Quantum Walks in Low Dimension

Naoto Namekata, Ryo Kobayashi, Takuya Naka +3 more

Quantum walks (QWs) simulate particle behaviors in the quantum mechanical regime, which exhibit time evolutions completely different from the classical (probabilistic) random works [1]. QWs offer quantum physical simulations and the novel approach to build quantum computers. Currently on-chip photonic devices implementing QWs have been intensively developed. QW photonic chips were realized using photonic circuits based on silica (SiO2) waveguide platforms [1,2]. Here, we demonstrate a QW chip based on the other waveguide platform, namely the long-range surface plasmon polariton (LR-SPP) waveguide [3]. The LR-SPP waveguide allows a single-polarization mode, and has relatively low optical losses. Moreover, the LR-SPP waveguide platform will realize two-dimensional waveguide lattice structures with a high reliability. Thus, the LR-SPP waveguide is the potential platform to construct large-scale and high-dimension QW simulators.

Kyohei Watanabe

Andreu Anglés-Castillo, Armando Pérez

We investigate the properties of a quantum walk which can simulate the behavior of a spin 1/2 particle in a model with an ordinary spatial dimension, and one extra dimension with warped geometry between two branes. Such a setup constitutes a [Formula: see text] dimensional version of the Randall-Sundrum model, which plays an important role in high energy physics. In the continuum spacetime limit, the quantum walk reproduces the Dirac equation corresponding to the model, which allows to anticipate some of the properties that can be reproduced by the quantum walk. In particular, we observe that the probability distribution becomes, at large time steps, concentrated near the "low energy" brane, and can be approximated as the lowest eigenstate of the continuum Hamiltonian that is compatible with the symmetries of the model. In this way, we obtain a localization effect whose strength is controlled by a warp coefficient. In other words, here localization arises from the geometry of the model, at variance with the usual effect that is originated from random irregularities, as in Anderson localization. In summary, we establish an interesting correspondence between a high energy physics model and localization in quantum walks.

Norio Konno

Giuseppe Di Molfetta

This manuscript gathers and subsumes a long series of works on using QW to simulate transport phenomena. Quantum Walks (QWs) consist of single and isolated quantum systems, evolving in discrete or continuous time steps according to a causal, shift-invariant unitary evolution in discrete space. We start reminding some necessary fundamentals of linear algebra, including the definitions of Hilbert space, tensor state, the definition of linear operator and then we briefly present the principles of quantum mechanics on which this thesis is grounded. After having reviewed the literature of QWs and the main historical approaches to their study, we then move on to consider a new property of QWs, the plasticity. Plastic QWs are those ones admitting both continuous time-discrete space and continuous spacetime time limit. We show that such QWs can be used to quantum simulate a large class of physical phenomena described by transport equations. We investigate this new family of QWs in one and two spatial dimensions, showing that in two dimensions, the PDEs we can simulate are more general and include dispersive terms. We show that the above results do not need to rely on the grid and we prove that such QW-based quantum simulators can be defined on 2-complex simplicia, i.e. triangular lattices. Finally, we extend the above result to any arbitrary triangulation, proving that such QWs coincide in the continuous limit to a transport equation on a general curved surface, including the curved Dirac equation in 2+1 spacetime dimensions.

Hao Luo, Peng Xue