scholarly journals Z2 Topological Order and Topological Protection of Majorana Fermion Qubits

2021 ◽  
Vol 6 (1) ◽  
pp. 11
Author(s):  
Rukhsan Ul Haq ◽  
Louis H. Kauffman
Keyword(s):  
Braid Group ◽  
Transverse Field ◽  
Majorana Fermion ◽  
Chain Model ◽  
Topological Phases ◽  
Topological Order ◽  
Group Representations ◽  
Broad Perspective ◽  
Topological Quantum ◽  
Symmetry Operators

The Kitaev chain model exhibits topological order that manifests as topological degeneracy, Majorana edge modes and Z2 topological invariant of the bulk spectrum. This model can be obtained from a transverse field Ising model(TFIM) using the Jordan–Wigner transformation. TFIM has neither topological degeneracy nor any edge modes. Topological degeneracy associated with topological order is central to topological quantum computation. In this paper, we explore topological protection of the ground state manifold in the case of Majorana fermion models which exhibit Z2 topological order. We show that there are at least two different ways to understand this topological protection of Majorana fermion qubits: one way is based on fermionic mode operators and the other is based on anti-commuting symmetry operators. We also show how these two different ways are related to each other. We provide a very general approach to understanding the topological protection of Majorana fermion qubits in the case of lattice Hamiltonians. We then show how in topological phases in Majorana fermion models gives rise to new braid group representations. So, we give a unifying and broad perspective of topological phases in Majorana fermion models based on anti-commuting symmetry operators and braid group representations of Majorana fermions as anyons.

Braiding and Majorana fermions

2017 ◽  
Vol 26 (09) ◽  
pp. 1743001 ◽  
Author(s):  
Louis H. Kauffman
Keyword(s):  
Quantum Computing ◽  
Clifford Algebra ◽  
General Result ◽  
Braid Group ◽  
Clifford Algebras ◽  
Majorana Fermions ◽  
Group Representations ◽  
Topological Quantum

In this paper, we study unitary braid group representations associated with Majorana fermions. Majorana fermions are represented by Majorana operators, elements of a Clifford algebra. The paper proves a general result about braid group representations associated with Clifford algebras and compares this result with the Ivanov braiding associated with Majorana operators and with other braiding representations associated with Majorana fermions such as the Fibonacci model for universal topological quantum computing.

scholarly journals Topological Order, Mixed States and Open Systems

2019 ◽  
Vol 26 (03) ◽  
pp. 1950012 ◽  
Author(s):  
Manuel Asorey ◽  
Paolo Facchi ◽  
Giuseppe Marmo
Keyword(s):  
Open Systems ◽  
Quantum Systems ◽  
Topological Phases ◽  
Topological Order ◽  
Mixed States ◽  
Quantum Matter ◽  
Physical Role ◽  
Topological Quantum ◽  
Pure States

The role of mixed states in topological quantum matter is less known than that of pure quantum states. Generalisations of topological phases appearing in pure states have received attention in the literature only quite recently. In particular, it is still unclear whether the generalisation of the Aharonov–Anandan phase for mixed states due to Uhlmann plays any physical role in the behaviour of the quantum systems. We analyse, from a general viewpoint, topological phases of mixed states and the robustness of their invariance. In particular, we analyse the role of these phases in the behaviour of systems with periodic symmetry and their evolution under the influence of an environment preserving its crystalline symmetries.

scholarly journals Characterizing Topological Order with Matrix Product Operators

2021 ◽  
Author(s):  
Mehmet Burak Şahinoğlu ◽  
Dominic Williamson ◽  
Nick Bultinck ◽  
Michaël Mariën ◽  
Jutho Haegeman ◽  
...  
Keyword(s):  
Ground State ◽  
Local Order ◽  
Virtual Level ◽  
Matrix Product ◽  
Topological Phases ◽  
Topological Order ◽  
Quantum Phases ◽  
Topological Features ◽  
Topological Quantum ◽  
Tensor Network

AbstractOne of the most striking features of gapped quantum phases that exhibit topological order is the presence of long-range entanglement that cannot be detected by any local order parameter. The formalism of projected entangled-pair states is a natural framework for the parameterization of gapped ground state wavefunctions which allows one to characterize topological order in terms of the virtual symmetries of the local tensors that encode the wavefunction. In their most general form, these symmetries are represented by matrix product operators acting on the virtual level, which leads to a set of algebraic rules characterizing states with topological quantum order. This construction generalizes the concepts of $${\mathsf {G}}$$ G - and twisted injectivity; the corresponding matrix product operators encode all topological features of the theory and provide a complete picture of the ground state manifold on the torus. We show how the string-net models of Levin and Wen fit within this formalism and in doing so provide a particularly intuitive interpretation of the pentagon equation for F-symbols as the pulling of matrix product operators through the string-net tensor network. Our approach paves the way to finding novel topological phases beyond string nets and elucidates the description of topological phases in terms of entanglement Hamiltonians and edge theories.

scholarly journals GENERALIZED YANG–BAXTER EQUATIONS AND BRAIDING QUANTUM GATES

2012 ◽  
Vol 21 (09) ◽  
pp. 1250087 ◽  
Author(s):  
REBECCA S. CHEN
Keyword(s):  
Braid Group ◽  
Knot Theory ◽  
Information Science ◽  
Quantum Gates ◽  
Quantum Computers ◽  
Baxter Equation ◽  
Group Representations ◽  
Ongoing Research ◽  
Topological Quantum ◽  
Mathematics And Physics

Solutions to the Yang–Baxter equation — an important equation in mathematics and physics — and their afforded braid group representations have applications in fields such as knot theory, statistical mechanics, and, most recently, quantum information science. In particular, unitary representations of the braid group are desired because they generate braiding quantum gates. These are actively studied in the ongoing research into topological quantum computing. A generalized Yang–Baxter equation was proposed a few years ago by Eric Rowell et al. By finding solutions to the generalized Yang–Baxter equation, we obtain new unitary braid group representations. Our representations give rise to braiding quantum gates and thus have the potential to aid in the construction of useful quantum computers.

scholarly journals Extraspecial two-Groups, generalized Yang-Baxter equations and braiding quantum gates

2010 ◽  
Vol 10 (7&8) ◽  
pp. 685-702
Author(s):  
E.C. Rowell ◽  
Y. Zhang ◽  
Y.-S. Wu ◽  
M.-L. Ge
Keyword(s):  
Quantum Computing ◽  
Braid Group ◽  
Quantum Error Correction ◽  
Unitary Representations ◽  
Quantum Gates ◽  
Group Representations ◽  
Unitary Evolution ◽  
Ghz States ◽  
Topological Quantum ◽  
Quantum Error

In this paper we describe connections among extraspecial 2-groups, unitary representations of the braid group and multi-qubit braiding quantum gates. We first construct new representations of extraspecial 2-groups. Extending the latter by the symmetric group, we construct new unitary braid representations, which are solutions to generalized Yang-Baxter equations and use them to realize new braiding quantum gates. These gates generate the GHZ (Greenberger-Horne-Zeilinger) states, for an arbitrary (particularly an \emph{odd}) number of qubits, from the product basis. We also discuss the Yang-Baxterization of the new braid group representations, which describes unitary evolution of the GHZ states. Our study suggests that through their connection with braiding gates, extraspecial 2-groups and the GHZ states may play an important role in quantum error correction and topological quantum computing.

scholarly journals Giant phonon anomalies in the proximate Kitaev quantum spin liquid α-RuCl3

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Haoxiang Li ◽  
T. T. Zhang ◽  
A. Said ◽  
G. Fabbris ◽  
D. G. Mazzone ◽  
...  
Keyword(s):  
Acoustic Phonon ◽  
Energy Scale ◽  
Quantum Spin ◽  
Majorana Fermion ◽  
Spin Liquid ◽  
Optical Phonons ◽  
Large Intensity ◽  
Topological Quantum ◽  
Topological State ◽  
Quantum Spin Liquid

AbstractThe Kitaev quantum spin liquid epitomizes an entangled topological state, for which two flavors of fractionalized low-energy excitations are predicted: the itinerant Majorana fermion and the Z2 gauge flux. It was proposed recently that fingerprints of fractional excitations are encoded in the phonon spectra of Kitaev quantum spin liquids through a novel fractional-excitation-phonon coupling. Here, we detect anomalous phonon effects in α-RuCl3 using inelastic X-ray scattering with meV resolution. At high temperature, we discover interlaced optical phonons intercepting a transverse acoustic phonon between 3 and 7 meV. Upon decreasing temperature, the optical phonons display a large intensity enhancement near the Kitaev energy, JK~8 meV, that coincides with a giant acoustic phonon softening near the Z2 gauge flux energy scale. These phonon anomalies signify the coupling of phonon and Kitaev magnetic excitations in α-RuCl3 and demonstrates a proof-of-principle method to detect anomalous excitations in topological quantum materials.

scholarly journals Finiteness for mapping class group representations from twisted Dijkgraaf–Witten theory

2018 ◽  
Vol 27 (06) ◽  
pp. 1850043 ◽  
Author(s):  
Paul P. Gustafson
Keyword(s):  
Braid Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Braid Groups ◽  
Fusion Category ◽  
Mapping Class ◽  
Group Representations ◽  
Colored Graphs ◽  
Finite Image ◽  
Spanning Set

We show that any twisted Dijkgraaf–Witten representation of a mapping class group of an orientable, compact surface with boundary has finite image. This generalizes work of Etingof et al. showing that the braid group images are finite [P. Etingof, E. C. Rowell and S. Witherspoon, Braid group representations from twisted quantum doubles of finite groups, Pacific J. Math. 234 (2008)(1) 33–42]. In particular, our result answers their question regarding finiteness of images of arbitrary mapping class group representations in the affirmative. Our approach is to translate the problem into manipulation of colored graphs embedded in the given surface. To do this translation, we use the fact that any twisted Dijkgraaf–Witten representation associated to a finite group [Formula: see text] and 3-cocycle [Formula: see text] is isomorphic to a Turaev–Viro–Barrett–Westbury (TVBW) representation associated to the spherical fusion category [Formula: see text] of twisted [Formula: see text]-graded vector spaces. The representation space for this TVBW representation is canonically isomorphic to a vector space of [Formula: see text]-colored graphs embedded in the surface [A. Kirillov, String-net model of Turaev-Viro invariants, Preprint (2011), arXiv:1106.6033 ]. By analyzing the action of the Birman generators [J. Birman, Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22 (1969) 213–242] on a finite spanning set of colored graphs, we find that the mapping class group acts by permutations on a slightly larger finite spanning set. This implies that the representation has finite image.

scholarly journals Hurwitz Spaces and Braid Group Representations

2004 ◽  
Vol 34 (3) ◽  
pp. 1005-1030
Author(s):  
Eric P. Klassen ◽  
Yaacov Kopeliovich
Keyword(s):  
Braid Group ◽  
Group Representations ◽  
Hurwitz Spaces

scholarly journals Unveiling Signatures of Topological Phases in Open Kitaev Chains and Ladders

Nanomaterials ◽  
2019 ◽  
Vol 9 (6) ◽  
pp. 894 ◽  
Author(s):  
Alfonso Maiellaro ◽  
Francesco Romeo ◽  
Carmine Antonio Perroni ◽  
Vittorio Cataudella ◽  
Roberta Citro
Keyword(s):  
Electrical Response ◽  
Differential Conductance ◽  
Model Parameters ◽  
Topological Phases ◽  
Topological Order ◽  
Topological Phase ◽  
Moderate Amount ◽  
Topological System ◽  
Isolated Systems

In this work, the general problem of the characterization of the topological phase of an open quantum system is addressed. In particular, we study the topological properties of Kitaev chains and ladders under the perturbing effect of a current flux injected into the system using an external normal lead and derived from it via a superconducting electrode. After discussing the topological phase diagram of the isolated systems, using a scattering technique within the Bogoliubov–de Gennes formulation, we analyze the differential conductance properties of these topological devices as a function of all relevant model parameters. The relevant problem of implementing local spectroscopic measurements to characterize topological systems is also addressed by studying the system electrical response as a function of the position and the distance of the normal electrode (tip). The results show how the signatures of topological order affect the electrical response of the analyzed systems, a subset of which being robust also against the effects of a moderate amount of disorder. The analysis of the internal modes of the nanodevices demonstrates that topological protection can be lost when quantum states of an initially isolated topological system are hybridized with those of the external reservoirs. The conclusions of this work could be useful in understanding the topological phases of nanowire-based mesoscopic devices.

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