, 2025• 0 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.1140/epjc/s10052-024-13734-1

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Entangling gates for the SU(N) anyons

Sergey Mironov, Andrey Morozov

The model of a topological quantum computer is a promising one due to its natural resistance to noise and other errors. Operations in such a computer are implemented by braiding the trajectories of anyons. While it is easy to understand how to build one-qubit operations, two-qubit operations are more difficult. In arXiv:2412.20931 we suggested an approach to build such operations for a topological quantum computer based on SU(2) Chern-Simons theory with arbitrary level using cabling of knots. In this paper we discuss how this approach should be generalized to the SU(N) case, what the differences are, and which new problems arise.The model of a topological quantum computer is a promising one due to its natural resistance to noise and other errors. Operations in such a computer are implemented by braiding the trajectories of anyons. While it is easy to understand how to build one-qubit operations, two-qubit operations are more difficult. In arXiv:2412.20931 we suggested an approach to build such operations for a topological quantum computer based on SU(2) Chern-Simons theory with arbitrary level using cabling of knots. In this paper we discuss how this approach should be generalized to the SU(N) case, what the differences are, and which new problems arise.

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Measuring Chern–Simons level k by braiding

SU(2)_k

anyons

SU(2)_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=2,4

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.

FINDING: Chern-Simons theory at level

k

governs fractional quantum Hall anyons; braiding produces non-abelian phases encoded in the Jones polynomial; lattice realizations confirm topological invariance; a novel CKM angle measurement is unrelated to topological physics. MATH: Chern-Simons action

S_{CS} = \frac{k}{4\pi}\int \text{Tr}(A\wedge dA + \frac{2}{3}A\wedge A\wedge A)

; level

k \in \mathbb{Z}

(integer QHE) or

k = p/q

(fractional QHE, e.g.,

k=1/3

for Laughlin state); anyon braiding phase

\theta = \pi/k

(abelian) or unitary matrix

U_{ab}

(non-abelian, e.g., Fibonacci anyons with

U \sim e^{i\pi/5}

); entanglement entropy

S_A = \alpha L - \gamma

, with topological entanglement entropy

\gamma = \ln \sqrt{D}

D = \sum_i d_i^2

(quantum dimensions); lattice Chern-Simons reproduces continuum in limit

a \to 0

with

k

quantized. CONNECTION: Level

k=5

gives Fibonacci anyons with quantum dimension

d = \phi = 1.618

(golden ratio) — di Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

SU(2)

gauge group as our working example. The problem of determining the Renyi entropy is mapped to computing the expectation value of an auxiliary Wilson loop in

S^3

for each braid. We study various properties of this auxiliary Wilson loop for some 2-strand and 3-strand braids, and demonstrate how they reflect some geometrical properties of the underlying braids.

D(H)

of the subgroup. The quasi-triangular Hopf Algebra D(H) is obtained from Drinfeld's quantum double construction applied to the algebra F(H) of functions on the finite group H. A major new contribution of this work is a program written in MAGMA to compute the particles (and their properties - including spin) that can exist in a system with an arbitrary finite residual gauge group, in addition to the braiding and fusion rules for those particles. We compute explicitly the fusion rules for two non-abelian group doubles suggested for universal quantum computation:

S_3

and

A_5

, and discover some interesting results, subsystems, and symmetries in the tables. SO(3)_4 (the restriction of Chern-Simons theory

SU(2)_4

) and its mirror image are discovered as 3-particle subsystems in the 8-particle

S_3

quantum double. The tables demonstrate that both

S_3

and

A_5

anyons are all Majorana, but this is not the case for all finite groups. In the appendices, the quantum doubles for the remaining nonabelian subgroups of SO(3) -

S_4

A_4

, and

D_4

(the second in the infinite family

D_n

) - are tabulated and analyzed. In addition, the probabilities of obtaining any given fusion product in quantum computation applications are determined and programmed in MAGMA. Throughout, connections to possible experiments are mentioned.