Quantum computing and post-quantum cryptography

One of the main practical implications of quantum mechanical theory is quantum computing, and therefore the quantum computer. Quantum computing (for example, with Shor’s algorithm) challenges the computational hardness assumptions, such as the factoring problem and the discrete logarithm problem, that anchor the safety of cryptosystems. So the scientific community is studying how to defend cryptography; there are two defense strategies: the quantum cryptography (which involves the use of quantum cryptographic algorithms on quantum computers) and the post-quantum cryptography (based on classical cryptographic algorithms, but resistant to quantum computers). For example, National Institute of Standards and Technology (NIST) is collecting and standardizing the post-quantum ciphers, as it established DES and AES as symmetric cipher standards, in the past. In this thesis an introduction on quantum mechanics was given, in order to be able to talk about quantum computing and to analyze Shor’s algorithm. The differences between quantum and post-quantum cryptography were then analyzed. Subsequently the focus was given to the mathematical problems assumed to be resistant to quantum computers. To conclude, post-quantum digital signature cryptographic algorithms selected by NIST were studied and compared in order to apply them in today’s life.

Quantum information and quantum communication theory in relation to black hole idealized mechanical models

Oggigiorno il concetto di informazione è diventato cruciale in fisica, pertanto, siccome la migliore teoria che abbiamo per compiere predizioni riguardo l'universo è la meccanica quantistica, assume una particolare importanza lo sviluppo di una versione quantistica della teoria dell'informazione. Questa centralità è confermata dal fatto che i buchi neri hanno entropia. Per questo motivo, in questo lavoro sono presentati elementi di teoria dell'informazione quantistica e della comunicazione quantistica e alcuni sono illustrati riferendosi a modelli quantistici altamente idealizzati della meccanica di buco nero. In particolare, nel primo capitolo sono forniti tutti gli strumenti quanto-meccanici per la teoria dell'informazione e della comunicazione quantistica. Successivamente, viene affrontata la teoria dell'informazione quantistica e viene trovato il limite di Bekenstein alla quantità di informazione chiudibile entro una qualunque regione spaziale. Tale questione viene trattata utilizzando un modello quantistico idealizzato della meccanica di buco nero supportato dalla termodinamica. Nell'ultimo capitolo, viene esaminato il problema di trovare un tasso raggiungibile per la comunicazione quantistica facendo nuovamente uso di un modello quantistico idealizzato di un buco nero, al fine di illustrare elementi della teoria. Infine, un breve sommario della fisica dei buchi neri è fornito in appendice.

Planck stars: theory and phenomenology

General Relativity (GR) is one of the greatest scientific achievements of the 20th century along with quantum theory. Despite the elegance and the accordance with experimental tests, these two theories appear to be utterly incompatible at fundamental level. Black holes provide a perfect stage to point out these difficulties. Indeed, classical GR fails to describe Nature at small radii, because nothing prevents quantum mechanics from affecting the high curvature zone, and because classical GR becomes ill-defined at r = 0 anyway. Rovelli and Haggard have recently proposed a scenario where a negative quantum pressure at the Planck scales stops and reverts the gravitational collapse, leading to an effective “bounce” and explosion, thus resolving the central singularity. This scenario, called Black Hole Fireworks, has been proposed in a semiclassical framework. The purpose of this thesis is twofold: - Compute the bouncing time by means of a pure quantum computation based on Loop Quantum Gravity; - Extend the known theory to a more realistic scenario, in which the rotation is taken into account by means of the Newman-Janis Algorithm.

Semiclassical analysis of systems of Schrödinger equations

The scalar Schrödinger equation models the probability density distribution for a particle to be found in a point x given a certain potential V(x) forming a well with respect to a fixed energy level E_0. Formally two real inversion points a,b exist such that V(a)=V(b)=E_0, V(x)<0 in (a,b) and V(x)>0 for xb. Following the work made by D.Yafaev and performing a WKB approximation we obtain solutions defined on specific intervals. The aim of the first part of the thesis is to find a condition on E, which belongs to a neighbourhood of E_0, such that it is an eigenvalue of the Schrödinger operator, obtaining in this way global and linear dependent solutions in L2. In quantum mechanics this condition is known as Bohr-Sommerfeld quantization. In the second part we define a Schrödinger operator referred to two potential wells and we study the quantization conditions on E in order to have a global solution in L2xL2 with respect to the mutual position of the potentials. In particular their wells can be disjoint,can have an intersection, can be included one into the other and can have a single point intersection. For these cases we refer to the works of A.Martinez, S. Fujiié, T. Watanabe, S. Ashida.

Computational study on the asymmetric aminocatalysed Michael addition reaction of cyclohexanone to trans–β–nitrostyrene

Asymmetric organocatalysed reactions are one of the most fascinating synthetic strategies which one can adopt in order to induct a desired chirality into a reaction product. From all the possible practical applications of small organic molecules in catalytic reaction, amine–based catalysis has attracted a lot of attention during the past two decades. The high interest in asymmetric aminocatalytic pathways is to account to the huge variety of carbonyl compounds that can be functionalized by many different reactions of their corresponding chiral–enamine or –iminium ion as activated nucleophile and electrophile, respectively. Starting from the employment of L–Proline, many useful substrates have been proposed in order to further enhance the catalytic performances of these reaction in terms of enantiomeric excess values, yield, conversion of the substrate and turnover number. In particular, in the last decade the use of chiral and quasi–enantiomeric primary amine species has got a lot of attention in the field. Contemporaneously, many studies have been carried out in order to highlight the mechanism through which these kinds of substrates induct chirality into the desired products. In this scenario, computational chemistry has played a crucial role due to the possibility of simulating and studying any kind of reaction and the transition state structures involved. In the present work the transition state geometries of primary amine–catalysed Michael addition reaction of cyclohexanone to trans–β–nitrostyrene with different organic acid cocatalysts has been studied through different computational techniques such as density functional theory based quantum mechanics calculation and force–field directed molecular simulations.

How to get an equation named after you (part I)

Abstract|IT Nelle lauree triennali le pubblicazioni scientifiche non vengono studiate. Per quanto possa rivelarsi un piccolo intervento, lo scopo di questa tesi è invece quello di rendere più accessibili ai pochi interessati alcune vecchie e polverose pubblicazioni. Inoltre, la presente tesi non dovrebbe essere trattata come un lavoro a se stante, ma come il primo mattone di un progetto che comprende tutte le maggiori pubblicazioni della storia. How to get an equation named after you (Part I) in particolare discute la serie di pubblicazioni del 1926 di Schrödinger "Quantisierung als Eigenwertproblem" e l’articolo del 1928 di Dirac "The Quantum Theory of the Electron", ovvero quei lavori dove le equazioni di Schrödinger e Dirac vennero per prime derivate. La serie di articoli del 1926 sommano ad un totale di oltre 100 pagine. Inizialmente Schrödinger dimostra come ciò su cui è basata la sua teoria possa spiegare correttamente fenomeni conosciuti come l’atomo di idrogeno e l’effetto Stark, per poi derivare la famosa equazione d’onda complessa del secondo ordine. I procedimenti matematici, in questi articoli, sono complicati e molte delle dimostrazioni non vengono mostrate oppure risultano inutilmente lunghe. La pubblicazione di Dirac invece ha principalmente a che fare con la derivazione dell’equazione, la sua generalizzazione e l’invarianza relativistica. Dimostra inoltre che tale equazione è compatibile con passate teorie. La lettura di Dirac è molto più sistematica, dato il largo utilizzo di dimostrazioni matematiche laddove Schrödinger avrebbe usato parole.

Variational Quantum Splines: Moving Beyond Linearity for Quantum Activation Functions

Activation functions within neural networks play a crucial role in Deep Learning since they allow to learn complex and non-trivial patterns in the data. However, the ability to approximate non-linear functions is a significant limitation when implementing neural networks in a quantum computer to solve typical machine learning tasks. The main burden lies in the unitarity constraint of quantum operators, which forbids non-linearity and poses a considerable obstacle to developing such non-linear functions in a quantum setting. Nevertheless, several attempts have been made to tackle the realization of the quantum activation function in the literature. Recently, the idea of the QSplines has been proposed to approximate a non-linear activation function by implementing the quantum version of the spline functions. Yet, QSplines suffers from various drawbacks. Firstly, the final function estimation requires a post-processing step; thus, the value of the activation function is not available directly as a quantum state. Secondly, QSplines need many error-corrected qubits and a very long quantum circuits to be executed. These constraints do not allow the adoption of the QSplines on near-term quantum devices and limit their generalization capabilities. This thesis aims to overcome these limitations by leveraging hybrid quantum-classical computation. In particular, a few different methods for Variational Quantum Splines are proposed and implemented, to pave the way for the development of complete quantum activation functions and unlock the full potential of quantum neural networks in the field of quantum machine learning.