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Physica Scripta

Neighborhood-history quantum walk

Asif Shakeel

Published Jan 28, 2019• 1 citation
Publisher

Abstract

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.

DOI
10.1088/1402-4896/ab0254

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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.
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.
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.
Discrete-time quantum walks (QWs) represent robust and versatile platforms for the controlled engineering of single particle quantum dynamics, and have attracted special attention due to their algorithmic applications in quantum information science. Even in their simplest 1D architectures, they display complex topological phenomena, which can be employed in the systematic study of topological quantum phase transitions [1]. Due to the exponential scaling in the number of resources required, most experimental realizations of QWs to date have been limited to single particles, with only a few implementations involving correlated quantum pairs. In this paper we study applications of QWs in the controlled dynamical engineering of entanglement in bipartite bosonic systems. We show that QWs can be employed in the transition from mode entanglement, where indistinguishability of the quantum particles plays a key role, to the standard type of entanglement associated with distinguishable particles. We also show that by carefully tailoring the steps in the QWs, as well as the initial state for the quantum walker, it is possible to preserve the entanglement content by topological protection. The underlying mechanism that allows for the possibility of both entanglement engineering and entanglement protection is the strong ‘spin–orbit’ coupling induced by the QW. We anticipate that the results reported here can be employed for the controlled emulation of quantum correlations in topological phases.
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.