We introduce a new type of sparse CSS quantum error correcting code based on the homology of hypermaps. Sparse quantum error correcting codes are of interest in the building of quantum computers due to their ease of implementation and the possibility of developing fast decoders for them. Codes based on the homology of embeddings of graphs, such as Kitaev's toric code, have been discussed widely in the literature and our class of codes generalize these. We use embedded hypergraphs, which are a generalization of graphs that can have edges connected to more than two vertices. We develop theorems and examples of our hypermap-homology codes, especially in the case that we choose a special type of basis in our homology chain complex. In particular, the most straightforward generalization of the m × m toric code to hypermap-homology codes gives us a [(3/2)m²,2, m] code as compared to the toric code which is a [2m²,2, m]code. Thus we can protect the same amount of quantum information, with the same error-correcting capability, using less physical qubits.

In this thesis we consider the properties of a class of Z2 topological phases on a two-dimensional square lattice. The ground states of Z2 topological order are generally degenerate on a periodic lattice, characterized by certain global Z2 quantum numbers. This property is important for application in quantum computing as the global quantum numbers can be used as protected qubits. It is therefore instrumental to construct and study Z2 topological order from a general framework. Our results in this thesis provide such a framework. It is based on the simplest and most illustrative Z2 topological order: the Toric Code (TC), which contains static and non-interacting anyonic quasiparticles e, m and ε. Building on this interpretation, in the first part of the thesis two exact mappings are presented from the spin Hilbert space to the Hilbert space of (e,m) and (e,ε). The mappings are derived on infinite, open, cylindrical and periodic lattices respectively. Mutual anyonic statistics as well as the effect of the global Z2 quantum numbers are taken into account. Due to the mutual anyonic statistics of the elementary excitations, the mappings turn out to be highly non-local. In addition, it is shown that the mapping to e and ε anyons can be carried over directly to the honeycomb lattice, where the anyons become visons and Majorana fermions in the Kitaev honeycomb model. The mappings allow one to rewrite any spin Hamiltonians as Hamiltonians of anyons. In the second part of the thesis, we construct a series of spin models which are mapped to Hamiltonians of free anyons. In particular, a series of Z2 topological phases `enriched by lattice translation symmetry' are constructed which are also topological superconductors of ε particles. Their properties can be analyzed generally using the duality and then the theory of topological superconductivity. In particular, their ground state degeneracy on a periodic lattice may depend on lattice size. For these phases a classification scheme is proposed, which generalizes classification by the integer Chern number. Some of the conclusions are then verified directly by exact solutions on the spin lattice. The emergent anyon statistics of e-particles in these phases is also analyzed by computing numerically the Berry phase of their motion on top of the superconducting vacua. For phases with C=0 yet still topologically non-trivial, we discover examples of `weak symmetry breaking': the e-lattice splits into two inequivalent sublattices which are exchanged by lattice translations. The e-particles on the two sublattices acquire mutual anyonic statistics. In topological phases with non-zero C, the mutual braiding of e is confirmed explicitly. In addition, the Berry phase due to background flux of each square unit cell is quantized depending on the underlying topology of the phases. This quantity is related to properties of the vison band in Kitaev materials. Lastly, the ZN (N>2) extension of Z2 topological order is discussed. Constructing the duality to `parafermions' in this case is much more complex. The difficulties of deriving such a mapping are pointed out and we only present exact solutions to certain Hamiltonians on the spin lattice.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Quantum bits, or qubits, are highly fragile due to interactions with the environment. The search for good protocols for protecting quantum information from decoherence is mandatory in order to make large-scale quantum computation possible. Most of the models proposed for this assume that correlations in the environment do not exist. Correlations can induce a time dependent error probability thus seriously damaging the quantum information over the time even if a quantum correction code is avaliable. In this way, we must taking into consideration possible physical limitations to fault-tolerant quantum computing. In this work we apply the Kernel Polynomial Method (KPM) to evaluate the density of states and fidelity decay of a L = 3 toric code without taking the lattice spin dynamics into account. The Hamiltonian model is based in a free bosonic environment and a spin-boson coupling, with two decoherence channels X and Z. A long-range, anisotropic interaction between spin pairs is then proposed as a correlated model. This correlation is directly related to the interaction strengh and range between spins. We show that the fidelity decay time scale depends on these parameters. / Os bits quânticos, ou qubits, são altamente sensíveis a interações com o ambiente. O estudo de protocolos visando proteger a informação quântica da descoerência é essencial para a implementação da computação quântica em larga escala. Boa parte dos modelos propostos para esta finalidade assume as correlações no ambiente como inexistentes. Estas podem induzir uma dependência temporal na probabilidade de erro, comprometendo efetivamente a confiabilidade da informação quântica ao longo do tempo, mesmo na presença de um código de correção. Sendo assim, devemos levar em consideração possíveis limitações físicas na computação quântica tolerante a falhas. Neste trabalho aplicamos o Método Kernel Polinomial (KPM) no cálculo da densidade de estados e do decaimento da fidelidade para o código tórico L = 3 sem considerar a dinâmica entre os spins da rede. O modelo Hamiltoniano utilizado consiste em um ambiente bosônico livre e um acoplamento spin-bóson, com dois canais de descoerência, X e Z. Uma interação efetiva de longo alcance, anisotrópica, entre todos os pares de spins da rede é então proposta como um modelo correlacionado. A correlação está diretamente associada à amplitude e ao alcance da interação entre os spins. Mostramos que a escala de tempo do decaimento da fidelidade depende destes fatores.

Topological order is a new field of research involving exotic physics. Among other things it has been suggested as a means for realising fault-tolerant quantum computation. Topological degeneracy, i.e. the ground state degeneracy of a topologically ordered state, is one of the quantities that have been used to characterize such states. Topological order has also been suggested as a possible quantum information storage. We study two-dimensional lattice models defined on a closed manifold, specifically on a torus, and find that these systems exhibit topological degeneracy proportional to the genus of the manifold on which they are defined. We also find that the addition of lattice dislocations increases the ground state degeneracy, a behaviour that can be interpreted as artificially increasing the genus of the manifold. We derive the fusion and braiding rules of the model, which are then used to calculate the braiding properties of the dislocations themselves. These turn out to resemble non-abelian anyons, a property that is important for the possibility to achieve universal quantum computation. One can also emulate lattice dislocations synthetically, by adding an external field. This makes them more realistic for potential experimental realisations. / Topologisk ordning är ett nytt område inom fysik som bland annat verkar lovande som verktyg för förverkligandet av kvantdatorer. En av storheterna som karakteriserar topologiska tillstånd är det totala antalet degenererade grundtillstånd, den topologiska degenerationen. Topologisk ordning har också föreslagits som ett möjligt sätt att lagra kvantdata. Vi undersöker tvådimensionella gittermodeller definierade på en sluten mångfald, specifikt en torus, och finner att dessa system påvisar topologisk degeneration som är proportionerlig mot mångfaldens topologiska genus. När dislokationer introduceras i gittret finner vi att grundtillståndets degeneration ökar, något som kan ses som en artificiell ökning av mångfaldens genus. Vi härleder sammanslagningsregler och flätningsregler för modellen och använder sedan dessa för att räkna ut flätegenskaperna hos själva dislokationerna. Dessa visar sig likna icke-abelska anyoner, en egenskap som är viktiga för möjligheten att kunna utföra universella kvantberäkningar. Det går också att emulera dislokationer i gittret genom att lägga på ett yttre fält. Detta gör dem mer realistiska för eventuella experimentella realisationer.