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arXiv:2306.12628 · Physical Review A

Probing fractal spatiotemporal inhomogeneity in a quantum walk

Marcelo A. Pires, Caio B. Naves, Diogo O. Soares-Pinto, Sı́lvio M. Duarte Queirós

Published Feb 16, 2024• 4 citations
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Abstract

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

DOI
10.1103/physreva.109.022217

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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
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.
Properties of one-dimensional discrete-time quantum walks (DTQWs) are sensitive to the presence of inhomogeneities in the substrate, which can be generated by defining position-dependent coin operators. Deterministic aperiodic sequences of two or more symbols provide ideal environments where these properties can be explored in a controlled way. Based on an exhaustive numerical study, this work discusses a two-coin model resulting from the construction rules that lead to the usual fractal Cantor set. Although the fraction of the less frequent coin [Formula: see text] as the size of the chain is increased, it leaves peculiar properties in the walker dynamics. They are characterized by the wave function, from which results for the probability distribution and its variance, as well as the entanglement entropy, were obtained. A number of results for different choices of the two coins are presented. The entanglement entropy has shown to be very sensitive to uncovering subtle quantum effects present in the model.
Abstract In large-scale quantum networks, quantum repeaters provide an efficient method to distribute entangled states among selected nodes for realizing long-distance and complicated quantum communications. However, extending quantum repeater protocols to high-dimensional quantum states in existing experiments faces great challenges. Owing to the feasible physical implementations of quantum walks, we proposed various basic modules applicable to quantum repeaters for distributing high-dimensional entangled states via quantum walks, including d-dimensional Bell states and multi-particle d-dimensional GHZ states. Furthermore, based on the above schemes, we provided a high-dimensional entanglement distribution scheme for arbitrary quantum tree networks. By searching for a Steiner tree in a quantum network, we can achieve high-dimensional entanglement distributions over an arbitrary quantum network. We constructed a quantum fractal network based on d-dimensional GHZ states and analyzed the quantum transport properties of continuous quantum walks in the network. Compared with the continuous quantum walk on the Sierpinski gasket, the quantum walk on the new fractal network spreads more widely within the same time frame. Finally, we conducted five experiments to implement various basic modules of 2-party or 3-party entanglement distribution schemes in a superconducting quantum processor. Our study can serve as a building block for constructing large and complex quantum networks.
We investigate the evolution dynamics of inhomogeneous discrete-time one-dimensional quantum walks displaying long-range correlations in both space and time. The associated quantum coin operators of internal states are built to exhibit random inhomogeneity distribution of long-range correlations embedded in the time evolution protocol through a fractional Brownian motion with spectrum following a power-law behavior, S(k)∼1/k^{ν}. From extensive numerical simulations with averages over a large number of independent realizations of the phases of quantum coins, the power-law correlated disorder encoded in the coin phases is shown to give rise to a wide variety of spreading patterns of the qubit states, from localized to subdiffusive, diffusive, and superdiffusive (including ballistic) behavior, depending on the relative strength of the parameters driving the correlation degree. Dispersion control is possible in one-dimensional discrete-time quantum walks by tuning the long-range correlation properties assigned to the inhomogeneous quantum coin operator.
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.
We model quantum transport, described by continuous-time quantum walks (CTQWs), on deterministic Sierpinski fractals, differentiating between Sierpinski gaskets and Sierpinski carpets, along with their dual structures. The transport efficiencies are defined in terms of the exact and the average return probabilities, as well as by the mean survival probability when absorbing traps are present. In the case of gaskets, localization can be identified already for small networks (generations). For carpets, our numerical results indicate a trend towards localization, but only for relatively large structures. The comparison of gaskets and carpets further implies that, distinct from the corresponding classical continuous-time random walk, the spectral dimension does not fully determine the evolution of the CTQW.