arXiv:2209.03822 · Journal of High Energy Physics

Topological quantum computation on supersymmetric spin chains

Indrajit Jana, Filippo Montorsi, Pramod Padmanabhan, Diego Trancanelli

Published Feb 27, 2023• 2 citations

Abstract

Abstract Quantum gates built out of braid group elements form the building blocks of topological quantum computation. They have been extensively studied in SU(2)k quantum group theories, a rich source of examples of non-Abelian anyons such as the Ising (k = 2), Fibonacci (k = 3) and Jones-Kauffman (k = 4) anyons. We show that the fusion spaces of these anyonic systems can be precisely mapped to the product state zero modes of certain Nicolai-like supersymmetric spin chains. As a result, we can realize the braid group in terms of the product state zero modes of these supersymmetric systems. These operators kill all the other states in the Hilbert space, thus preventing the occurrence of errors while processing information, making them suitable for quantum computing.

DOI: 10.1007/jhep02(2023)251

Related papers

Topological Quantum Computing via Non-Abelian Anyon Braiding and Majorana Modes — E8 Intelligence Research

Andrew Stewart Caldin

Jul 22, 20260 citations

Universal Gates from Braiding and Fusing Anyons on Quantum Hardware

Chiu Fan Bowen Lo, Anasuya Lyons, Dan Gresh +8 more

quant-pharXiv:2601.20956NatureJan 30, 20260 citations

Topological Quantum Computing with Non-Abelian Anyons and Braid Group Operations — E8 Intelligence Research

Andrew Stewart Caldin

Jul 25, 20260 citations

Topological Quantum Computation via Non-Abelian Anyon Braid Group Representations — E8 Intelligence Research

Andrew Stewart Caldin

Oct 10, 20260 citations

Diagonal Coset Approach to Topological Quantum Computation with Fibonacci Anyons

Lachezar S. Georgiev, Ludmil K. Hadjiivanov, Grigori Matein

arXiv:2404.01779Apr 2, 20240 citations

Topological Quantum Computing: Braid Groups, Majorana Modes, and Microsoft's Majorana 1 Chip — E8 Intelligence Research

Andrew Stewart Caldin

Sep 10, 20260 citations

arXiv:2209.03822 · Journal of High Energy Physics

Topological quantum computation on supersymmetric spin chains

Indrajit Jana, Filippo Montorsi, Pramod Padmanabhan, Diego Trancanelli

Published Feb 27, 2023• 2 citations

arXiv HTML PDF

Abstract

DOI: 10.1007/jhep02(2023)251

Related papers

Topological Quantum Computing via Non-Abelian Anyon Braiding and Majorana Modes — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Topological quantum computing uses non-Abelian anyons (Majorana zero modes) for fault-tolerant qubits, with braiding operations encoding quantum gates. | MATH: Braid group $B_n$ generators satisfy $\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}$ ; Majorana operators $\gamma_i$ satisfy $\{\gamma_i, \gamma_j\} = 2\delta_{ij}$ ; fusion rules for Ising anyons: $\sigma \times \sigma = 1 + \psi$ , $\sigma \times \psi = \sigma$ , $\psi \times \psi = 1$ . | CONNECTION: Braid group is deeply linked to the golden ratio via the Temperley-Lieb algebra at $q = e^{i\pi/5}$ , where the Jones polynomial yields $\tau = (1+\sqrt{5})/2 \approx 1.618$ ; Fibonacci anyons have fusion rules $\tau \times \tau = 1 + \tau$ , directly encoding the golden ratio. | DEPTH: 9 — Non-Abelian statistics and braiding are foundational to topological quantum computation, with direct geometric harmony in Fibonacci anyon models. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Topological quantum computing uses non-Abelian anyons (Majorana zero modes) for fault-tolerant qubits, with braiding operations encoding quantum gates. | MATH: Braid group $ B_n $ generators satisfy $ \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} $; Majorana operators $ \gamma_i $ satisfy $ \{\gamma_i, \gamma_j\} = 2\delta_{ij} $; fusion rules for Ising anyons: $ \sigma \times \sigma = 1 + \psi $, $ \sigma \times \psi = \sigma $, $ \psi \times \psi = 1 $. | CONNECTION: Braid group is deeply linked to the golden ratio via the Temperley-Lieb algebra at $ q = e^{i\pi/5} $, where the Jones polynomial yields $ \tau = (1+\sqrt{5})/2 \approx 1.618 $; Fibonacci anyons have fusion rules $ \tau \times \tau = 1 + \tau $, directly encoding the golden ratio. | DEPTH: 9 — Non-Abelian statistics and braiding are foundational to topological quantum computation, with direct geometric harmony in Fibonacci anyon models. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Jul 22, 20260 citations

Universal Gates from Braiding and Fusing Anyons on Quantum Hardware

Chiu Fan Bowen Lo, Anasuya Lyons, Dan Gresh +8 more

Topological quantum computation encodes quantum information in the internal fusion space of non-Abelian anyonic quasiparticles, whose braiding implements logical gates. This goes beyond Abelian topological order (TO) such as the toric code, as its anyons lack internal structure. However, the simplest non-Abelian generalizations of the toric code do not support universality via braiding alone. Here we demonstrate that such minimally non-Abelian TOs can be made universal by treating anyon fusion as a computational primitive. We prepare a 54-qubit TO wavefunction associated with the smallest non-Abelian group, $S_3$ , on Quantinuum's H2 quantum processor. This phase of matter exhibits cyclic anyon fusion rules, known to underpin universality, which we evidence by trapping a single non-Abelian anyon on the torus. We encode logical qutrits in the nonlocal fusion space of non-Abelian fluxes and, by combining an entangling braiding operation with anyon charge measurements, realize a universal topological gate set and read-out, which we further demonstrate by topologically preparing a magic state. This work establishes $S_3$ TO as simple enough to be prepared efficiently, yet rich enough to enable universal topological quantum computation.Topological quantum computation encodes quantum information in the internal fusion space of non-Abelian anyonic quasiparticles, whose braiding implements logical gates. This goes beyond Abelian topological order (TO) such as the toric code, as its anyons lack internal structure. However, the simplest non-Abelian generalizations of the toric code do not support universality via braiding alone. Here we demonstrate that such minimally non-Abelian TOs can be made universal by treating anyon fusion as a computational primitive. We prepare a 54-qubit TO wavefunction associated with the smallest non-Abelian group, $S_3$, on Quantinuum's H2 quantum processor. This phase of matter exhibits cyclic anyon fusion rules, known to underpin universality, which we evidence by trapping a single non-Abelian anyon on the torus. We encode logical qutrits in the nonlocal fusion space of non-Abelian fluxes and, by combining an entangling braiding operation with anyon charge measurements, realize a universal topological gate set and read-out, which we further demonstrate by topologically preparing a magic state. This work establishes $S_3$ TO as simple enough to be prepared efficiently, yet rich enough to enable universal topological quantum computation.

quant-pharXiv:2601.20956NatureJan 30, 20260 citations

Topological Quantum Computing with Non-Abelian Anyons and Braid Group Operations — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Topological quantum computing uses non-Abelian anyons (Majorana zero modes) to encode qubits in braiding operations, immune to local decoherence. | MATH: Braid group $B_n$ generators $\sigma_i$ satisfy $\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}$ ; anyon fusion rules: $\sigma \times \sigma = 1 + \psi$ (Ising model); Majorana operators $\gamma_i$ with $\{\gamma_i, \gamma_j\} = 2\delta_{ij}$ ; topological qubit encoded in parity $i\gamma_1\gamma_2 = \pm 1$ . | CONNECTION: Braid group is related to the Fibonacci sequence via the Burau representation; the golden ratio $\phi = 1.618$ appears in the quantum dimension of Fibonacci anyons: $d_\tau = \phi$ . The Ising anyon quantum dimension is $\sqrt{2}$ , linking to $\sqrt{2} \approx 1.414$ , close to 1.4142 (no direct golden ratio). Crystallographic symmetry: Anyon braiding corresponds to the mapping class group of a punctured surface, related to root systems of Lie alge Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Topological quantum computing uses non-Abelian anyons (Majorana zero modes) to encode qubits in braiding operations, immune to local decoherence. | MATH: Braid group $ B_n $ generators $ \sigma_i $ satisfy $ \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} $; anyon fusion rules: $ \sigma \times \sigma = 1 + \psi $ (Ising model); Majorana operators $ \gamma_i $ with $ \{\gamma_i, \gamma_j\} = 2\delta_{ij} $; topological qubit encoded in parity $ i\gamma_1\gamma_2 = \pm 1 $. | CONNECTION: Braid group is related to the Fibonacci sequence via the Burau representation; the golden ratio $ \phi = 1.618 $ appears in the quantum dimension of Fibonacci anyons: $ d_\tau = \phi $. The Ising anyon quantum dimension is $ \sqrt{2} $, linking to $ \sqrt{2} \approx 1.414 $, close to 1.4142 (no direct golden ratio). Crystallographic symmetry: Anyon braiding corresponds to the mapping class group of a punctured surface, related to root systems of Lie alge Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Jul 25, 20260 citations

Topological Quantum Computation via Non-Abelian Anyon Braid Group Representations — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Anyon braiding forms the computational substrate for topological quantum computation, with non-Abelian statistics encoded in braid group representations rather than local qubit states. | MATH: Braid group $B_n$ 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.comFINDING: Anyon braiding forms the computational substrate for topological quantum computation, with non-Abelian statistics encoded in braid group representations rather than local qubit states. | MATH: Braid group $B_n$ 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

Oct 10, 20260 citations

Diagonal Coset Approach to Topological Quantum Computation with Fibonacci Anyons

Lachezar S. Georgiev, Ludmil K. Hadjiivanov, Grigori Matein

We investigate a promising conformal field theory realization scheme for topological quantum computation based on the Fibonacci anyons, which are believed to be realized as quasiparticle excitations in the $\mathbb{Z}_3$ 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.We investigate a promising conformal field theory realization scheme for topological quantum computation based on the Fibonacci anyons, which are believed to be realized as quasiparticle excitations in the $\mathbb{Z}_3$ 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.

arXiv:2404.01779Apr 2, 20240 citations

Topological Quantum Computing: Braid Groups, Majorana Modes, and Microsoft's Majorana 1 Chip — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Topological quantum computing leverages non-Abelian anyons (Majorana zero modes) for fault-tolerant braid-based computation, with Microsoft's Majorana 1 chip claiming hardware realization. | MATH: Braid group B_n generators σ_i satisfy σ_iσ_{i+1}σ_i = σ_{i+1}σ_iσ_{i+1} and σ_iσ_j = σ_jσ_i for |i−j|≥2; Fibonacci anyons yield fusion space dimension φ (golden ratio) via F-matrix with entries involving φ^{-1/2}; Majorana modes obey γ_i†=γ_i, {γ_i,γ_j}=2δ_ij; topological charge conservation enforces fusion rules; Jones polynomial evaluated at q=e^{2πi/5} (or e^{2πi/k} for SU(2)_k Chern-Simons) encodes braid invariants. | CONNECTION: Fibonacci anyon model has quantum dimension d = φ = (1+√5)/2 ≈ 1.618, with F-matrix elements ∝ 1/φ ≈ 0.618; braid group representations correspond to Temperley-Lieb algebra with loop value δ = φ+φ^{-1} = √5 ≈ 2.236; SU(2)_k level k=3 gives φ; crystallographic root system A_2 (hexagonal lattice) underlies SU(3) anyons; base-60 appears in modular data S,T Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Topological quantum computing leverages non-Abelian anyons (Majorana zero modes) for fault-tolerant braid-based computation, with Microsoft's Majorana 1 chip claiming hardware realization. | MATH: Braid group B_n generators σ_i satisfy σ_iσ_{i+1}σ_i = σ_{i+1}σ_iσ_{i+1} and σ_iσ_j = σ_jσ_i for |i−j|≥2; Fibonacci anyons yield fusion space dimension φ (golden ratio) via F-matrix with entries involving φ^{-1/2}; Majorana modes obey γ_i†=γ_i, {γ_i,γ_j}=2δ_ij; topological charge conservation enforces fusion rules; Jones polynomial evaluated at q=e^{2πi/5} (or e^{2πi/k} for SU(2)_k Chern-Simons) encodes braid invariants. | CONNECTION: Fibonacci anyon model has quantum dimension d = φ = (1+√5)/2 ≈ 1.618, with F-matrix elements ∝ 1/φ ≈ 0.618; braid group representations correspond to Temperley-Lieb algebra with loop value δ = φ+φ^{-1} = √5 ≈ 2.236; SU(2)_k level k=3 gives φ; crystallographic root system A_2 (hexagonal lattice) underlies SU(3) anyons; base-60 appears in modular data S,T Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Sep 10, 20260 citations

FINDING: Topological quantum computing uses non-Abelian anyons (Majorana zero modes) for fault-tolerant qubits, with braiding operations encoding quantum gates. | MATH: Braid group

B_n

generators satisfy

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

; Majorana operators

\gamma_i

satisfy

\{\gamma_i, \gamma_j\} = 2\delta_{ij}

; fusion rules for Ising anyons:

\sigma \times \sigma = 1 + \psi

\sigma \times \psi = \sigma

\psi \times \psi = 1

. | CONNECTION: Braid group is deeply linked to the golden ratio via the Temperley-Lieb algebra at

q = e^{i\pi/5}

, where the Jones polynomial yields

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

; Fibonacci anyons have fusion rules

\tau \times \tau = 1 + \tau

, directly encoding the golden ratio. | DEPTH: 9 — Non-Abelian statistics and braiding are foundational to topological quantum computation, with direct geometric harmony in Fibonacci anyon models. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

S_3

, on Quantinuum's H2 quantum processor. This phase of matter exhibits cyclic anyon fusion rules, known to underpin universality, which we evidence by trapping a single non-Abelian anyon on the torus. We encode logical qutrits in the nonlocal fusion space of non-Abelian fluxes and, by combining an entangling braiding operation with anyon charge measurements, realize a universal topological gate set and read-out, which we further demonstrate by topologically preparing a magic state. This work establishes

S_3

TO as simple enough to be prepared efficiently, yet rich enough to enable universal topological quantum computation.

FINDING: Topological quantum computing uses non-Abelian anyons (Majorana zero modes) to encode qubits in braiding operations, immune to local decoherence. | MATH: Braid group

B_n

generators

\sigma_i

satisfy

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

; anyon fusion rules:

\sigma \times \sigma = 1 + \psi

(Ising model); Majorana operators

\gamma_i

with

\{\gamma_i, \gamma_j\} = 2\delta_{ij}

; topological qubit encoded in parity

i\gamma_1\gamma_2 = \pm 1

. | CONNECTION: Braid group is related to the Fibonacci sequence via the Burau representation; the golden ratio

\phi = 1.618

appears in the quantum dimension of Fibonacci anyons:

d_\tau = \phi

. The Ising anyon quantum dimension is

\sqrt{2}

, linking to

\sqrt{2} \approx 1.414

, close to 1.4142 (no direct golden ratio). Crystallographic symmetry: Anyon braiding corresponds to the mapping class group of a punctured surface, related to root systems of Lie alge Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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.