arXiv:1210.7929 · Physical Review B

Measurement-only topological quantum computation via tunable interactions

Parsa Bonderson

Published Jan 10, 2013• 61 citations

Abstract

I examine, in general, how tunable interactions may be used to perform anyonic teleportation and generate braiding transformations for non-Abelian anyons. I explain how these methods are encompassed by the ``measurement-only'' approach to topological quantum computation. The physically most relevant example of Ising anyons or Majorana zero modes is considered in detail, particularly in the context of Majorana nanowires.

DOI: 10.1103/physrevb.87.035113

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arXiv:1210.7929 · Physical Review B

Measurement-only topological quantum computation via tunable interactions

Parsa Bonderson

Published Jan 10, 2013• 61 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevb.87.035113

Related papers

Computing with quantum knots: Marjorana modes, non-Abelian anyons, and topological quantum computation

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I will discuss the revolutionary new concept of topological quantum computation, which is fault-tolerant at the hardware level with no need, in principle, of any quantum error correction protocols. Errors simply do not occur since the physical qubits and the computation steps are protected against decoherence by non-local topological correlations in the underlying physical system. The key idea is non-Abelian statistics of the quasiparticles (called 'anyons' as opposed to fermions or bosons), where the space-time braiding of the anyons around each other, i.e. quantum 'knots', form topologically protected quantum gate operations. I will describe in detail the theoretical principles guiding the experimental search for the appropriate topological phases of matter where such non-Abelian anyons, which are low-dimensional solid state versions of the elusive and exotic Majorana fermions hypothesized seventy-five years ago, may exist. I will critically discuss the recent experimental claims of observing the Majorana modes in semiconductor nanowire structures following earlier theoretical proposals, outlining the future developments which would be necessary to eventually build a topological quantum computer.I will discuss the revolutionary new concept of topological quantum computation, which is fault-tolerant at the hardware level with no need, in principle, of any quantum error correction protocols. Errors simply do not occur since the physical qubits and the computation steps are protected against decoherence by non-local topological correlations in the underlying physical system. The key idea is non-Abelian statistics of the quasiparticles (called 'anyons' as opposed to fermions or bosons), where the space-time braiding of the anyons around each other, i.e. quantum 'knots', form topologically protected quantum gate operations. I will describe in detail the theoretical principles guiding the experimental search for the appropriate topological phases of matter where such non-Abelian anyons, which are low-dimensional solid state versions of the elusive and exotic Majorana fermions hypothesized seventy-five years ago, may exist. I will critically discuss the recent experimental claims of observing the Majorana modes in semiconductor nanowire structures following earlier theoretical proposals, outlining the future developments which would be necessary to eventually build a topological quantum computer.

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A paradigm to build a quantum computer, based on topological invariants is highlighted. The identities in the ensemble of knots, links and braids originally discovered in relation to topological quantum field theory are shown: how they define Artin braid group -- the mathematical basis of topological quantum computation (TQC). Vector spaces of TQC correspond to associated strings of particle interactions, and TQC operates its calculations on braided strings of special physical quasiparticles -- anyons -- with non-Abelian statistics. The physical platform of TQC is to use the topological quantum numbers of such small groups of anyons as qubits and to perform operations on these qubits by exchanging the anyons, both within the groups that form the qubits and, for multi-qubit gates, between groups. By braiding two or more anyons, they acquire up a topological phase or Berry phase similar to that found in the Aharonov-Bohm effect. Topological matter such as fractional quantum Hall systems and novel discovered topological insulators open the way to form system of anyons -- Majorana fermions -- with the unique property of encoding and processing quantum information in a naturally fault-tolerant way. In the topological insulators, due to its fundamental attribute of topological surface state occurrence of the bound, Majorana fermions are generated at its heterocontact with superconductors. One of the key operations of TQC -- braiding of non-Abelian anyons: it is illustrated how it can be implemented in one-dimensional topological isolator wire networks.A paradigm to build a quantum computer, based on topological invariants is highlighted. The identities in the ensemble of knots, links and braids originally discovered in relation to topological quantum field theory are shown: how they define Artin braid group -- the mathematical basis of topological quantum computation (TQC). Vector spaces of TQC correspond to associated strings of particle interactions, and TQC operates its calculations on braided strings of special physical quasiparticles -- anyons -- with non-Abelian statistics. The physical platform of TQC is to use the topological quantum numbers of such small groups of anyons as qubits and to perform operations on these qubits by exchanging the anyons, both within the groups that form the qubits and, for multi-qubit gates, between groups. By braiding two or more anyons, they acquire up a topological phase or Berry phase similar to that found in the Aharonov-Bohm effect. Topological matter such as fractional quantum Hall systems and novel discovered topological insulators open the way to form system of anyons -- Majorana fermions -- with the unique property of encoding and processing quantum information in a naturally fault-tolerant way. In the topological insulators, due to its fundamental attribute of topological surface state occurrence of the bound, Majorana fermions are generated at its heterocontact with superconductors. One of the key operations of TQC -- braiding of non-Abelian anyons: it is illustrated how it can be implemented in one-dimensional topological isolator wire networks.

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

Andrew Stewart Caldin

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Jul 25, 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

B_n

generators

\sigma_i

satisfy

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

and

\sigma_i \sigma_j = \sigma_j \sigma_i

for

|i-j|>1

. Anyonic exchange yields unitary matrices

U(\sigma_i)

with non-commuting phases

e^{i\theta}

, where

\theta

depends on topological charge (e.g., Ising anyons:

\theta = \pi/8

for

\sigma

particles). Fibonacci anyons: dimension

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

, with fusion rules

\tau \times \tau = 1 + \tau

. Majorana zero modes:

\gamma_i^\dagger = \gamma_i

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

, parity

P = i^k \prod \gamma_i

. | CONNECTION: Fibonacci anyon dimension

d = \phi = 1.618

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

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