We compute rigorously the ground and equilibrium states for Kitaev's model in 2D, both the finite and infinite version, using an analogy with the 1D Ising ferromagnet. Next, we investigate the structure of the reduced dynamics in the presence of thermal baths in the Markovian regime. Special attention is paid to the dynamics of the topological freedoms which have been proposed for storing quantum information.
We compute rigorously the ground and equilibrium states for Kitaev's model in 2D, both the finite and infinite version, using an analogy with the 1D Ising ferromagnet. Next, we investigate the structure of the reduced dynamics in the presence of thermal baths in the Markovian regime. Special attention is paid to the dynamics of the topological freedoms which have been proposed for storing quantum information.
We analyse stability of the four-dimensional Kitaev model — a candidate for scalable quantum memory — in finite temperature within the weak coupling Markovian limit. It is shown that, below a critical temperature, certain topological qubit observables X and Z possess relaxation times exponentially long in the size of the system. Their construction involves polynomial in system size algorithm which uses as an input the results of measurements performed on all individual spins. We also discuss the drawbacks of such candidate for quantum memory and mention the implications of the stability of qubit for statistical mechanics.
The ability to store quantum information without recourse to constant feedback processes would yield a significant advantage for future implementations of quantum information processing. In this Letter, limitations of the prototypical model, the Toric code in two dimensions, are elucidated along with a sufficient condition for overcoming these limitations. Specifically, the interplay between Hamiltonian perturbations and dynamically occurring noise is considered as a system in its ground state is brought into contact with a thermal reservoir. This proves that when utilizing the Toric code on N2 qubits in a 2D lattice as a quantum memory, the information cannot be stored for a time O(N). In contrast, the 2D Ising model protects classical information against the described noise model for exponentially long times. The results also have implications for the robustness of braiding operations in topological quantum computation.
Statistical mechanics characterizes systems in or near equilibrium using in terms of a handful of "state" variables, e.g. temperature, rather than infinitely many degrees of freedom. Statistical physics describes the expansion of the early universe, aspects of black holes, and most fruitfully, phases of matter and their properties. Quantum considerations have improved this understanding over time and revealed new phenomena, especially in complicated "strongly correlated" systems. Topological phases of matter, e.g., are of both fundamental and practical interest: these phases cannot be distinguished locally, unlike ice and water, which also allows them to store and process quantum information in a "fault-tolerant" manner, recently proposed for application to quantum computation. However, above zero temperature, thermal effects can overwrite this information.Recent experiments on isolated systems have raised fundamental questions and revealed new routes to quantum computing. We now know that entanglement, generated dynamically as a quantum state evolves, "hides" local information about the past, producing familiar equilibrium states, described by a temperature. However, many systems do not thermalize: strong disorder can lead to MBL, which supports numerous phenomena forbidden in equilibrium and can protect quantum information at infinite temperature. In particular, both MBL and thermal systems are robust phases of matter, with a novel, athermal phase transition between them. This thesis begins with an overview of MBL and thermalization, followed by an overview of exactly soluble quantum systems. We then turn to an important result in the field by this author: we introduce the first nontrivial example of an integrable Floquet model and comment on its solution and salient features. We then discuss how integrable models can provide insight into quantum thermalization, e.g. in terms of entanglement growth and demonstrating that conserved charges diffuse. We then investigate thermalization away from the integrable limit, also known as "quantum chaos." We review the standard techniques in this field and, briefly, several important results, before reproducing work by this author establishing definitively the long-conjectured result that the onset of thermalization in the presence of a conserved charge is governed by diffusion of said charge. We then investigate the interplay of conventional and topological order with nonequilibrium phase structure, with applications to quantum computation in mind. We review localization-protected quantum order in several models. We then investigate two models with non-Abelian symmetry, and show that MBL in such models can only realize if the symmetry breaks spontaneously to an Abelian subgroup. Finally, we conclude by examining open quantum systems, where we find several counterintuitive results that show that baths can, in some cases, enhance localization in certain systems, which may have use in realizing quantum computation.
We prove a no-go theorem for storing quantum information in equilibrium systems. Namely, quantum information cannot be stored in a system with time-independent Hamiltonian interacting with heat bath of temperature T>0 during time that grows with the number of used qubits. We prove it by showing, that storing quantum information for macroscopic time would imply existence of perpetuum mobile of the second kind. The general results are illustrated by the Kitaev model of quantum memory. In contrast, classical information can be stored in equilibrium states for arbitrary long times. We show how it is possible via phase-transition type phenomena. Our result shows that there is a fundamental difference between quantum and classical information in {\it physical} terms.
Developing quantum systems which are robust against noise are of prime importance to the realisation of quantum technologies. Without fault tolerance, we will never be able to preserve delicate quantum states for macroscopic time scales, a necessary requirement in the construction of scalable quantum computer. Topologically ordered models offer very beautiful mechanisms for preserving quantum states. In such models, quantum information is encoded globally over a degenerate topological Hilbert space which offers a natural robustness against local environmental noise. Remarkably, topological order is present in certain realistic condensed-matter systems. This provides a platform for accessing topological order in a laboratory. Moreover, considering condensed-matter systems enables us to combine topological features with other physical effects to enhance their behaviour. In this Thesis we study the physics of topologically ordered lattice models with defects. We seek practical applications of lattice defects for the realisation of a fault-tolerant quantum computer. The first result we present shows that anyonic data of point-like lattice defects, called twists, can be found by measuring the entanglement of the ground state of the host system. The data we learn relates to the capacity of a twist defect to perform quantum computational tasks. The second result in this Thesis concerns the dynamics of the qudit toric code model with line-like defects coupled to a thermal bath. We show we can entropically inspire fragile glassy dynamics in the system. Such dynamics qualitatively improve the coherence times of quantum information encoded in the ground state of the lattice. A novel way of achieving fault-tolerant quantum computation is by producing and manipulating twist defects of topologically ordered systems. In many ways, this paradigm of topological quantum computation is analogous to quantum computation using anyons. The first area of study in this Thesis extends the analogy between anyons and twists. Specifically, we show that the anyonic data of twists in Kitaev's toric code model can be extracted using topological entanglement entropy calculations in the same way as the same data can be extracted from anyons. We show this using a rigorously solvable lattice model as an example to produce exact analytic results. In particular, our results show we can extract the quantum dimension of a twist, and that we can study the quantum dimension of their fusion products. We compare the obtained results with the anyonic data of the Ising anyon model, further probing an analogy drawn previously in the literature. The second result presented in this Thesis shows that the application of lattice defects can introduce novel dynamics to a two-dimensional topologically ordered quantum memory where excitations carry different masses. A two-dimensional topological model which supports an anyon model with a splitting structure allowed by its fusion channels should be able to entropically achieve high-energy excitation configurations before quantum information encoded in its ground state decoheres. We introduce a grid of lattice defects to a local two-dimensional Hamiltonian model, which, when coupled to a thermal bath, will dynamically steer the excitations into high energy configurations with high probability within a suitable temperature regime. We demonstrate the proliferating dynamics using numerical simulations in a low-temperature regime where we show polynomial improvements in coherence times by increasing the system size for small system sizes, as well as coherence times which scale weakly super exponentially with the inverse temperature of the bath. The dynamics we demonstrate provide the first example of a system which entropically steers excitations into high-energy configurations. The dynamics we demonstrate may lead to the development of experimentally tractable architectures for low dimensional quantum memories. The study of thermal stability in this Thesis requires the development of methods of correcting topologically ordered lattices which have suffered errors. Namely, we require a decoding algorithm for the qudit generalisation of Kitaev's toric code. Further work presented in this Thesis compares rigorously two different decoding algorithms which use renormalisation-group techniques to process classical syndrome information about noise suffered by the topological code. In particular, we study the improvement in their thresholds as the local dimension of its physical systems increase. Our numerical results enable us to analyse and identify the limitations of the different methods of decoding.