TQCにおけるモジュラーS行列をめぐって

Jun Ichi Fujii

Published Feb 29, 2016• 0 citations

Abstract

The term ‘TQC' means the topological quantum computation, which is one of remarkable theories to realize quantum computers. This theory is related to various fields; theory of quantum groups, Lie algebras, Hopf algebras and conformal field or string theory. But there are few textbooks which readers of other fields can read easily. We wrote a survay[10]easy to observe from the basics of TQC theory to basic quantum gates for quantum computers. But we cannot explain modular S-matrices that is one of the keys in TQC in the preceding survey. So, in this paper, we try to express total basic issue including theories around modular S-matrix. In the many papers for TQC, there are some subtle mistakes and we correct them in this paper. At first we introduce formulae for modular S-matrix in the axiomatic (categorical) approach. Secondly we give an important model for TQC; spin network model based on the Temperly-Lieb-Jones algebra. Finally we mention another important model suk(2). In each levels, Fibannaci anyon is a basic example to explain them.

DOI: 10.32287/td00028752

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TQCにおけるモジュラーS行列をめぐって

Jun Ichi Fujii

Published Feb 29, 2016• 0 citations

Publisher

Abstract

DOI: 10.32287/td00028752

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Mar 19, 20260 citations

N^k_{ij}

S

-matrix (modular transformation

\tau \to -1/\tau

T

-matrix (diagonal,

\theta_a = e^{2\pi i h_a}

topological spins), satisfying

S^2 = C

(charge conjugation),

(ST)^3 = e^{2\pi i c/8} S^2

(where

c

is the chiral central charge). The entries of

S

and

T

lie in a cyclotomic field

\mathbb{Q}(\zeta_N)

\zeta_N = e^{2\pi i/N}

. For the Fibonacci anyon MTC:

S = \frac{1}{\sqrt{\phi+2}}\begin{pmatrix} 1 & \phi \\ \phi & -1 \end{pmatrix}

T = \text{diag}(1, e^{4\pi i/5})

, where

\phi = (1+\sqrt{5})/2

. | CONNECTION: **Direct hit**: The Fibonacci MTC's

S

-matrix contains

\phi = 1.618

and

\phi+2 = \phi^2+1 = 3.618

, whose inverse square root yields

1/\sqrt{3.618} \approx 0.5257

. The golden ratio appears intrinsically. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

L \in \{0, 1\}

of a Hopf soliton pair:

|0\rangle

corresponds to an unlinked pair,

|1\rangle

to a linked pair. Since linking number is an integer topological invariant, the qubit is passively protected against decoherence --- errors require reconnection events exponentially suppressed at energies below the soliton mass gap

\Delta

. The SU(2)

_1

Chern-Simons theory governing the Hopf vacuum (Paper V) provides the R-matrix

R = e^{i\pi/4}

for soliton exchange, yielding Abelian (semionic) braiding. While SU(2)

_1

alone is not universal, the 3+1D loop braid group provides additional operations (leapfrogging) beyond 2+1D braiding, and universality can be achieved via magic state distillation. We construct a gate set from soliton self-linking (single-qubit rotations) and inter-pair braiding (entangling gates), showing that topological protection yields error rates exponentially small in the ratio of system size to correlation length. Physical implementations include liquid crystal Hopfions, BEC vortex knots, and photonic topological states. This is a theoretical proposal: no experimental realization currently exists. > [!warning] Speculative Content > This paper is a theoretical proposal. The identification of Hopf soliton pairs as a TQC platform is new and untested. Standard TQC theory (Kitaev, Freedman et al., Nayak et al.) is well-established; the application to Hopf solitons is speculative. We clearly distinguish established results from proposals throughout. Keywords physics quantum computing topology anyons braiding TQC hopf solitons Type Preprint License CC BY 4.0 Date 2026-03-22 Subject Theoretical Physics DOI 10.5281/zenodo.19108219 © 2026 Alexander Novickis. Licensed under Creative Commons Attribution 4.0 International.