Topological Quantum Computation : An Analysis of an Anyon Model Based on Quantum Double Symmetries

There exists various suggestions for building a functional and a fault-tolerant large-scale quantum computer. Topological quantum computation is a more exotic suggestion, which makes use of the properties of quasiparticles manifest only in certain two-dimensional systems. These so called anyons exhibit topological degrees of freedom, which, in principle, can be used to execute quantum computation with intrinsic fault-tolerance. This feature is the main incentive to study topological quantum computation. The objective of this thesis is to provide an accessible introduction to the theory. In this thesis one has considered the theory of anyons arising in two-dimensional quantum mechanical systems, which are described by gauge theories based on so called quantum double symmetries. The quasiparticles are shown to exhibit interactions and carry quantum numbers, which are both of topological nature. Particularly, it is found that the addition of the quantum numbers is not unique, but that the fusion of the quasiparticles is described by a non-trivial fusion algebra. It is discussed how this property can be used to encode quantum information in a manner which is intrinsically protected from decoherence and how one could, in principle, perform quantum computation by braiding the quasiparticles. As an example of the presented general discussion, the particle spectrum and the fusion algebra of an anyon model based on the gauge group S_3 are explicitly derived. The fusion algebra is found to branch into multiple proper subalgebras and the simplest one of them is chosen as a model for an illustrative demonstration. The different steps of a topological quantum computation are outlined and the computational power of the model is assessed. It turns out that the chosen model is not universal for quantum computation. However, because the objective was a demonstration of the theory with explicit calculations, none of the other more complicated fusion subalgebras were considered. Studying their applicability for quantum computation could be a topic of further research.Toimivan ja virheitä sietävän makroskooppisen kvanttitietokoneen rakentamiseksi on esitetty erilaisia ehdotuksia. Topologinen kvanttilaskenta on yksi eksoottisemmista ehdotuksista, jossa käytetään hyväksi eräissä kaksiulotteisissa kvanttimekaanisissa systeemeissä esiintyvien kvasihiukkasten ominaisuuksia. Motivaatio tällaisten systeeminen tutkimiseen tulee siitä, että näillä yleisemmin anyoneiksi kutsutuilla hiukkasilla esiintyy topologisia vapausasteita, joita voidaan ainakin periaatteessa käyttää virheitä hyvin sietävään kvanttilaskentaan. Tämän työn päämääränä on tarjota lähestyttävä johdatus topologisen kvanttilaskennan teoriaan. Työssä tarkastellut anyonit esiintyvät kvanttimekaanisissa systeemeissä, joita voidaan kuvata kvanttituplasymmetrisillä mittakenttäteorioilla. Kvasihiukkasten välillä osoitetaan olevan topologisia vuorovaikutuksia, sekä niiden osoitetaan kantavan systeemin topologiasta seuraavia kvanttilukuja. Näiden kvanttilukujen yhteenlasku ei ole yksikäsitteistä, vaan niitä kuvaavat anyonimallia karakterisoivat fuusiosäännöt. Työssä tarkastellaan yleisellä tasolla millaisia vapausasteita näihin fuusiosääntöihin liittyy ja miten niitä voidaan käyttää sekä tallentamaan, että käsittelemään kvantti-informaatiota tavalla, joka ei ole altis dekoherenssille. Esimerkkinä yleisestä tarkastelusta työssä esitetään anyonimalli, joka pohjautuu S_3 mittaryhmään. Mallille johdetaan yksityiskohtaisesti siinä esiintyvät kvasihiukkaset, sekä niiden fuusiosäännöt. Fuusiosäännöistä valitaan sulkeutuva alijoukko, jota käytetään havainnollistavana mallina topologisesta kvanttilaskennasta. Laskennan eri vaiheet käydään yksityiskohtaisesti läpi ja mallin laskennallista tehoa arvioidaan. Osoittautuu, että valittu malli ei salli universaalia kvanttilaskentaa, mutta koska päämäärä kuitenkin oli topologisen kvanttilaskennan teorian esittäminen mahdollisimman yksiselitteisellä ja lähestyttävällä tavalla, muita S_3 anyonien fuusiosääntöjen sallimia malleja ei tarkastella. Niiden soveltuvuuden tutkiminen kvanttilaskentaan voisi olla mahdollinen jatkotutkimuksen aihe

Kitaev spin models from topological nanowire networks

We show that networks of topological nanowires can realize the physics of exactly solvable Kitaev spin models with two-body interactions. This connection arises from the description of the low-energy theory of both systems in terms of a tight-binding model of Majorana modes. In Kitaev spin models the Majorana description provides a convenient representation to solve the model, whereas in an array of topological nanowires it arises, because the physical Majorana modes localized at wire ends permit tunnelling between wire ends and across different Josephson junctions. We explicitly show that an array of junctions of three wires -- a setup relevant to topological quantum computing with nanowires -- can realize the Yao-Kivelson model, a variant of Kitaev spin models on a decorated honeycomb lattice. Translating the results from the latter, we show that the network can be constructed to give rise to collective states characterized by Chern numbers \nu = 0, +/-1 and +/-2, and that defects in an array can be associated with vortex-like quasi-particle excitations. Finally, we analyze the stability of the collective states as well as that of the network as a quantum information processor. We show that decoherence inducing instabilities, be them due to disorder or phase fluctuations, can be understood in terms of proliferation of the vortex-like quasi-particles.Comment: 15 pages, 9 figure

A hierarchy of exactly solvable spin-1/2 chains with so(N)_1 critical points

We construct a hierarchy of exactly solvable spin-1/2 chains with so(N)_1 critical points. Our construction is based on the framework of condensate-induced transitions between topological phases. We employ this framework to construct a Hamiltonian term that couples N transverse field Ising chains such that the resulting theory is critical and described by the so(N)_1 conformal field theory. By employing spin duality transformations, we then cast these spin chains for arbitrary N into translationally invariant forms that all allow exact solution by the means of a Jordan-Wigner transformation. For odd N our models generalize the phase diagram of the transverse field Ising chain, the simplest model in our hierarchy. For even N the models can be viewed as longer ranger generalizations of the XY chain, the next model in the hierarchy. We also demonstrate that our method of constructing spin chains with given critical points goes beyond exactly solvable models. Applying the same strategy to the Blume-Capel model, a spin-1 generalization of the Ising chain in a generic magnetic field, we construct another critical spin-1 chain with the predicted CFT describing the criticality.Comment: 24 pages, 5 figures; v2: minor changes and added reference

Topological phase transitions driven by gauge fields in an exactly solvable model

We demonstrate the existence of a new topologically ordered phase in Kitaev's honeycomb lattice model. This new phase appears due to the presence of a vortex lattice and it supports chiral Abelian anyons. We characterize the phase by its low-energy behavior that is described by a distinct number of Dirac fermions. We identify two physically distinct types of topological phase transitions and obtain analytically the critical behavior of the extended phase space. The Fermi surface evolution associated with the transitions is shown to be due to the Dirac fermions coupling to chiral gauge fields. Finally, we describe how the new phase can be understood in terms of interactions between the anyonic vortices.Comment: 5 pages, 5 figures, published versio

Interacting non-Abelian anyons in an exactly solvable lattice model

In this thesis, we study the non-Abelian anyons that emerge as vortices in Ki-taev's honeycomb spin lattice model. By generalizing the solution of the model, we explicity demonstrate the non-Abelian fusion rules and the braid statistics that charaterize the anyons. This is based on showing the presence of vortices leads to zero modes in the spectrum. These can acquire finite energy due to short range vortex-vortex interactions. By studying the spectral evolution as a function of the vortex seperation, we unambigously identify the zero modes with the fusion degrees of freedom of non-Abelian anyons. To calculate the non-Abelian statistics, we show how the vortex transport can be implemented through local manipulation of the couplings. This enables us to employ the eigenstates of the model to simulate a process where a vortex winds around another. The corresponding evolution of the degenerate ground state space is given by a Berry phase, which under suitable conditions coincides with the statistics. By considering a range of finite size systems, we find a physical regime where the Berry phase gives the predicted statistics of the anoyonic vortices with high fidelity. Finally we study the full-vortex sector of the model and find that is supports a previously undiscovered topological phase. This new phase emerges from the phase with non-Abelian anyons due to their interactions. To study the transitions between the different topological phases appearing in the model, we consider Fermi surface, whose topology captures the long-range properties. Each phase is found to be characterized by a distinct number of Fermi points, with the number depending on distinct global Hamiltonian symmetries. To study how the Fermi surfaces evolve into each other at phase transitions, we consider the low-energy field theory that is described by Dirac fermions. We show that phase transition driving perturbations translate to a coupling to chiral gauge fields, that always lead to Fermi point transport. By studying this transport, we obtain analytically the extended phase space of the model and its properties

Condensate-induced transitions and critical spin chains

We show that condensate-induced transitions between two-dimensional topological phases provide a general framework to relate one-dimensional spin models at their critical points. We demonstrate this using two examples. First, we show that two well-known spin chains, namely the XY chain and the transverse field Ising chain with only next-nearest-neighbor interactions, differ at their critical points only by a non-local boundary term and can be related via an exact mapping. The boundary term constrains the set of possible boundary conditions of the transverse field Ising chain, reducing the number of primary fields in the conformal field theory that describes its critical behavior. We argue that the reduction of the field content is equivalent to the confinement of a set of primary fields, in precise analogy to the confinement of quasiparticles resulting from a condensation of a boson in a topological phase. As the second example we show that when a similar confining boundary term is applied to the XY chain with only next-nearest-neighbor interactions, the resulting system can be mapped to a local spin chain with the u(1)_2 x u(1)_2 critical behavior predicted by the condensation framework.Comment: 5 pages, 1 figure; v2: several minor textual change

Perturbed vortex lattices and the stability of nucleated topological phases

We study the stability of nucleated topological phases that can emerge when interacting non-Abelian anyons form a regular array. The studies are carried out in the context of Kitaev's honeycomb model, where we consider three distinct types of perturbations in the presence of a lattice of Majorana mode binding vortices -- spatial anisotropy of the vortices, dimerization of the vortex lattice and local random disorder. While all the nucleated phases are stable with respect to weak perturbations of each kind, strong perturbations are found to result in very different behavior. Anisotropy of the vortices stabilizes the strong-pairing like phases, while dimerization can recover the underlying non-Abelian phase. Local random disorder, on the other hand, can drive all the nucleated phases into a gapless thermal metal state. We show that all these distinct behavior can be captured by an effective staggered tight-binding model for the Majorana modes. By studying the pairwise interactions between the vortices, i.e. the amplitudes for the Majorana modes to tunnel between vortex cores, the locations of phase transitions and the nature of the resulting states can be predicted. We also find that due to oscillations in the Majorana tunneling amplitude, lattices of Majorana modes may exhibit a Peierls-like instability, where a dimerized configuration is favored over a uniform lattice. As the nature of the nucleated phases depends only on the Majorana tunneling, our results apply also to other system supporting localized Majorana mode arrays, such as Abrikosov lattices in p-wave superconductors, Wigner crystals in Moore-Read fractional quantum Hall states or arrays of topological nanowires.Comment: 13 pages, 4 pages of appendices, 24 figures. Published versio

Realizing All so(N)1 Quantum Criticalities in Symmetry Protected Cluster Models

We show that all so(N)1 universality class quantum criticalities emerge when one-dimensional generalized cluster models are perturbed with Ising or Zeeman terms. Each critical point is described by a low-energy theory of N linearly dispersing fermions, whose spectrum we show to precisely match the prediction by so(N)1 conformal field theory. Furthermore, by an explicit construction we show that all the cluster models are dual to nonlocally coupled transverse field Ising chains, with the universality of the so(N)1 criticality manifesting itself as N of these chains becoming critical. This duality also reveals that the symmetry protection of cluster models arises from the underlying Ising symmetries and it enables the identification of local representations for the primary fields of the so(N)1 conformal field theories. For the simplest and experimentally most realistic case that corresponds to the original one-dimensional cluster model with local three-spin interactions, our results show that the su(2)2≃so(3)1 Wess-Zumino-Witten model can emerge in a local, translationally invariant, and Jordan-Wigner solvable spin-1/2 model