History dependent discrete time quantum walks (QWs) are often studied for their lattice traversal properties. A particular model in the literature uses the state of a memory qubit at each site to record visits and to control the dynamics of the walk. We generalize this model to the neighborhood-history quantum walk (NHQW), in which the walk dynamics and the state of the memory qubits in a neighborhood of the particle's position are interdependent. To demonstrate it, we construct an NHQW on a one-dimensional lattice, with a simple neighborhood. Several dynamically interesting history dependent QWs can be realized as single-particle sectors of quantum lattice gas automata (QLGA). In contrast, the NHQW constructed in this paper is realized as a single-particle sector of the more general quantum cellular automaton (QCA). The complexity of the NHQW dynamics presents a promising avenue toward richer walk strategies and a potentially useful model of QWs for the Noisy Intermediate-Scale Quantum (NISQ) era of quantum computing. It also modifies QWs to conceivably allow for modeling fundamental physics incorporating quantum field interactions with particles.
History dependent discrete time quantum walks (QWs) are often studied for their lattice traversal properties. A particular model in the literature uses the state of a memory qubit at each site to record visits and to control the dynamics of the walk. We generalize this model to the neighborhood-history quantum walk (NHQW), in which the walk dynamics and the state of the memory qubits in a neighborhood of the particle's position are interdependent. To demonstrate it, we construct an NHQW on a one-dimensional lattice, with a simple neighborhood. Several dynamically interesting history dependent QWs can be realized as single-particle sectors of quantum lattice gas automata (QLGA). In contrast, the NHQW constructed in this paper is realized as a single-particle sector of the more general quantum cellular automaton (QCA). The complexity of the NHQW dynamics presents a promising avenue toward richer walk strategies and a potentially useful model of QWs for the Noisy Intermediate-Scale Quantum (NISQ) era of quantum computing. It also modifies QWs to conceivably allow for modeling fundamental physics incorporating quantum field interactions with particles.
Abstract History dependent discrete time quantum walks (QWs) are often studied for their lattice traversal properties. A particular model in the literature uses the state of a memory qubit at each site to record visits and to control the dynamics of the walk. We generalize this model to the neighborhood-history quantum walk (NHQW), in which the walk dynamics and the state of the memory qubits in a neighborhood of the particle’s position are interdependent. To demonstrate it, we construct an NHQW on a one-dimensional lattice, with a simple neighborhood. Several dynamically interesting history dependent QWs can be realized as single-particle sectors of quantum lattice gas automata. In contrast, the NHQW constructed in this paper is realized as a single-particle sector of the more general quantum cellular automaton. The complexity of the NHQW dynamics presents a promising avenue toward richer walk strategies and a potentially useful model of QWs for the Noisy Intermediate-Scale Quantum era of quantum computing. It also modifies QWs to conceivably allow for modeling fundamental physics incorporating quantum field interactions with particles.
Quantum Random Walks (QRW) were first defined as one-particle sectors of Quantum Lattice Gas Automata (QLGA). Recently, they have been generalized to include history dependence, either on previous coin (internal, i.e., spin or velocity) states or on previous position states. These models have the goal of studying the transition to classicality, or more generally, changes in the performance of quantum walks in algorithmic applications. We show that several history dependent QRW can be identified as one-particle sectors of QLGA. This provides a unifying conceptual framework for these models in which the extra degrees of freedom required to store the history information arise naturally as geometrical degrees of freedom on the lattice.
Quantum Random Walks (QRW) were first defined as one-particle sectors of Quantum Lattice Gas Automata (QLGA). Recently, they have been generalized to include history dependence, either on previous coin (internal, i.e., spin or velocity) states or on previous position states. These models have the goal of studying the transition to classicality, or more generally, changes in the performance of quantum walks in algorithmic applications. We show that several history dependent QRW can be identified as one-particle sectors of QLGA. This provides a unifying conceptual framework for these models in which the extra degrees of freedom required to store the history information arise naturally as geometrical degrees of freedom on the lattice.
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.
A discrete-time quantum walk (QW) is essentially a unitary operator driving the evolution of a single particle on the lattice. Some QWs have familiar physics PDEs as their continuum limit. Some slight generalization of them (allowing for prior encoding and larger neighbourhoods) even have the curved spacetime Dirac equation, as their continuum limit. In the (1+1)− dimensional massless case, this equation decouples as scalar transport equations with tunable speeds. We characterise and construct all those QWs that lead to scalar transport with tunable speeds. The local coin operator dictates that speed; we provide concrete techniques to tune the speed of propagation, by making use only of a finite number of coin operators-differently from previous models, in which the speed of propagation depends upon a continuous parameter of the quantum coin. The interest of such a discretization is twofold : to allow for easier experimental implementations on the one hand, and to evaluate ways of quantizing the metric field, on the other.
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.