arXiv:2306.16062

Dyck Paths and Topological Quantum Computation

Vivek Kumar Singh, Sinha, Akash, Pramod Padmanabhan, Indrajit Jana

Published Jun 28, 2023• 0 citations

Abstract

The fusion basis of Fibonacci anyons supports unitary braid representations that can be utilized for universal quantum computation. We show a mapping between the fusion basis of three Fibonacci anyons, $\{|1\rangle, |τ\rangle\}$, and the two length 4 Dyck paths via an isomorphism between the two dimensional braid group representations on the fusion basis and the braid group representation built on the standard $(2,2)$ Young diagrams using the Jones construction. This correspondence helps us construct the fusion basis of the Fibonacci anyons using Dyck paths as the number of standard $(N,N)$ Young tableaux is the Catalan number, $C_N$ . We then use the local Fredkin moves to construct a spin chain that contains precisely those Dyck paths that correspond to the Fibonacci fusion basis, as a degenerate set. We show that the system is gapped and examine its stability to random noise thereby establishing its usefulness as a platform for topological quantum computation. Finally, we show braidwords in this rotated space that efficiently enable the execution of any desired single-qubit operation, achieving the desired level of precision($\sim 10^{-3}$).

DOI: 10.48550/arxiv.2306.16062

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arXiv:2306.16062

Dyck Paths and Topological Quantum Computation

Vivek Kumar Singh, Sinha, Akash, Pramod Padmanabhan, Indrajit Jana

Published Jun 28, 2023• 0 citations

arXiv HTML PDF

Abstract

\{|1\rangle, |τ\rangle\}

, and the two length 4 Dyck paths via an isomorphism between the two dimensional braid group representations on the fusion basis and the braid group representation built on the standard

(2,2)

Young diagrams using the Jones construction. This correspondence helps us construct the fusion basis of the Fibonacci anyons using Dyck paths as the number of standard

(N,N)

Young tableaux is the Catalan number,

C_N

. We then use the local Fredkin moves to construct a spin chain that contains precisely those Dyck paths that correspond to the Fibonacci fusion basis, as a degenerate set. We show that the system is gapped and examine its stability to random noise thereby establishing its usefulness as a platform for topological quantum computation. Finally, we show braidwords in this rotated space that efficiently enable the execution of any desired single-qubit operation, achieving the desired level of precision(

\sim 10^{-3}

DOI: 10.48550/arxiv.2306.16062

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arXiv:2307.01892Jul 4, 20230 citations

\{|1\rangle, |τ\rangle\}

, and the two length 4 Dyck paths via an isomorphism between the two dimensional braid group representations on the fusion basis and the braid group representation built on the standard

(2,2)

Young diagrams using the Jones construction. This correspondence helps us construct the fusion basis of the Fibonacci anyons using Dyck paths as the number of standard

(N,N)

Young tableaux is the Catalan number,

C_N

\sim 10^{-3}

B_n

generators

\sigma_i

satisfy

\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}

(Yang–Baxter equation); non-Abelian anyons carry unitary representations

U(\sigma_i)

with

U(\sigma_i)U(\sigma_j)=U(\sigma_j)U(\sigma_i)

for

|i-j|\geq 2

. Fibonacci anyons: fusion rule

\tau \times \tau = 1 + \tau

, Hilbert space dimension grows as Fibonacci sequence

F_n

, golden ratio

\phi = (1+\sqrt{5})/2 \approx 1.618

appears as asymptotic dimension growth rate. Topological protection arises from degeneracy of ground states on genus-

g

surfaces: dimension

= k^g

for level-

k

Chern–Simons theory. | CONNECTION: The Fibonacci anyon fusion space dimension ratio

F_{n+1}/F_n \to 1.618

— direct geometric harmony link. Braid group relations ar Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

\mathbb{Z}_3

parafermion fractional quantum Hall state in the second Landau level with filling factor

ν=12/5

. These anyons are non-Abelian and are known to be capable of universal topological quantum computation. The quantum information is encoded in the fusion channels of pairs of such non-Abelian anyons and is protected from noise and decoherence by the topological properties of these systems.The quantum gates are realized by braiding of these anyons. We propose here an implementation of the

n

-qubit topological quantum register in terms of

2n+2

Fibonacci anyons. The matrices emerging from the anyon exchanges, i.e. the generators of the braid group for one qubit are derived from the coordinate wave functions of a large number of electron holes and 4 Fibonacci anyons which can furthermore be represented as correlation functions in

\mathbb{Z}_3

parafermionic two-dimensional conformal field theory. The representations of the braid groups for more than 4 anyons are obtained by fusing pairs of anyons before braiding, thus reducing eventually the system to 4 anyons.

\mathcal{B}

B n acting on n -point conformal blocks of Fibonacci anyons. The matrices of braid group generators are displayed explicitly for all n ≤ 8. A simple recursion formula makes it possible to extend without efforts the construction to any n . Finally, we construct

\mathcal{N}

N qubit computational spaces in terms of conformal blocks of

2\mathcal{N}

2 N + 2 Fibonacci anyons.

SU(2)_k

. Braiding non-abelian anyons is the essence of TQC, offering a topologically protected implementation of quantum gates. However, obtaining elementary braid matrix representations starting from the fusion and rotation matrices of a specific anyon model is {theoretically guarenteed but no numerical method is given, especially for systems with numerous anyons and complex fusion patterns. Our proposed method addresses this challenge, first in the special case of sparse encoding, allowing for the inclusion of an arbitrary number of anyons per qudit, {and in the general case}. This is accomplished by introducing two methods, one is based on a novel braiding move we call knitting, {the second introduces more general algorithm which is optimal in number of required moves}. The method plays a key role in a broad topological quantum circuit simulator, enabling the examination and study of complex quantum circuits within the TQC framework. Importantly, it proves effective across various anyonic models, accommodating diverse fusion rules. We validate the method by simulating an approximated CNOT gate and present a first-of-a-kind GHZ state simulation on five qubits using three Fibonacci anyons per qubit.