No description available.

Quantum many-body phenomena have been a focal point of the physics community for the last several decades. From material science and chemistry to model systems and quantum computing, diverse problems share mathematical description and challenges. A key roadblock in many subfields is the exponential increase in problem size with increasing number of quantum constituents. Therefore, development of efficient compression and approximation methods is the only way to move forward. Parameterized models coming from the field of machine learning have successfully been applied to very large classical problems where data is abundant, leveraging recent advances in high-performance computing. In this thesis, state-of-the-art methods relying on such models are applied to the quantum many-body problem in two distinct ways: from first principles and data-driven, as described in chapter 1. In chapters 2 and 3, the framework of quantum Monte Carlo is used to efficiently manipulate variational approximations of many-body states, obtaining non-equilibrium states occurring in quantum circuits and real-time dynamics of large systems. In chapters 4 and 5, simulated synthetic data is used to train surrogate models that enhance original methods, allowing for computations that would otherwise be out of reach for conventional solvers. In all cases, a computational advantage is established when using machine learning methods to compress different versions of the quantum many-body problem. Each chapter is concluded by proposing extensions and novel applications of new compressed representation of the problem.

Device-independent cryptography provides security in various tasks whilst removing an assumption that cryptographers previously thought of as crucial -- complete trust in the machinations of their experimental apparatus. The theory of Bell inequalities as a proof of indeterminism within nature allows for secure device-independent schemes requiring neither trust in the cryptographers' devices nor reliance on the completeness of quantum mechanics. However, the extreme paranoia attributable to the relaxed assumptions within device independence requires an explicit consideration of the previously assumed ability of the experimenters to freely make random choices. This thesis addresses the so-called `free will loophole', presenting Bell tests and associated cryptographic protocols robust against adversarial manipulation of the random number generators with which measurements in a Bell test are selected. We present several quantitative measures for this experimental free will, otherwise known as measurement dependence. We discuss how an eavesdropper maliciously preprogramming the experimenters' untrusted devices can falsely simulate the violation of a Bell inequality. We also bound the amount of Bell violation achievable within a certain degree of measurement dependence. This analysis extends to device-independent randomness expansion, bounding the guessing probability and estimating the amount of privacy amplification required to distil private randomness. The protocol is secure against either arbitrary no-signalling or quantum adversaries. We also consider device-independent key distribution, studying adversarial models that exploit the free will loophole. Finally, we examine a model correlated between the random number generators and Bell devices across multiple runs of a Bell test. This enables an explicit exposition of the optimal cheating strategy and how the correlations manifest themselves within this strategy. We prove that there remain Bell violations for a sufficiently high, yet non-maximal degree of measurement dependence which cannot be simulated by a classical attack, regardless of how many runs of the experiment those choices are correlated over.

We investigate the simulations of the the Schrödinger equation using the onedimensional quantum lattice Boltzmann (QLB) scheme and the irregular behaviour of solution. We isolate error due to approximation of the Schrödinger solution with the non-relativistic limit of the Dirac equation and numerical error in solving the Dirac equation. Detailed analysis of the original scheme showed it to be first order accurate. By discretizing the Dirac equation consistently on both sides we derive a second order accurate QLB scheme with the same evolution algorithm as the original and requiring only a one-time unitary transformation of the initial conditions and final output. We show that initializing the scheme in a way that is consistent with the non-relativistic limit supresses the oscillations around the Schrödinger solution. However, we find the QLB scheme better suited to simulation of relativistic quantum systems governed by the Dirac equation and apply it to the Klein paradox. We reproduce the quantum tunnelling results of previous research and show second order convergence to the theoretical wave packet transmission probability. After identifying and correcting the error in the multidimensional extension of the original QLB scheme that produced asymmetric solutions, we expand our second order QLB scheme to multiple dimensions. Next we use the QLB scheme to simulate Klein tunnelling of massless charge carriers in graphene, compare with theoretical solutions and study the dependence of charge transmission on the incidence angle, wave packet and potential barrier shape. To do this we derive a representation of the Dirac-like equation governing charge carriers in graphene for the one-dimensional QLB scheme, and derive a two-dimensional second order graphene QLB scheme for more accurate simulation of wave packets. We demonstrate charge confinement in a graphene device using a configuration of multiple smooth potential barriers, thereby achieving a high ratio of on/off current with potential application in graphene field effect transistors for logic devices. To allow simulation in magnetic or pseudo-magnetic fields created by deformation of graphene, we expand the scheme to include vector potentials. In addition, we derive QLB schemes for bilayer graphene and the non-linear Dirac equation governing Bose-Einstein condensates in hexagonal optical lattices.

The American Projectivist verse of Jack Spicer, Charles Olson, and Robert Duncan contains within its poetics many epistemological assumptions shared by quantum physics. These assumptions exist in three broad categories: perception, process, and wholeness. In physics, the epistemology of perception has been profoundly altered by the Heisenberg Uncertainty Relation, which creates a symbiotic relationship between the observer and the observed. At least one photon of light is necessary to observe an electron; one photon is sufficient to alter the electron's momentum or position; therefore, a physicist affects an electron's "fate" in the act of observing it. Similarly, in Projectivist poetics, the perceptions of the reader are often enlisted to help "compose" the poem which is offered to him in "pieces," or, as in Robert Duncan's poetry especially, in self-reflexive segments. By "self-reflexive," we further mean that the Projectivist poem often "mirrors itself" as an electron "mirrors itself" as wave or as particle, while it is paradoxically both. A Projectivist poem may pause halfway through and "unravel" itself, i.e., study its own etymology. The reader thus must participate in "putting the poem back together," as the physicist participates in the phenomena he observes. The second epistemological model in physics and poetry stresses becoming, rather than being. Matter at the subatomic level has been defined as energy-in-flux. Similarly, the Projectivist poems of Charles Olson especially often exist as "fields" with no syntactical beginnings or endings. Moreover, the "I" of the Maximus Poems is often seen in a perpetual process of becoming the world of spacetime in the poems, creating a system similar to the being-and-becoming model of particle-and-field in quantum mechanics. Third, wholeness is a premise governing poetry and physics separately and together. Jack Spicer's thematics blend matter and consciousness, as "love and death matter/Matter as wave and particle." Similarly, Robert Duncan's poetics describes a "dancing organization between personal and cosmic identity." In physics, wholeness is seen primarily in an "implicate order" which attempts to overturn the old paradigms of fragmentation and connect matter and consciousness, including language, as interrelated systems of information.

This thesis contains four related papers which study different aspects of double constructions for braided Hopf algebras. The main result is a categorical action of a braided version of the Drinfeld center on a Heisenberg analogue, called the Hopf center. Moreover, an application of this action to the representation theory of rational Cherednik algebras is considered. Chapter 1 : In this chapter, the Drinfeld center of a monoidal category is generalized to a class of mixed Drinfeld centers. This gives a unified picture for the Drinfeld center and a natural Heisenberg analogue. Further, there is an action of the former on the latter. This picture is translated to a description in terms of Yetter-Drinfeld and Hopf modules over quasi-bialgebras in a braided monoidal category. Via braided reconstruction theory, intrinsic definitions of braided Drinfeld and Heisenberg doubles are obtained, together with a generalization of the result of Lu (1994) that the Heisenberg double is a 2-cocycle twist of the Drinfeld double for general braided Hopf algebras. Chapter 2 : In this chapter, we present an approach to the definition of multiparameter quantum groups by studying Hopf algebras with triangular decomposition. Classifying all of these Hopf algebras which are of what we call weakly separable type, we obtain a class of pointed Hopf algebras which can be viewed as natural generalizations of multiparameter deformations of universal enveloping algebras of Lie algebras. These Hopf algebras are instances of a new version of braided Drinfeld doubles, which we call asymmetric braided Drinfeld doubles. This is a generalization of an earlier result by Benkart and Witherspoon (2004) who showed that two-parameter quantum groups are Drinfeld doubles. It is possible to recover a Lie algebra from these doubles in the case where the group is free and the parameters are generic. The Lie algebras arising are generated by Lie subalgebras isomorphic to sl2. Chapter 3 : The universal enveloping algebra <i>U</i>(tr_n) of a Lie algebra associated to the classical Yang-Baxter equation was introduced in 2006 by Bartholdi-Enriquez-Etingof-Rains where it was shown to be Koszul. This algebra appears as the A_<i>n</i>-1 case in a general class of braided Hopf algebras in work of Bazlov-Berenstein (2009) for any complex reection group. In this chapter, we show that the algebras corresponding to the series <i>B_n</i> and <i>D_n</i>, which are again universal enveloping algebras, are Koszul. This is done by constructing a PBW-basis for the quadratic dual. We further show how results of Bazlov-Berenstein can be used to produce pairs of adjoint functors between categories of rational Cherednik algebra representations of different rank and type for the classical series of Coxeter groups. Chapter 4 : Quantum groups can be understood as braided Drinfeld doubles over the group algebra of a lattice. The main objects of this chapter are certain braided Drinfeld doubles over the Drinfeld double of an irreducible complex reflection group. We argue that these algebras are analogues of the Drinfeld-Jimbo quantum enveloping algebras in a setting relevant for rational Cherednik algebra. This analogy manifests itself in terms of categorical actions, related to the general Drinfeld-Heisenberg double picture developed in Chapter 2, using embeddings of Bazlov and Berenstein (2009). In particular, this work provides a class of quasitriangular Hopf algebras associated to any complex reflection group which are in some cases finite-dimensional.

This thesis studies the categorical formalisation of quantum computing, through the prism of type theory, in a three-tier process. The first stage of our investigation involves the creation of the dagger lambda calculus, a lambda calculus for dagger compact categories. Our second contribution lifts the expressive power of the dagger lambda calculus, to that of a quantum programming language, by adding classical control in the form of complementary classical structures and dualisers. Finally, our third contribution demonstrates how our lambda calculus can be applied to various well known problems in quantum computation: Quantum Key Distribution, the quantum Fourier transform, and the teleportation protocol.

Models of geometry that are intrinsically quantum-mechanical in nature arise from the recoupling theory of space-time symmetry groups. Roger Penrose constructed such a model from SU(2) recoupling in his theory of spin networks; he showed that spin measurements in a classical limit are necessarily consistent with a three-dimensional Euclidian vector space. T. Regge and G. Ponzano expressed the semi-classical limit of this spin model in a form resembling a path integral of the Einstein-Hilbert action in three Euclidian dimensions. This thesis gives new proofs of the Penrose spin geometry theorem and of the Regge-Ponzano decomposition theorem. We then consider how to generalize these two approaches to other groups that give rise to new models of quantum geometries. In particular, we show how to construct quantum models of four-dimensional relativistic space-time from the re-coupling theory of the Poincare group.

This work is about diagrammatic languages, how they can be represented, and what they in turn can be used to represent. More specifically, it focuses on representations and applications of string diagrams. String diagrams are used to represent a collection of processes, depicted as "boxes" with multiple (typed) inputs and outputs, depicted as "wires". If we allow plugging input and output wires together, we can intuitively represent complex compositions of processes, formalised as morphisms in a monoidal category. While string diagrams are very intuitive, existing methods for defining them rigorously rely on topological notions that do not extend naturally to automated computation. The first major contribution of this dissertation is the introduction of a discretised version of a string diagram called a string graph. String graphs form a partial adhesive category, so they can be manipulated using double-pushout graph rewriting. Furthermore, we show how string graphs modulo a rewrite system can be used to construct free symmetric traced and compact closed categories on a monoidal signature. The second contribution is in the application of graphical languages to quantum information theory. We use a mixture of diagrammatic and algebraic techniques to prove a new classification result for strongly complementary observables. Namely, maximal sets of strongly complementary observables of dimension D must be of size no larger than 2, and are in 1-to-1 correspondence with the Abelian groups of order D. We also introduce a graphical language for multipartite entanglement and illustrate a simple graphical axiom that distinguishes the two maximally-entangled tripartite qubit states: GHZ and W. Notably, we illustrate how the algebraic structures induced by these operations correspond to the (partial) arithmetic operations of addition and multiplication on the complex projective line. The third contribution is a description of two software tools developed in part by the author to implement much of the theoretical content described here. The first tool is Quantomatic, a desktop application for building string graphs and graphical theories, as well as performing automated graph rewriting visually. The second is QuantoCoSy, which performs fully automated, model-driven theory creation using a procedure called conjecture synthesis.

Neste trabalho, apresentamos um método que permite explicitar a arbitrariedade contida nas soluções das equações de onda relativísticas, na presença de certos tipos de campos eletromagnéticos externos. Esta arbitrariedade está relacionada com a existência de uma transformação, com a qual podemos reduzir o número de variáveis presentes na equação original. Através desta transformação, criamos uma representação, a qual permite obter novos conjuntos de soluções exatas e construir a função de evolução para a equação de Klein-Gordon. Como resultado, apresentamos novos conjuntos de soluções, estacionárias e não-estacionárias, para o problema em um campo magnético constante e uniforme e a combinação deste campo com um campo elétrico longitudinal. / We demonstrate how one can describe explicitly the present arbitrariness in solutions of relativistic wave equations in external electromagnetic fields of special form. This arbitrariness is connected to the existence of a transformation, which reduces effectively the number of variables in the initial equations. Then we use the corresponding representations to construct new sets of exact solutions, which may have a physical interest, and to construct the evolution function to the Klein-Gordon equation. As resulted, we present new sets of stationary and nonstationary solutions in magnetic field and in some superpositions of electric and magnetic fields.