We model the transport of an unknown quantum state on one dimensional qubit lattices by means of a quantum cellular automata (QCA) evolution. We do this by first introducing a class of discrete noisy dynamics, in the first excitation sector, in which a wide group of classical stochastic dynamics is embedded within the more general formalism of quantum operations. We then extend the Hilbert space of the system to accommodate a global vacuum state, thus allowing for the transport of initial on-site coherences besides excitations, and determine the dynamical constraints that define the class of noisy QCA in this subspace. We then study the transport performance through numerical simulations, showing that for some instances of the dynamics perfect quantum state transfer is attainable. Our approach provides one with a natural description of both unitary and open quantum evolutions, where the homogeneity and locality of interactions allow one to take into account several forms of quantum noise in a plausible scenario.
We model the transport of an unknown quantum state on one dimensional qubit lattices by means of a quantum cellular automata (QCA) evolution. We do this by first introducing a class of discrete noisy dynamics, in the first excitation sector, in which a wide group of classical stochastic dynamics is embedded within the more general formalism of quantum operations. We then extend the Hilbert space of the system to accommodate a global vacuum state, thus allowing for the transport of initial on-site coherences besides excitations, and determine the dynamical constraints that define the class of noisy QCA in this subspace. We then study the transport performance through numerical simulations, showing that for some instances of the dynamics perfect quantum state transfer is attainable. Our approach provides one with a natural description of both unitary and open quantum evolutions, where the homogeneity and locality of interactions allow one to take into account several forms of quantum noise in a plausible scenario.
We model the transport of an unknown quantum state on one dimensional qubit lattices by means of a quantum cellular automata evolution. We do this by first introducing a class of discrete noisy dynamics, in the first excitation sector, in which a wide group of classical stochastic dynamics is embedded within the more general formalism of quantum operations. We then extend the Hilbert space of the system to accommodate a global vacuum state, thus allowing for the transport of initial on-site coherences besides excitations, and determine the dynamical constraints that define the class of noisy quantum cellular automata in this subspace. We then study the transport performance through numerical simulations, showing that for some instances of the dynamics perfect quantum state transfer is attainable. Our approach provides one with a natural description of both unitary and open quantum evolutions, where the homogeneity and locality of interactions allow one to take into account several forms of quantum noise in a plausible scenario.
We introduce a class of noisy quantum cellular automata on a qubit lattice that includes all classical Markov chains, as well as maps where quantum coherence between sites is allowed to build up over time. We apply such a construction to the problem of excitation transfer through 1D lattices, and compare the performance of classical and quantum dynamics with equal local transition probabilities. Our discrete approach has the merits of stripping down the complications of the open system dynamics, of clearly isolating coherent effects, and of allowing for an exact treatment of conditional dynamics, all while capturing a rich variety of dynamical behaviors.
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.
The quantum walk is a subset of classical random walks.Considerable research has lately been conducted on quantum walks in relation to quantum computing.The walk may be thought of as a type of quantum cellular automata.In a broader sense, there is a one-to-one link between them.Furthermore, we look at another type of quantum cellular automata that may be thought of as a quantum version of stochastic cellular automata.The quantum walk formalism is a widely used and quite practical paradigm for modelling quantum systems such as Dirac equation simulations, discrete dynamics in both the low and high energy regimes, and the development of a multitude of quantum algorithms.The walker in a QW moves in a quantum superposition of pathways, and the ensuing interference serves as the foundation for a broad range of quantum algorithms, including quantum search, graph isomorphism issues, and rating nodes in a network.To generate a block representation of n-dimensional quantum cellular automata, we apply the following general result to them.As a result, we have demonstrated that their generic, axiomatic formulation nevertheless produces a unifying, operational representation of them.Some non-trivial ramifications are investigated, such as the fact that bijective non-reversible CA are not physical as closed systems, or that quantum information can move faster than classical informationwithin some specified dynamics.Finally, we present n-dimensional quantum cellular automata that can simulate anything else.This means that the initial configuration and local transition rule of any one-dimensional QCA may be stored within the universal QCA's initial configuration.The simulated QCA will then correspond to many phases of the universal QCA.In the sense that each cell in the simulated QCA is represented as a group of nearby cells in the universal QCA, the simulation preserves the topology.Because the encoding is linear, it bears no cost of the calculation.
Quantum cellular automata are important tools in understanding quantum dynamics, thanks to their simple and effective list of rules. Here we consider a class of noisy, one-dimensional quantum cellular automata that allow one to shift from unitary dynamics to completely positive maps, investigating the evolution of coherence as well as the notion of reversibility in such a setting. To this aim, we associate an approximate reverse automaton to each noisy automaton, and assess its effect, and we define an irreversibility time based on the distance from the maximally mixed state, which is shown to be the only attractor of the automaton map in the presence of dephasing. Our analysis illustrates the interplay between unitary and noisy dynamics.
A quantum cellular automaton (QCA) is a unitary discrete-time evolution of a quantum many-body system, which can be described as a dynamics system , where A is any matrix and is a unitary operator representing the global evolution of QCA. And QCAs are characterized by locality (or causality). Among them, quantum circuits are the special classes of QCAs, which use unitary operators to control and transform the initial quantum state of qubit into a final quantum state, quantum circuits are special and widely studied in quantum information. A group is a set of elements combined with an operation, which satisfies associativity, identity and invertibility. This article shows that group theory provides the right language for studying the relation between QCAs and quantum circuits among them.