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QUANTUM RANDOM WALK ON FRACTALS

Kai Zhao

Published Jan 1, 2018• 0 citations
OpenAlex

Abstract

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.

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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 consider discrete time quantum random walks on a Sierpinski gasket. We study the hitting probability as the level of the fractal graph goes to infinity in terms of their localization exponents βw, total variation exponents δw, and relative entropy exponents ηw. We define and solve the amplitude Green functions recursively when the level of the fractal graph goes to infinity. We obtain exact recursive formulas for the amplitude Green functions, on which the hitting probabilities and expectation of the first-passage time are calculated, and using the recursive formula with the aid of Monte Carlo integration, we evaluate their numerical values. We also show that when the level of the fractal graph goes to infinity, with probability 1, the quantum random walks will return to the origin, i.e., the quantum walks on a Sierpinski gasket are recurrent.
We introduce a framework to analyze quantum algorithms with the renormalization group (RG). To this end, we present a detailed analysis of the real-space RG for discrete-time quantum walks on fractal networks and show how deep insights into the analytic structure as well as generic results about the long-time behavior can be extracted. The RG flow for such a walk on a dual Sierpinski gasket and a Migdal-Kadanoff hierarchical network is obtained explicitly from elementary algebraic manipulations, after transforming the unitary evolution equation into Laplace space. Unlike for classical random walks, we find that the long-time asymptotics for the quantum walk requires consideration of a diverging number of Laplace poles, which we demonstrate exactly for the closed-form solution available for the walk on a one-dimensional loop. In particular, we calculate the probability of the walk to overlap with its starting position, which oscillates with a period that scales as NdwQ/df{N}^{{d}_{w}^{Q}/{d}_{f}}NdwQ​/df​ with system size NNN. While the largest Jacobian eigenvalue ${\ensuremath{\lambda}}_{1}$ of the RG flow merely reproduces the fractal dimension, ${d}_{f}={log}_{2}{\ensuremath{\lambda}}_{1}$, the asymptotic analysis shows that the second Jacobian eigenvalue ${\ensuremath{\lambda}}_{2}$ becomes essential to determine the dimension of the quantum walk via ${d}_{w}^{Q}={log}_{2}\sqrt{{\ensuremath{\lambda}}_{1}{\ensuremath{\lambda}}_{2}}$. We trace this fact to delicate cancellations caused by unitarity. We obtain identical relations for other networks, although the details of the RG analysis may exhibit surprisingly distinct features. Thus, our conclusions---which trivially reproduce those for regular lattices with translational invariance with df=d{d}_{f}=ddf​=d and dwQ=1{d}_{w}^{Q}=1dwQ​=1---appear to be quite general and likely apply to networks beyond those studied here.
We investigate the transport and entanglement properties exhibited by quantum walks with coin operators concatenated in a space-time fractal structure. Inspired by recent developments in photonics, we choose the paradigmatic Sierpinski gasket. The 0-1 pattern of the fractal is mapped into an alternation of the generalized Hadamard-Fourier operators. This two-state coin-operator approach overcomes the intricacies caused by the utilization of high-dimensional coin operators required in prior studies of discrete-time quantum walks on fractals. In fulfilling the blank space on the analysis of the impact of inhomogeneity in quantum walk properties, specifically, fractal deterministic inhomogeneity, our results show a robust effect of entanglement enhancement as well as an interesting road to superdiffusive spreading with a tunable scaling exponent attaining robust superdiffusion, subballistic though. Explicitly, with this fractal approach it is possible to obtain an increase in quantum entanglement with reduced impact on standard quantum walk theoretical spreading. Alongside those features, we analyze further properties such as the degree of interference and visibility. The present model corresponds to an application of fractals in an experimentally viable setting, namely, the building block for the construction of photonic patterned structures. Published by the American Physical Society 2024
The spatial search problem on regular lattice structures in integer number of dimensions $d\ensuremath{\ge}2$ has been studied extensively, using both coined and coinless quantum walks. The relativistic Dirac operator has been a crucial ingredient in these studies. Here, we investigate the spatial search problem on fractals of noninteger dimensions. Although the Dirac operator cannot be defined on a fractal, we construct the quantum walk on a fractal using the flip-flop operator that incorporates a Klein-Gordon mode. We find that the scaling behavior of the spatial search is determined by the spectral (and not the fractal) dimension. Our numerical results have been obtained on the well-known Sierpinski gaskets in two and three dimensions.
Quantum walks, both discrete (coined) and continuous time, form the basis of several recent quantum algorithms. Here we use numerical simulations to study the properties of discrete, coined quantum walks. We investigate the variation in the entanglement between the coin and the position of the particle by calculating the entropy of the reduced density matrix of the coin. We consider both dynamical evolution and asymptotic limits for coins of dimensions from two to eight on regular graphs. For low coin dimensions, quantum walks which spread faster (as measured by the mean square deviation of their distribution from uniform) also exhibit faster convergence towards the asymptotic value of the entanglement between the coin and particle's position. For high-dimensional coins, the DFT coin operator is more efficient at spreading than the Grover coin. We study the entanglement of the coin on regular finite graphs such as cycles, and also show that on complete bipartite graphs, a quantum walk with a Grover coin is always periodic with period four. We generalize the 'glued trees' graph used by Childs et al (2003 Proc. STOC , pp 59–68) to higher branching rate (fan out) and verify that the scaling with branching rate and with tree depth is polynomial.
We analyze discrete-time quantum walks on Sierpinski gaskets using a flip-flop shift operator with the Grover coin. We obtain the scaling of two important physical quantities: The mean-square displacement, and the mixing time as function of the number of points. The Sierpinski gasket is a fractal that lacks translational invariance and the results differ from those described in the literature for ordinary lattices. We find that the diffusion scaling depends on the initial location. Averaged over all initial locations, our simulation obtain an exponent very similar to the classical diffusion.