Path-integral solution of the one-dimensional Dirac quantum cellular automaton

Quantum cellular automata have been recently considered as a fundamental approach to quantum field theory, resorting to a precise automaton, linear in the field, for the Dirac equation in one dimension. In such linear case a quantum automaton is isomorphic to a quantum walk, and a convenient formulation can be given in terms of transition matrices, leading to a new kind of discrete path integral that we solve analytically in terms of Jacobi polynomials versus the arbitrary mass parameter.Comment: 5 page

Path-sum solution of the Weyl Quantum Walk in 3+1 dimensions

We consider the Weyl quantum walk in 3+1 dimensions, that is a discrete-time walk describing a particle with two internal degrees of freedom moving on a Cayley graph of the group $\mathbb Z^3$, that in an appropriate regime evolves according to Weyl's equation. The Weyl quantum walk was recently derived as the unique unitary evolution on a Cayley graph of $\mathbb Z^3$ that is homogeneous and isotropic. The general solution of the quantum walk evolution is provided here in the position representation, by the analytical expression of the propagator, i.e. transition amplitude from a node of the graph to another node in a finite number of steps. The quantum nature of the walk manifests itself in the interference of the paths on the graph joining the given nodes. The solution is based on the binary encoding of the admissible paths on the graph and on the semigroup structure of the walk transition matrices.Comment: 13 page

Solutions of a two-particle interacting quantum walk

We study the solutions of the interacting Fermionic cellular automaton introduced in Ref. [Phys Rev A 97, 032132 (2018)]. The automaton is the analogue of the Thirring model with both space and time discrete. We present a derivation of the two-particles solutions of the automaton, which exploits the symmetries of the evolution operator. In the two-particles sector, the evolution operator is given by the sequence of two steps, the first one corresponding to a unitary interaction activated by two-particle excitation at the same site, and the second one to two independent one-dimensional Dirac quantum walks. The interaction step can be regarded as the discrete-time version of the interacting term of some Hamiltonian integrable system, such as the Hubbard or the Thirring model. The present automaton exhibits scattering solutions with nontrivial momentum transfer, jumping between different regions of the Brillouin zone that can be interpreted as Fermion-doubled particles, in stark contrast with the customary momentum-exchange of the one dimensional Hamiltonian systems. A further difference compared to the Hamiltonian model is that there exist bound states for every value of the total momentum, and even for vanishing coupling constant. As a complement to the analytical derivations we show numerical simulations of the interacting evolution.Comment: 16 pages, 6 figure

Scattering and perturbation theory for discrete-time dynamics

We present a systematic treatment of scattering processes for quantum systems whose time evolution is discrete. First we define and show some general properties of the scattering operator, in particular the conservation of quasi-energy which is defined only modulo $2 \pi$. Then we develop two perturbative techniques for the power series expansion of the scattering operator, the first one analogous to the iterative solution of the Lippmann-Schwinger equation, the second one to the Dyson series of perturbative Quantum Field Theory. Our framework can be applied to a wide class of quantum simulators, like quantum walks and quantum cellular automata. As a case study we analyse the scattering properties of a one-dimensional cellular automaton with locally interacting fermions.Comment: 11 pages, 1 figur

The future of Cybersecurity in Italy: Strategic focus area

This volume has been created as a continuation of the previous one, with the aim of outlining a set of focus areas and actions that the Italian Nation research community considers essential. The book touches many aspects of cyber security, ranging from the definition of the infrastructure and controls needed to organize cyberdefence to the actions and technologies to be developed to be better protected, from the identification of the main technologies to be defended to the proposal of a set of horizontal actions for training, awareness raising, and risk management

Il Futuro della Cybersecurity in Italia: Ambiti Progettuali Strategici

Il presente volume nasce come continuazione del precedente, con l’obiettivo di delineare un insieme di ambiti progettuali e di azioni che la comunità nazionale della ricerca ritiene essenziali a complemento e a supporto di quelli previsti nel DPCM Gentiloni in materia di sicurezza cibernetica, pubblicato nel febbraio del 2017. La lettura non richiede particolari conoscenze tecniche; il testo è fruibile da chiunque utilizzi strumenti informatici o navighi in rete. Nel volume vengono considerati molteplici aspetti della cybersecurity, che vanno dalla definizione di infrastrutture e centri necessari a organizzare la difesa alle azioni e alle tecnologie da sviluppare per essere protetti al meglio, dall’individuazione delle principali tecnologie da difendere alla proposta di un insieme di azioni orizzontali per la formazione, la sensibilizzazione e la gestione dei rischi. Gli ambiti progettuali e le azioni, che noi speriamo possano svilupparsi nei prossimi anni in Italia, sono poi accompagnate da una serie di raccomandazioni agli organi preposti per affrontare al meglio, e da Paese consapevole, la sfida della trasformazione digitale. Le raccomandazioni non intendono essere esaustive, ma vanno a toccare dei punti che riteniamo essenziali per una corretta implementazione di una politica di sicurezza cibernetica a livello nazionale. Politica che, per sua natura, dovrà necessariamente essere dinamica e in continua evoluzione in base ai cambiamenti tecnologici, normativi, sociali e geopolitici. All’interno del volume, sono riportati dei riquadri con sfondo violetto o grigio; i primi sono usati nel capitolo introduttivo e nelle conclusioni per mettere in evidenza alcuni concetti ritenuti importanti, i secondi sono usati negli altri capitoli per spiegare il significato di alcuni termini tecnici comunemente utilizzati dagli addetti ai lavori. In conclusione, ringraziamo tutti i colleghi che hanno contribuito a questo volume: un gruppo di oltre 120 ricercatori, provenienti da circa 40 tra Enti di Ricerca e Università, unico per numerosità ed eccellenza, che rappresenta il meglio della ricerca in Italia nel settore della cybersecurity. Un grazie speciale va a Gabriella Caramagno e ad Angela Miola che hanno contribuito a tutte le fasi di produzione del libro. Tra i ringraziamenti ci fa piacere aggiungere il supporto ottenuto dai partecipanti al progetto FILIERASICURA

Analytical solutions of the Dirac Quantum Cellular Automata

The thesis focuses on the analytical and the numerical study of the Weyl and Dirac Quantum Cellular Automata (QCAs). A QCA describes the unitary local evolution of a denumerable collection of quantum systems in mutual interaction. The Weyl QCA can be derived under the assumptions of linearity, unitarity, locality, homogeneity and isotropy and one can show that it recovers the relativistic evolution of the Weyl equation in the limit of small wave-vectors. Under these assumptions a QCA can be described in terms of a set of so-called transition matrices. In the specific case of the Weyl automaton the transition matrices generate a semigroup allowing for the explicit computation of the propagator in position space as a sum over paths. The computation of the propagator is simplified by encoding the lattice paths as binary strings, so that the problems reduces to a combinatorial problem for the binary strings. The thesis also addresses the Thirring QCA, featuring an on-site interaction. We provide a starting point for the study of the perturbation theory in this case

High-dimensional methods for quantum homodyne tomography

We provide optimized recursion relations for homodyne tomography. We improve previous methods by mitigating the divergences intrinsic in the calculation of the pattern functions used previously, and detail how to implement the data analysis through Monte Carlo simulations. Our refinements are necessary for the reconstruction of excited quantum states which populate a high-dimensional subspace of the electromagnetic field Hilbert space. (C) 2022 Elsevier B.V. All rights reserved