This thesis is split into two parts, which are united in the sense that they involve applying ideas from quantum information to fundamental physics. The first part is focused on examining discrete-time models in quantum computation (discrete-time quantum walks and quantum cellular automata) as discretized models of relativistic systems. One of the results here is a theorem demonstrating that a large class of discrete-time quantum walks have relativistic dynamics in the continuum limit. Additionally, the problem of fermion doubling for these models is investigated, and it is seen that the problem can be circumvented in two dimensional space. This was already known for one dimensional systems. Another result involves taking the limits of causal free field theories in discrete spacetime to recover continuum field theories, something that is not straightforward because of the nontrivial nature of the vacuum in quantum field theory. Additionally, it is shown that general systems of fermions evolving causally in discrete spacetime can be represented by quantum cellular automata, which makes them efficiently simulable by quantum computers. A related result is that quantum cellular automata composed of fermions are equivalent to regular quantum cellular automata. In the second part of this thesis, the focus is on the foundations of statistical physics. The main result of this part is a general bound on the time it takes a quantum system to effectively reach equilibrium. The discussion also includes a practical definition of equilibration that takes our measurement capabilities into account. Finally, the nature of the equilibrium state is also discussed, with a focus on initial state independence, which relates to the important question of when the equilibrium state is a Gibbs state.
This thesis is split into two parts, which are united in the sense that they involve applying ideas from quantum information to fundamental physics. The first part is focused on examining discrete-time models in quantum computation (discrete-time quantum walks and quantum cellular automata) as discretized models of relativistic systems. One of the results here is a theorem demonstrating that a large class of discrete-time quantum walks have relativistic dynamics in the continuum limit. Additionally, the problem of fermion doubling for these models is investigated, and it is seen that the problem can be circumvented in two dimensional space. This was already known for one dimensional systems. Another result involves taking the limits of causal free field theories in discrete spacetime to recover continuum field theories, something that is not straightforward because of the nontrivial nature of the vacuum in quantum field theory. Additionally, it is shown that general systems of fermions evolving causally in discrete spacetime can be represented by quantum cellular automata, which makes them efficiently simulable by quantum computers. A related result is that quantum cellular automata composed of fermions are equivalent to regular quantum cellular automata. In the second part of this thesis, the focus is on the foundations of statistical physics. The main result of this part is a general bound on the time it takes a quantum system to effectively reach equilibrium. The discussion also includes a practical definition of equilibration that takes our measurement capabilities into account. Finally, the nature of the equilibrium state is also discussed, with a focus on initial state independence, which relates to the important question of when the equilibrium state is a Gibbs state.
Discrete time quantum lattice systems recently have come into the focus of quantum computation because they provide a versatile tool for many different applications and they are potentially implementable in current experimental realizations. In this thesis we study the fundamental structures of such quantum lattice systems as well as consequences of experimental imperfections. Essentially, there are two models of discrete time quantum lattice systems, namely quantum cellular automata and quantum walks, which are quantum versions of their classical counterparts, i.e., cellular automata and random walks. In both cases, the dynamics acts locally on the lattice and is usually also translationally invariant. The main difference between these structures is that quantum cellular automata can describe the dynamics of many interacting particles, where quantum walks describe the evolution of a single particle. The first part of this thesis is devoted to quantum cellular automata. We characterize one-dimensional quantum cellular automata in terms of an index theory up to local deformations. Further, we characterize in detail a subclass of quantum cellular automata by requiring that Pauli operators are mapped to Pauli operators. This structure can be understood in terms of certain classical cellular automata. The second part of this thesis is concerned with quantum walks. We identify a quantum walk with the one-particle sector of a quantum cellular automaton. We also establish an index theory for quantum walks and we discuss decoherent quantum walks, i.e., the behavior of quantum walks with experimental imperfections.
Can we reduce Quantum Field Theory (QFT) to a quantum computation? Can physics be simulated by a quantum computer? Do we believe that a quantum field is ultimately made of a numerable set of quantum systems that are unitarily interacting? A positive answer to these questions corresponds to substituting QFT with a theory of quantum cellular automata (QCA), and the present work is examining this hypothesis. These investigations are part of a large research program on a quantum-digitalization of physics, with Quantum Theory as a special theory of information, and Physics as emergent from the same quantum-information processing. A QCA-based QFT has tremendous potential advantages compared to QFT, being quantum ab-initio and free from the problems plaguing QFT due to the continuum hypothesis. Here I will show how dynamics emerges from the quantum processing, how the QCA can reproduce the Dirac-field phenomenology at large scales, and the kind of departures from QFT that that should be expected at a Planckscale discreteness. I will introduce the notions of linear field quantum automaton and local-matrix quantum automaton, in terms of which I will provide the solution to the Feynman's problem about the possibility of simulating a Fermi field with a quantum computer.
Discretizing spacetime is often a natural step towards modelling physical systems. For quantum systems, if we also demand a strict bound on the speed of information propagation, we get quantum cellular automata (QCAs). These originally arose as an alternative paradigm for quantum computation, though more recently they have found application in understanding topological phases of matter and have} been proposed as models of periodically driven (Floquet) quantum systems, where QCA methods were used to classify their phases. QCAs have also been used as a natural discretization of quantum field theory, and some interesting examples of QCAs have been introduced that become interacting quantum field theories in the continuum limit. This review discusses all of these applications, as well as some other interesting results on the structure of quantum cellular automata, including the tensor-network unitary approach, the index theory and higher dimensional classifications of QCAs.
Can we reduce Quantum Field Theory (QFT) to a quantum computation? Can physics be simulated by a quantum computer? Do we believe that a quantum field is ultimately made of a numerable set of quantum systems that are unitarily interacting? A positive answer to these questions corresponds to substituting QFT with a theory of quantum cellular automata (QCA), and the present work is examining this hypothesis. These investigations are part of a large research program on a "quantum-digitalization" of physics, with Quantum Theory as a special theory of information, and Physics as emergent from the same quantum-information processing. A QCA-based QFT has tremendous potential advantages compared to QFT, being quantum "ab-initio" and free from the problems plaguing QFT due to the continuum hypothesis. Here I will show how dynamics emerges from the quantum processing, how the QCA can reproduce the Dirac-field phenomenology at large scales, and the kind of departures from QFT that that should be expected at a Planck-scale discreteness. I will introduce the notions of linear field quantum automaton and local-matrix quantum automaton, in terms of which I will provide the solution to the Feynman's problem about the possibility of simulating a Fermi field with a quantum computer.
Quantum cellular automata consist in arrays of identical finite-dimensional quantum systems, evolving in discrete-time steps by iterating a unitary operator G. Moreover the global evolution G is required to be causal (it propagates information at a bounded speed) and translation-invariant (it acts everywhere the same). Quantum cellular automata provide a model/architecture for distributed quantum computation. More generally, they encompass most of discrete-space discrete-time quantum theory. We give an overview of their theory, with particular focus on structure results; computability and universality results; and quantum simulation results.
In the thesis, models of quantum systems characterized by discrete time dynamics are presented and applied to different problems. More specifically, aspects of excitation transport, quantum state transfer and thermalization are investigated mainly within the framework of quantum cellular automata (QCA), which are discrete time evolution quantum systems defined on a regular lattice of, in this case, qubits. The peculiarities of QCA, together with the restriction to the first excitation sector of the Hilbert space, make it possible to define a transport dynamics on a one dimensional lattice that encompasses all classical Markov chains, as well as maps where quantum coherence between sites can build up over time. This allows for the possibility to make a direct comparison between a fully classical dynamics and a dynamics affected by quantum effects, which is of relevance both in the field of quantum biology, especially regarding the transport of electronic excitations in photosynthetic systems, and in quantum computing when dealing with the transport of an unknown quantum state. In a quantum thermodynamics context, instead, QCA can serve the purpose to validate arguments of tipicality that have been brought forward in recent years in order to demonstrate the thermalization of closed quantum systems (i.e. the tendency of any small subsystems to evolve towards the maximally mixed state). Numerical results concerning this issue are presented. Finally, a modelization of scattering-like processes, where the scattering consists of random unitary interactions between an environment and part of a quantum bipartite system is discussed. Relevant properties of the resulting state such as its mean purity and the correlations between the interacting and the “sheltered” part of the system can be analitically evaluated, and their fluctuations bounded.