Topological Code Architectures for Quantum Computation

This dissertation is concerned with quantum computation using many-body quantum systems encoded in topological codes. The interest in these topological systems has increased in recent years as devices in the lab begin to reach the fidelities required for performing arbitrarily long quantum algorithms. The most well-studied system, Kitaev\u27s toric code, provides both a physical substrate for performing universal fault-tolerant quantum computations and a useful pedagogical tool for explaining the way other topological codes work. In this dissertation, I first review the necessary formalism for quantum information and quantum stabilizer codes, and then I introduce two families of topological codes: Kitaev\u27s toric code and Bombin\u27s color codes. I then present three chapters of original work. First, I explore the distinctness of encoding schemes in the color codes. Second, I introduce a model of quantum computation based on the toric code that uses adiabatic interpolations between static Hamiltonians with gaps constant in the system size. Lastly, I describe novel state distillation protocols that are naturally suited for topological architectures and show that they provide resource savings in terms of the number of required ancilla states when compared to more traditional approaches to quantum gate approximation

Quantum cryptanalysis in the RAM model: Claw-finding attacks on SIKE

We introduce models of computation that enable direct comparisons between classical and quantum algorithms. Incorporating previous work on quantum computation and error correction, we justify the use of the gate-count and depth-times-width cost metrics for quantum circuits. We demonstrate the relevance of these models to cryptanalysis by revisiting, and increasing, the security estimates for the Supersingular Isogeny Diffie--Hellman (SIDH) and Supersingular Isogeny Key Encapsulation (SIKE) schemes. Our models, analyses, and physical justifications have applications to a number of memory intensive quantum algorithms

Entropy decay for Davies semigroups of a one dimensional quantum lattice

Given a finite-range, translation-invariant commuting system Hamiltonians on a spin chain, we show that the Davies semigroup describing the reduced dynamics resulting from the joint Hamiltonian evolution of a spin chain weakly coupled to a large heat bath thermalizes rapidly at any temperature. More precisely, we prove that the relative entropy between any evolved state and the equilibrium Gibbs state contracts exponentially fast with an exponent that scales logarithmically with the length of the chain. Our theorem extends a seminal result of Holley and Stroock [40] to the quantum setting, up to a logarithmic overhead, as well as provides an exponential improvement over the non-closure of the gap proved by Brandao and Kastoryano [43]. This has wide-ranging applications to the study of many-body in and out-of-equilibrium quantum systems. Our proof relies upon a recently derived strong decay of correlations for Gibbs states of one dimensional, translation-invariant local Hamiltonians, and tools from the theory of operator spaces.Depto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasFALSEunpu

Fases topológicas de la materia y sistemas cuánticos abiertos

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Físicas, Departamento de Física Teórica I Física Teórica II (Métodos Matemáticos de la física), leída el 02/11/2016. Tesis formato europeo (compendio de artículos)The growing field of topological orders has been extensively studied both form the communities of condensed matter and quantum simulation. However, very little is known about the fate of topological order in the presence of disturbing effects such as external noise or dissipation. In the first part of this thesis, we start by studying how the edge states of a topological insulator become unstable when interacting with thermal baths. Motivated by these results, we generalise the notion of Chern insulators from the well-known Hamiltonian case to Liouvillian dynamics. We achieve this goal by defining a new topological witness that is still related to the quantum Hall conductivity at finite temperature. The mixed character of edge states is also well captured by our formalism, and explicit models for topological insulators and dissipative channels are considered. Additionally, we find new topological phases that remain quantised at finite temperature. The construction is based on the Uhlmann phase, a geometric quantum phase defined for general density matrices. Using this new tool, we are able to characterise topological insulators and superconductors at finite temperature both in one and two spatial dimensions. From the experimental side, we propose a state-independent protocol to measure the topological Uhlmann phase in the context of quantum simulation. Symmetry-protected topological orders have traditionally emerged from shortrange interactions. It remains very much unknown what the role played by longrange interactions is, within the physics of these topological systems. In the second part of this thesis, we analyse how topological superconducting phases are affected by the inclusion of long-range couplings. Remarkably, we unveil new topological quasi-particles due to long-range interactions, that were absent in short-range models. We also study how topological invariants are modified by the presence of long-range effects. In the appendix section of the thesis, we explore new numerical methods for driven-dissipative phase transitions. We consider quantum systems with a dissipative term driving the system into a non-equilibrium steady state. The inclusion of short-range fluctuations out-of-equilibrium deeply modifies the shape of the phase-diagram, something never observed in equilibrium thermodynamics.Una transición de fase es una transformación entre dos estados de la materia con propiedades físicas diferentes, por ejemplo cuando el agua líquida se convierte en hielo. Tradicionalmente, la física de las transiciones de fase ha sido perfectamente descrita por la teoría de Landau. Esta teoría propone la existencia de un parámetro de orden local que es capaz de distinguir entre dos fases distintas. Además, al atravesar la transición de fase se rompe espontáneamente una simetría del sistema. A partir de los años 80 se empezaron a encontrar un tipo de transiciones de fase que no estaban bien descritas por la teoría de Landau. Estas fases de la materia se denominan órdenes topológicos y constituyen el principal objeto de esta tesis doctoral. Para estas transiciones no existe un parámetro de orden local que pueda distinguir entre fases con propiedades físicas distintas. Por el contrario, vienen caracterizadas por un parámetro de orden global que es capaz de retener la información topológica del sistema. La otra principal diferencia con respecto a las transiciones de orden, descritas por la teoría de Landau, es el papel que juegan las simetrías. En las transiciones de fase topológicas, cuando se cambia de una fase a otra, no se rompe ninguna simetría. De manera adicional, las fases topológicas de la materia vienen caracterizadas por un conjunto de propiedades distintivas: (1) el estado fundamental está separado por un gap del resto de excitaciones y está degenerado, (2) el sistema presenta estados gapless localizados en el borde, (3) las excitaciones son anyones con estadística exótica, etc...Depto. de Física TeóricaFac. de Ciencias FísicasTRUEunpu

Artificial Intelligence in Materials Science: Applications of Machine Learning to Extraction of Physically Meaningful Information from Atomic Resolution Microscopy Imaging

Materials science is the cornerstone for technological development of the modern world that has been largely shaped by the advances in fabrication of semiconductor materials and devices. However, the Moore’s Law is expected to stop by 2025 due to reaching the limits of traditional transistor scaling. However, the classical approach has shown to be unable to keep up with the needs of materials manufacturing, requiring more than 20 years to move a material from discovery to market. To adapt materials fabrication to the needs of the 21st century, it is necessary to develop methods for much faster processing of experimental data and connecting the results to theory, with feedback flow in both directions. However, state-of-the-art analysis remains selective and manual, prone to human error and unable to handle large quantities of data generated by modern equipment. Recent advances in scanning transmission electron and scanning tunneling microscopies have allowed imaging and manipulation of materials on the atomic level, and these capabilities require development of automated, robust, reproducible methods.Artificial intelligence and machine learning have dealt with similar issues in applications to image and speech recognition, autonomous vehicles, and other projects that are beginning to change the world around us. However, materials science faces significant challenges preventing direct application of the such models without taking physical constraints and domain expertise into account.Atomic resolution imaging can generate data that can lead to better understanding of materials and their properties through using artificial intelligence methods. Machine learning, in particular combinations of deep learning and probabilistic modeling, can learn to recognize physical features in imaging, making this process automated and speeding up characterization. By incorporating the knowledge from theory and simulations with such frameworks, it is possible to create the foundation for the automated atomic scale manufacturing

Quantum simulations with ultracold atoms: beyond standard optical lattices

Many outstanding problems in quantum physics, such as high-Tc superconductivity or quark confinement, are still - after decades of research - awaiting commonly accepted explanations. One reason is that such systems are often difficult to control, show an intermingling of several effects, or are not easily accessible to measurement. To arrive at a deeper understanding of the physics at work, researchers typically derive simplified models designed to capture the most striking phenomena of the system under consideration. However, due to the exponential complexity of Hilbert space, even some of the simplest of such models pose formidable challenges to analytical and numerical calculations. In 1982, Feynman proposed to solve such quantum models with experimental simulation on a physically distinct, specifically engineered quantum system [Int. J. Theor.Phys. 21, 467]. Designed to be governed by the same underlying equations as the original model, it is hoped that direct measurements on these so called quantum simulators (QSs) will allow to gather deep insights into outstanding problems of physics and beyond. In this thesis, we identify four requirements that a useful QS has to fulfill, relevance, control, reliability, and efficiency. Focusing on these, we review the state of the art of two popular approaches, digital QSs (i.e., special purpose quantum computers) and analog QSs (devices with always-on interactions). Further, focusing on possibilities to increase control over QSs, we discuss a scheme to engineer quantum correlations between mesoscopic numbers of spinful particles in optical lattices. This technique, based on quantum polarization spectroscopy, may be useful for state preparation and quantum information protocols. Additionally, employing several analytical and numerical methods for the calculation of many-body ground states, we demonstrate the variety of condensed-matter problems that can be attacked with QSs consisting of ultracold ions or neutral atoms in optical lattices. The chosen examples, some of which have already been realized in experiment, include such diverse settings as frustrated antiferromagnetism, quantum phase transitions in exotic lattice geometries, topological insulators, non-Abelian gauge-fields, orbital order of ultracold Fermions, and systems with long-range interactions. The experimental realization of all of these models requires techniques which go beyond standard optical lattices, e.g., time-periodic driving of lattices with exotic geometry, loading ultracold atoms into higher bands, or immersing trapped ions into an optical lattice. The chosen models, motivated by important open questions of quantum physics, pose difficult problems for classical computers, but they may be amenable in the near future to quantum simulation with ultracold atoms or ions. While the experimental control over relevant models has increased dramatically in the last years, the reliability and efficiency of QSs has received considerably less attention. As a second important part of this thesis, we emphasize the need to consider these aspects under realistic experimental conditions. We discuss specific situations where terms that have typically been neglected in the description of the QS introduce systematic errors and even lead to novel physics. Further, we characterize in a generic example the influence of quenched disorder on an analog QS. Its performance for simulating universal behavior near a quantum phase transition seems satisfactory for low disorder. Moreover, our results suggest a connection between the reliability and efficiency of a QS: it works less reliable exactly in those interesting regimes where classical calculations are less efficient. If QSs fulfill all of our four requirements, they may revolutionize our approach to quantum-mechanical problems, allowing to solve the behavior of complex Hamiltonians, and to design nano-scale materials and chemical compounds from the ground up