Homomorphisms and Structural Properties of Relational Systems

Two main topics are considered: The characterisation of finite homomorphism dualities for relational structures, and the splitting property of maximal antichains in the homomorphism order.Comment: PhD Thesis, 77 pages, 14 figure

Normalizer Circuits and Quantum Computation

(Abridged abstract.) In this thesis we introduce new models of quantum computation to study the emergence of quantum speed-up in quantum computer algorithms. Our first contribution is a formalism of restricted quantum operations, named normalizer circuit formalism, based on algebraic extensions of the qubit Clifford gates (CNOT, Hadamard and $\pi/4$-phase gates): a normalizer circuit consists of quantum Fourier transforms (QFTs), automorphism gates and quadratic phase gates associated to a set $G$, which is either an abelian group or abelian hypergroup. Though Clifford circuits are efficiently classically simulable, we show that normalizer circuit models encompass Shor's celebrated factoring algorithm and the quantum algorithms for abelian Hidden Subgroup Problems. We develop classical-simulation techniques to characterize under which scenarios normalizer circuits provide quantum speed-ups. Finally, we devise new quantum algorithms for finding hidden hyperstructures. The results offer new insights into the source of quantum speed-ups for several algebraic problems. Our second contribution is an algebraic (group- and hypergroup-theoretic) framework for describing quantum many-body states and classically simulating quantum circuits. Our framework extends Gottesman's Pauli Stabilizer Formalism (PSF), wherein quantum states are written as joint eigenspaces of stabilizer groups of commuting Pauli operators: while the PSF is valid for qubit/qudit systems, our formalism can be applied to discrete- and continuous-variable systems, hybrid settings, and anyonic systems. These results enlarge the known families of quantum processes that can be efficiently classically simulated. This thesis also establishes a precise connection between Shor's quantum algorithm and the stabilizer formalism, revealing a common mathematical structure in several quantum speed-ups and error-correcting codes.Comment: PhD thesis, Technical University of Munich (2016). Please cite original papers if possible. Appendix E contains unpublished work on Gaussian unitaries. If you spot typos/omissions please email me at JLastNames at posteo dot net. Source: http://bit.ly/2gMdHn3. Related video talk: https://www.perimeterinstitute.ca/videos/toy-theory-quantum-speed-ups-based-stabilizer-formalism Posted on my birthda

The Diophantine problem in Chevalley groups

In this paper we study the Diophantine problem in Chevalley groups $G_\pi (\Phi,R)$, where $\Phi$ is an indecomposable root system of rank $> 1$, $R$ is an arbitrary commutative ring with $1$. We establish a variant of double centralizer theorem for elementary unipotents $x_\alpha(1)$. This theorem is valid for arbitrary commutative rings with $1$. The result is principle to show that any one-parametric subgroup $X_\alpha$, $\alpha \in \Phi$, is Diophantine in $G$. Then we prove that the Diophantine problem in $G_\pi (\Phi,R)$ is polynomial time equivalent (more precisely, Karp equivalent) to the Diophantine problem in $R$. This fact gives rise to a number of model-theoretic corollaries for specific types of rings.Comment: 44 page

Contextuality in foundations and quantum computation

Contextuality is a key concept in quantum theory. We reveal just how important it is by demonstrating that quantum theory builds on contextuality in a fundamental way: a number of key theorems in quantum foundations can be given a unifi ed presentation in the topos approach to quantum theory, which is based on contextuality as the common underlying principle. We review existing results and complement them by providing contextual reformulations for Stinespring's and Bell's theorem. Both have a number of consequences that go far beyond the evident confirmation of the unifying character of contextuality in quantum theory. Complete positivity of quantum channels is already encoded in contexts, nonlocality arises from a notion of composition of contexts, and quantum states can be singled out among more general non-signalling correlations over the composite context structure by a notion of time orientation in subsystems, thus solving a much discussed open problem in quantum information theory. We also discuss nonlocal correlations under the generalisation to orthomodular lattices and provide generalised Bell inequalities in this setting. The dominant role of contextuality in quantum foundations further supports a recent hypothesis in quantum computation, which identifi es contextuality as the resource for the supposed quantum advantage over classical computers. In particular, within the architecture of measurement-based quantum computation, the resource character of nonlocality and contextuality exhibits rather clearly. We study contextuality in this framework and generalise the strong link between contextuality and computation observed in the qubit case to qudit systems. More precisely, we provide new proofs of contextuality as well as a universal implementation of computation in this setting, while emphasising the crucial role played by phase relations between measurement eigenstates. Finally, we suggest a fine-grained measure for contextuality in the form of the number of qubits required for implementation in the non-adaptive, deterministic case.Open Acces

Many valued logics: interpretations, representations and applications

2015 - 2016This thesis, as the research activity of the author, is devoted to establish new connections and to strengthen well-established relations between different branches of mathematics, via logic tools. Two main many valued logics, logic of balance and L ukasiewicz logic, are considered; their associated algebraic structures will be studied with different tools and these techniques will be applied in social choice theory and artificial neural networks. The thesis is structured in three parts. Part I The logic of balance, for short Bal(H), is introduced. It is showed: the relation with `-Groups, i.e. lattice ordered abelian groups (Chapter 2); a functional representation (Chapter 3); the algebraic geometry of the variety of `-Groups with constants (Chapter 4). Part II A brief historical introduction of L ukasiewicz logic and its extensions is provided. It is showed: a functional representation via generalized states (Chapter 5); a non-linear model for MV-algebras and a detailed study of it, culminating in a categorical theorem (Chapter 6). Part III Applications to social choice theory and artificial neural network are presented. In particular: preferences will be related to vector lattices and their cones, recalling the relation between polynomials and cones studied in Chapter 4; multilayer perceptrons will be elements of non-linear models introduced in Chapter 6 and networks will take advantages from polynomial completeness, which is studied in Chapter 2. We are going to present: in Sections 1.2 and 1.3 all the considered structures, our approach to them and their (possible) applications; in Section 1.4 a focus on the representation theory for `-Groups and MV-algebras. Note that: algebraic geometry for `-Groups provides a modus operandi which turns out to be useful not only in theoretical field, but also in applications, opening (we hope) new perspectives and intuitions, as we made in this first approach to social theory; non-linear models here presented and their relation to neural networks seem to be very promising, giving both intuitive and formal approach to many concrete problems, for instance degenerative diseases or distorted signals. All these interesting topics will be studied in future works of the author. [edited by author]Questa tesi, come l’attivit`a di ricerca dell’autore, `e dedicata a stabilire nuove connessioni e a rafforzare le relazioni ben consolidate tra diversi settori della matematica, attraverso strumenti logici. Sono considerate due principali logiche a piu` valori, logic of balance e L ukasiewicz logic; le loro strutture algebriche associate verranno studiate con strumenti diversi e queste tecniche saranno applicate nella teoria della scelta sociale e nelle reti neurali artificiali. La tesi `e strutturata in tre parti. Part I Viene introdotta la Logic of balance. Viene mostrato: la relazione con `-Groups, gruppi abeliani ordinati reticolarmente (Chapter 2); una rappresentazione funzionale (Chapter 3); geometria algebrica della variet`a degli `-Groups con costanti (Chapter 4). Part II Viene fornita una breve introduzione storica della logica di L ukasiewicz e delle sue estensioni. Viene mostrato: una rappresentazione funzionale tramite stati generalizzati (Chapter 5); Un modello non lineare per le MV-algebre e uno studio dettagliato di esso, culminando in un teorema categoriale (Chapter 6). Part III Sono presentate applicazioni alla teoria delle scelte sociali e delle rete neurali artificiali. In particolare: le preferenze saranno correlate ai reticoli vettoriali e ai loro coni, richiamando la relazione tra polinomi e coni studiati nel Capitolo 4; I multilayer perceptrons saranno elementi di modelli non lineari introdotti nel Capitolo 6 e le reti prenderanno vantaggi dalla completezza polinomiale, studiata nel Capitolo 2. La geometria algebrica per gli `-Groups fornisce un modus operandi che risulta utile non solo nel campo teorico, ma anche nelle applicazioni, aprendo (speriamo) nuove prospettive e intuizioni, come abbiamo fatto in questo primo approccio alla teoria sociale; I modelli non lineari qui presentati e la loro relazione con le reti neurali sembrano molto promettenti, offrendo un approccio intuitivo e formale a molti problemi concreti, ad esempio malattie degenerative o segnali distorti. Tutti questi argomenti saranno oggetto di studio in opere future dell’autore. [a cura dell'autore]XV n.s. (XXIX

Prospects for Declarative Mathematical Modeling of Complex Biological Systems

Declarative modeling uses symbolic expressions to represent models. With such expressions one can formalize high-level mathematical computations on models that would be difficult or impossible to perform directly on a lower-level simulation program, in a general-purpose programming language. Examples of such computations on models include model analysis, relatively general-purpose model-reduction maps, and the initial phases of model implementation, all of which should preserve or approximate the mathematical semantics of a complex biological model. The potential advantages are particularly relevant in the case of developmental modeling, wherein complex spatial structures exhibit dynamics at molecular, cellular, and organogenic levels to relate genotype to multicellular phenotype. Multiscale modeling can benefit from both the expressive power of declarative modeling languages and the application of model reduction methods to link models across scale. Based on previous work, here we define declarative modeling of complex biological systems by defining the operator algebra semantics of an increasingly powerful series of declarative modeling languages including reaction-like dynamics of parameterized and extended objects; we define semantics-preserving implementation and semantics-approximating model reduction transformations; and we outline a "meta-hierarchy" for organizing declarative models and the mathematical methods that can fruitfully manipulate them