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Journal of Physics A Mathematical and General

Quantum random walks on diamond-type crystals and carbon nanotubes

Nicolae Cotfas

Published Jun 27, 2001• 2 citations
Publisher

Abstract

A class of graphs including models for diamond-type crystals and carbon nanotubes is defined, and some results concerning random walks and symmetry transformations are presented. These graphs may be useful discrete approximants in a path integral approach to physical phenomena occurring in diamond-type crystals and carbon nanotubes. Some details concerning a possible application of Gudder's discrete quantum mechanics to these crystals are analysed.

DOI
10.1088/0305-4470/34/27/301

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A class of graphs including models for diamond-type crystals and carbon nanotubes is defined, and some results concerning random walks and symmetry transformations are presented. These graphs may be useful discrete approximants in a path integral approach to physical phenomena occurring in diamond-type crystals and carbon nanotubes. Some details concerning a possible application of Gudder's discrete quantum mechanics to these crystals are analysed.
Discrete quantum walks are quantum analogues of classical random walks. They are an important tool in quantum computing and a number of algorithms can be viewed as discrete quantum walks, in particular Grover's search algorithm. These walks are constructed on an underlying graph, and so there is a relation between properties of walks and properties of the graph. This book studies the mathematical problems that arise from this connection, and the different classes of walks that arise. Written at a level suitable for graduate students in mathematics, the only prerequisites are linear algebra and basic graph theory; no prior knowledge of physics is required. The text serves as an introduction to this important and rapidly developing area for mathematicians and as a detailed reference for computer scientists and physicists working on quantum information theory.
We present numerical studies of quantum walks on C60{\text{C}}_{60}C60​ and related graphene structures to investigate their transport properties. Also known as a honeycomb lattice, the lattice formed by carbon atoms in the graphene phase can be rolled up to form nanotubes of various dimensions. Graphene nanotubes have many important applications, some of which rely on their unusual electrical conductivity and related properties. Quantum walks on graphs provide an abstract setting in which to study such transport properties independent of the other chemical and physical properties of a physical substance. They can thus be used to further the understanding of mechanisms behind such properties. We find that nanotube structures are significantly more efficient in transporting a quantum walk than cycles of equivalent size, provided the symmetry of the structure is respected in how they are used. We find faster transport on zigzag nanotubes compared to armchair nanotubes, which is unexpected given that for the actual materials the armchair nanotube is metallic, while the zigzag is semiconducting.
Quantum walks are the quantum mechanical analogue of classical random walks. Discrete-time quantum walks have been introduced and studied mostly on the line Z or higher dimensional space Z d but rarely defined on graphs with fractal dimensions because the coin operator depends on the position and the Fourier transform on the fractals is not defined. Inspired by its nature of classical walks, different quantum walks will be defined by choosing different shift and coin operators. When the coin operator is uniform, the results of classical walks will be obtained upon measurement at each step. Moreover, with measurement at each step, our results reveal more information about the classical random walks. In this dissertation, two graphs with fractal dimensions will be considered. The first one is Sierpinski gasket, a degree-4 regular graph with Hausdorff di- mension of df = ln 3/ ln 2. The second is the Cantor graph derived like Cantor set, with Hausdorff dimension of df = ln 2/ ln 3. The definitions and amplitude functions of the quantum walks will be introduced. The main part of this dissertation is to derive a recursive formula to compute the amplitude Green function. The exiting probability will be computed and compared with the classical results. When the generation of graphs goes to infinity, the recursion of the walks will be investigated and the convergence rates will be obtained and compared with the classical counterparts.
We propose a class of continuous-time quantum walk models on graphs induced by a certain class of discrete-time quantum walk models with a parameter [Formula: see text]. Here the graph treated in this paper can be applied to both finite and infinite cases. The induced continuous-time quantum walk is an extended version of the (free) discrete-Schrödinger equation driven by the normalized Laplacian: the element of the weighted Hermitian takes not only a scalar value but also a matrix value depending on the underlying discrete-time quantum walk. We show that each discrete-time quantum walk with an appropriate setting of the parameter [Formula: see text] in the long time limit identifies with its induced continuous-time quantum walk and gives the running time for the discrete-time to approximate the induced continuous-time quantum walk with a small error [Formula: see text]. We also investigate the detailed spectral information on the induced continuous-time quantum walk.
In this paper, we focus on discrete-time random walks (DTRWs) on the path graph with reflecting walls. Also we consider the corresponding continuous-time random walks (CTRWs), discrete-time quantum walks (DTQWs) and continuous-time quantum walks (CTQWs). We give formulas for the return probabilities of these four walks (DTRWs, CTRWs, DTQWs, CTQWs) at the boundary point of the path graph and discuss their relationships. Furthermore, by considering the Ehrenfest model, we derive the scaling limit theorems for the return probabilities of both the CTRW and the CTQW.
Motivated by the immense success of random walk and Markov chain methods in the design of classical algorithms, we consider {\em quantum\/} walks on graphs. We analyse in detail the behaviour of unbiased quantum walk on the line, with the example of a typical walk, the ``Hadamard walk''''. In particular, we show that after~ttt time steps, the probability distribution on the line induced by the Hadamard walk is almost uniformly distributed over the interval~[−t/2,  t/2][-t/\sqrt{2},\;t/\sqrt{2}][−t/2​,t/2​]. This implies that the same walk defined on the circle mixes in {\em linear\/} time. This is in direct contrast with the quadratic mixing time for the corresponding classical walk. We conclude by indicating how our techniques may be applied to more general graphs.