Abstract
Constructing a discrete model like a cellular automaton is a powerful method for understanding various dynamical systems. However, the relationship between the discrete model and its continuous analogue is, in general, nontrivial. As a quantummechanical cellular automaton, a discretetime quantum walk is defined to include various quantum dynamical behavior. Here we generalize a discretetime quantum walk on a line into the feedforward quantum coin model, which depends on the coin state of the previous step. We show that our proposed model has an anomalous slow diffusion characterized by the porousmedium equation, while the conventional discretetime quantum walk model shows ballistic transport.
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Introduction
Cellular automata – discrete models that follow a set of rules^{1} – have been analyzed in various dynamical systems in physics, as well as in computational models and theoretical biology; wellknown examples include crystal growth and the BelousovZhabotinsky reaction. To simulate quantum mechanical phenomena, Feynman^{2} proposed a quantum cellular automaton (the Feynman checkerboard). This model, defined in the general case by Meyer^{3}, is known as the discretetime quantum walk (DTQW). Since the DTQW on a graph is a model of a universal quantum computation^{4,5}, it is of great utility, especially in quantum information^{6,7,8,9}. Furthermore, the DTQW has been demonstrated experimentally in various physical systems^{10,11,12,13,14,15,16,17,18,19,20,21,22,23,24} to reveal quantum nature under dynamical systems.
As the cellular automaton can be mapped to various differential equations by taking the continuous limit, some DTQW models can be mapped to the Dirac equation^{25,26,27}, the spatially discretized Schrödinger equation^{28,29}, the KleinGordon equation^{27,30}, or various other differential equations^{31,32}. These equations have ballistic transport properties, which are reflected mathematically in the onedimensional (1D) DTQW with a time and spatialindependent coin operator, i.e. a 1D homogeneous DTQW^{33}. We consider here the 1D DTQW model. Physically, the standard deviation of the homogeneous DTQW is σ(t) ~ t, whereas the unbiased classical random walk has a standard deviation of .
In the homogeneous DTQW, the time evolution of a quantum particle (walker) is given by a unitary operator U defined on the composite Hilbert space , where is the walker Hilbert space and is the twodimensional coin Hilbert space. For a unitary operator U, the quantum state evolves in each time step t by
with
where the upper (lower ) component corresponds to the left (right) coin state at the jth site at time step t. As an example, the time evolution of the DTQW is given by
The jth site probability at time step t is given by and is satisfied for each time step t.
As a generalization of Eq. (3), we define a DTQW with a feedforward quantum coin described by
with the sitedependent rate function
which incorporates the nearestneighbor interactions. Since this quantum coin depends on the probability distribution of the coin states on the nearestneighbor sites at the previous step, this model is called a feedforward DTQW. It is remarked that the feedforward DTQW is one of the nonlinear DTQW models. Note that if we set the rate function to g = cos θ, which is time and site independent, then the model in Eq. (4) reduces to the homogeneous model in Eq. (3). We will show that our proposed feedforward DTQW is experimentally feasible. Furthermore, we will show that this model shows the anomalous diffusion as introduced below.
One of the famous anomalous diffusion equations is the porous medium equation (PME)^{34}, defined by
where the real parameter m > 1 characterizes the degree of porosity of the porous medium. It is known that the PME can be derived from three physical equations for the density ρ, pressure p and velocity v of the gas flow: the equation of continuity, ∂ρ/∂t + ∇ · (ρv) = 0; Darcy's law, v ∝ −∇p; and the equation of state for a polytropic gas, p ∝ ρ^{ν}, where ν is the polytropic exponent and m = ν + 1. One of the peculiar features of the PME is the socalled finite propagation, which implies the appearance of a free boundary separating the positive region (p > 0) from the empty region (p = 0).
A wellknown solution of the PME is the BarenblattPattle (BP) one^{35}; it is selfsimilar and its total mass is conserved during evolution. The evolutionary behavior of the BP solution was recently studied in the context of generalized entropies and information geometry^{36}. The BP solution can also be expressed by Tsallis' onerealparameter (q) generalization of a Gaussian function, i.e., the qGaussian^{37}. In the case of 1D space, the BP solution is
with q = 2 − m. Here, is a positive parameter that characterizes the width of the qGaussian at time t and is similar to the variance in a standard Gaussian. In other words, the parameter σ_{q}(t) characterizes the spread of the qGaussian distribution^{38,39};
which reduces to in the limit of q → 1. Note that in the same limit, the qGaussian reduces to the standard Gaussian, and the PME reduces to the standard heat equation ∂p/∂t = ∂^{2}p/∂x^{2}.
In this paper, we analyze a specific feedforward DTQW with an experimental proposal using the polarized state and optical mode. We show numerically that the probability distributions of the feedforward DTQW model have anomalous diffusion characterized by σ_{q}_{= 0.5}(t) ~ t^{0.4}. These dynamics are consistent with the time evolution of the selfsimilar solution^{35} of the PME, which is known to describe well the anomalous diffusion of an isotropic gas through a porous medium. Furthermore, we show analytically that the interference terms in our model help the speedup of the associated Markovian model but does not help the quadratic speedup like the homogeneous DTQW does^{40}. Note that although anomalous diffusion was found numerically in a nonlinear model^{41}, an aperiodic timedependent coin model^{42} and the historydependent coin^{43} from the time dependence of the variance σ_{q}_{= 1}(t), the partial differential equation (PDE) corresponding to their models have not derived due to the lack of the numerical step (about 100 step). Therefore, we have not yet revealed the origin of the anomalous diffusion in the DTQW.
Results
Experimental proposal of feedforward DTQW
We propose an optical implementation of the feedforward DTQW. In the simple optical implementation of the homogeneous DTQW, the walker space uses the spatial mode and the coin space does the polarized state. The shift uses the polarized beam splitter and the quantum coin uses the quarterwave, halfwave and quarterwave plates, which can arbitrarily rotate the polarized state in the Poincaré sphere. This was experimentally done in Refs. 10,11,12,16,17,18,19,20,21,22.
Let us construct the feedforward system of the quantum coin. The detectors put at each path to evaluate the probability distribution of the coin state and . Since our proposed quantum coin depends on and , we can calculate the coin operator at the jth site. According to the Jones calculation^{44} to satisfy Eq. (4), we control the angels of the quarterwave, halfwave and quarterwave plates for each path. This can be taken as the quantum coin operator with the feedforward. This is depicted in Fig. 1. In what follows, we consider the long time time evolution of the feedforward DTQW.
Numerical results of feedforward DTQW with anomalous diffusion
To study the time evolution of the feedforward DTQW model, the initial state should have nonzero coin states at the nearestneighbor sites. This can be easily understood by considering the following example. Let us take (, ) as the only nonzero initial state. In this case, the rate is , because there is no neighboring state. From the map in Eq. (4), we see that the nonzero states at t = 1 are and . This gives and we see that the only nonzero state is at t = 2. This state at t = 2 only differs in sign (or phase) from the initial state. Thus if the initial state is concentrated at a single site, no spreading occurs; the state only oscillates around the initial site.
Figure 2 (A) shows a typical probability distribution of the feedforward DTQW after a longtime evolution. See the Supplementary Movie for more details. The initial state was set as . We note that the probability distribution diffuses very slowly and does not approach a Gaussian. These features are often observed in anomalous diffusion. It is also remarked that such behavior has not yet seen in DTQWs with the positiondependent coin^{45,46,47,48}, which show the localization property.
We performed longtime numerical simulations of the feedforward DTQW model [Eq. (4)] for up to t ~ 10^{8} steps. To study the asymptotic behavior, we take running averages of the numerical solutions to reduce the influence of multiple spikes. The averaged data were fitted with the qGaussian of Eq. (7) to determine the corresponding qgeneralized standard deviation σ_{q}(t), as shown in Fig. 2 (B). We note that the averaged data at each time step are well fitted by the qGaussian with q = 0.5.
The longtime evolution of σ_{q}(t), plotted in Fig. 2 (C), reveals that the time evolution of the feedforward DTQW model is well characterized by σ_{q}_{= 0.5}(t) ~ t^{0.4}, which is the same time dependency for q = 0.5 of the PME [Eq. (8)].
Analytical derivation of anomalous diffusion in the associated Markov model of feedforward DTQW
The relationship between our model and the PME can be explored using the decomposition method of Romanelli et al.^{40,49}, in which the unitary evolution of a DTQW model is decomposed into Markovian and interference terms. We obtain the following map for both coin distributions and :
where the two terms including are interference terms and is the real part of a complex number z.
Neglecting the interference terms and introducing the abbreviations and , we get the associated Markovian model;
The numerical simulation of the associated Markovian model is performed under initial conditions of and the typical probability distribution shown in Fig. 3 (A) is well fitted by the qGaussian with q = 0.0. Furthermore, Fig. 3 (B) shows that the time evolution of σ_{q}(t) of the associated Markovian model is well fitted to σ_{q}_{= 0.0}(t) ~ t^{0.33}, which again is the same time dependency as the PME for q = 0.
It is known that the classical Markovian model, i.e. one without the interference terms of the homogeneous DTQW, satisfies the standard heat equation in the continuous limit. Consequently, the associated asymptotic probability distribution is a standard Gaussian. This implies that the ballistic transport property of the homogeneous DTQW comes from the interference term^{40}. We thus consider the continuous limit^{50} of the associated Markovian model.
We introduce the density ρ(x, t) and current j(x, t) as
where Δx is the difference of the nearestneighbor sites. Taking a Taylor expansion of Eq. (10), we get
in the diffusion limit, i.e., the quantity (Δx)^{2}/Δt remains constant (set to unity here for simplicity) as Δt, Δx → 0 with the onestep time difference Δt. In a similar manner, by expanding Eq. (11) and taking the diffusion limit, we obtain
which implies a breakdown in Fick's first law (j ∝ −∂ρ/∂x) and is the hallmark of anomalous diffusion. By substituting Eq. (14) into Eq. (13), we obtain the following nonlinear PDE:
Evaluating the asymptotic solution of this nonlinear PDE, after a longtime evolution, ρ(x, t) becomes much less than unity. As the rough approximation in this longtime limit, we have 1 − ρ ≈ 1 and ρ^{2} ≈ 0 and Eq. (15) is thus well approximated by
which is nothing but the PME in Eq. (6) with m = 2 (q = 0). We thus conclude that the approximated asymptotic solution of Eq. (15) is a qGaussian with q = 0. In addition, we can show that this result is mathematically valid by applying the asymptotic Lie symmetry method^{51} (see Method). This method can give an equivalence between the asymptotic solution of the PDE and the analyticallysolved one of the other PDE without analytically solving this PDE. Therefore, the associated Markovian model exhibits anomalous diffusion described by the PME in Eq. (6) with m = 2. This implies that the interference term of our model leads to the speedup of the quantum walker σ_{q}_{= 0.5} ~ t^{0.4} compared to the associated Markovian model σ_{q}_{= 0} ~ t^{1/3} and makes the zigzag shape around the qGaussian distribution.
In summary, we have proposed a feedforward DTQW model Eq. (4) in which the coin operator depends on the coin states of the nearestneighbor sites. We show that this model is experimentally feasible. Our feedforward DTQW model asymptotically satisfies the PME for m = 1.5 (q = 0.5) and exhibits anomalous slow diffusion σ_{q}_{= 0.5}(t) ~ t^{0.4} from the probability distribution and the time dependency of the standard deviation defined in the qGaussian distribution.
Discussion
In this section, we show that our results after the longtime numerical simulations have no initial coin dependence and that the interference term can be taken as the noise source in addition to the PME. First, while the above analysis uses the only fixed initial coin states as , we numerically confirm that there is almost no dependence of the initial coin state except for the trivial cases as follows. We have performed the several numerical simulations for the initial state specified by and with the realparameter β and γ ranging from 0 to 1. Note that the trivial cases, β = 0.5, γ = 0 and β = 0, γ = 0.5, lead to the localization of the probability distribution for any time and we cannot define the parameter q for the trivial initial states. Figure 4 shows the numerical evaluation of the parameter q of qGaussian distribution from the data at the two different time steps t = 10^{6} and t = 10^{7}, under the assumption to satisfy the stationary solution of the PME [Eqs. (7) and (8)]. The evaluated qparameters for the various initial states are except for the trivial cases. Therefore, we can conclude that our nonlinear model shows the anomalous slow diffusion to satisfy the PME with () without the initial state dependence.
Finally, let us consider the difference between the probability distribution of our model and the qGaussian distribution with q = 0.5, as shown in Fig. 2 (B); the power spectrum of this difference exhibits a white noise as shown in Fig. 5. This power spectrum divided by the physical time scale t^{0.4} may remain finite in the asymptotic case, which suggests that our nonlinear model may be mapped to the stochastic PME, i.e. the PME plus a white noise term, in the continuous limit. This stochasticity must come from the interference term. The problem of extracting the stochasticity from a deterministic process has been discussed in another context, that of Mori's noise^{52}. Further analysis of this model may reveal the origin of the stochasticity. This is interesting as a purely mathematical problem of a stochastic nonlinear partial differential equation and for showing the relationship between the discrete model and its continuous limit.
Methods
In what follows, the solution of Eq. (15) is asymptotically identical to the solution of Eq. (16). This is mathematically equivalent to showing that the probability distribution
is invariant under an asymptotic Lie symmetry^{51} of the nonlinear partial differential equation (15). In other words,
In Eq. (17), Z(σ_{q}_{= 0}) = 4σ_{q}_{= 0}/3 is the normalization factor and in what follows, the argument of this function is omitted where possible and ∂_{t}ρ is denoted as ρ_{t} for simplicity.
We follow the asymptotic Lie symmetry method and notations in Ref. 51. Under an infinitesimal transformation with the generator
that is
the function ρ(x, t) is mapped to a new function , with
By applying this to the probability distribution Eq. (17), we see that the transformation X with ξ = −x leaves Eq. (17) invariant if and only if
Note that τ = η · t remains unrestricted at this stage because ρ^{(q = 0)}(x) does not explicitly depend on time t. Conversely, the function ρ(x) is invariant under for any τ if and only if ρ(x) is of the form given in Eq. (17).
Following the general procedure for a Lie group analysis of differential equations^{53}, the second prolongation of X is described by
The coefficients Ψ_{t}, Ψ_{x} and Ψ_{xx} are defined as follows. Under an infinitesimal transformation of X, the partial derivatives are transformed as , and . We then readily obtain
The coefficients Ψ^{t}, Ψ^{x} and Ψ^{xx} are then obtained by applying the prolongation formula (2.39) from Ref. 52:
We note that Eq. (18) can be written as
with
The asymptotic Lie symmetry condition
with
can be written in the following compact form:
When the condition in Eq. (30) is fulfilled, each A_{k}(k = 0, 1, 2, 3) function must vanish separately in the asymptotic limit
implying that the variance σ_{q}_{= 0} also becomes infinity in the asymptotic limit from Eq. (17);
The function A_{3} can be expressed as
which must be nonzero as σ_{q}_{= 0} → ∞, unless we choose
Making this choice, X becomes
and A_{3} reduces to
Thus, A_{3} → 0 as σ_{q}_{= 0} → ∞.
In a similar manner, A_{0}, A_{1} and A_{2} are given by
and all become zero as σ_{q}_{= 0} → ∞. Therefore, we conclude that the distribution in Eq. (17) is an invariant solution for the transformation X of Eq. (37), which is an asymptotic symmetry for large x of the nonlinear partial differential equation Eq. (18).
References
von Neumann, J. The general and logical theory of automata, in Cerebral Mechanisms in Behavior: The Hixon Symposium Jeffress, L. A. (Ed.), pp. 1–41 (John Wiley and Sons, New York, NY, 1951).
Feynman, R. P. Spacetime approach to nonrelativistic quantum mechanics. Rev. Mod. Phys. 20, 367–387 (1948).
Meyer, D. From quantum cellular automata to quantum lattice gases. J. Stat. Phys. 85, 551–574 (1996).
Lovett, N. B. et al. Universal quantum computation using the discretetime quantum walk. Phys. Rev. A 81, 042330 (2010).
Childs, A., Gosset, D. & Webb, Z. Universal Computation by Multiparticle Quantum Walk. Science 339, 791–794 (2013).
Kempe, J. Quantum random walks  an introductory overview. Contemp. Phys. 44, 307–327 (2003).
VenegasAndraca, S. E. Quantum walks: a comprehensive review. Quant. Inf. Proc. 11, 1015–1106 (2012).
Kitagawa, T. Topological phenomena in quantum walks: elementary introduction to the physics of topological phases. Quant. Inf. Proc. 11, 1107–1148 (2012).
Shikano, Y. From Discrete Time Quantum Walk to Continuous Time Quantum Walk in Limit Distribution. J. Comput. Theor. Nanosci. 10, 1558–1570 (2013).
Do, B. et al. Experimental realization of a quantum quincunx by use of linear optical elements. J. Opt. Soc. Am. B 22, 499–504 (2005).
Zhang, P. et al. Demonstration of onedimensional quantum random walks using orbital angular momentum of photons. Phys. Rev. A 75, 052310 (2007).
Perets, H. B. et al. Realization of Quantum Walks with Negligible Decoherence in Waveguide Lattices. Phys. Rev. Lett. 100, 170506 (2008).
Karski, M. et al. Quantum Walk in Position Space with Single Optically Trapped Atoms. Science 325, 174–177 (2009).
Peruzzo, A. et al. Quantum walks of correlated particles. Science 329, 1500–1503 (2010).
Zähringer, F. et al. Realization of a Quantum Walk with One and Two Trapped Ions. Phys. Rev. Lett. 104, 100503 (2010).
Schreiber, A. et al. Photons Walking the Line: A Quantum Walk with Adjustable Coin Operations. Phys. Rev. Lett. 104, 050502 (2010).
Kitagawa, T. et al. Observation of topologically protected bound states in photonic quantum walks. Nat. Comm. 3, 882 (2012).
Schreiber, A. et al. A 2D Quantum Walk Simulation of TwoParticle Dynamics. Science 336, 55–58 (2012).
Sansoni, L. et al. TwoParticle BosonicFermionic Quantum Walk via Integrated Photonics. Phys. Rev. Lett. 108, 010502 (2012).
Crespi, A. et al. Anderson localization of entangled photons in an integrated quantum walk. Nat. Photon. 7, 322–328 (2013).
Jeong, Y.C. et al. Experimental realization of a delayedchoice quantum walk. Nat. Comm. 4, 2471 (2013).
Xue, P. et al. Observation of quasiperiodic dynamics in a onedimensional quantum walk of single photons in space. eprint: arXiv:1312.0123 (2013).
Fukuhara, T. et al. Microscopic observation of magnon bound states and their dynamics. Nature 502, 76–79 (2013).
Manouchehri, K. & Wang, J. Physical Implementation of Quantum Walks (Springer, Berlin, 2014).
Strauch, F. W. Relativistic effects and rigorous limits for discrete and continuoustime quantum walks. J. Math. Phys. 48, 082102 (2007).
Sato, F. & Katori, M. Dirac equation with an ultraviolet cutoff and a quantum walk. Phys. Rev. A 81, 012314 (2010).
Chandrashekar, C. M., Banerjee, S. & Srikanth, R. Relationship between quantum walks and relativistic quantum mechanics. Phys. Rev. A 81, 062340 (2010).
Chisaki, K., Konno, N., Segawa, E. & Shikano, Y. Crossovers induced by discretetime quantum walks. Quant. Inf. Comp. 11, 741–760 (2011).
Childs, A. M. On the Relationship Between Continuous and DiscreteTime Quantum Walk. Comm. Math. Phys. 294, 581–603 (2010).
di Molfetta, G. & Debbasch, F. Discrete time Quantum Walks: continuous limit and symmetries. J. Math. Phys. 53, 123302 (2012).
Knight, P., Roldán, E. & Sipe, J. E. Propagating Quantum Walks: the origin of interference structures. J. Mod. Opt. 51, 1761–1777 (2004).
de Valcarcél, G. J., Roldán, E. & Romanelli, A. Tailoring discrete quantum walk dynamics via extended initial conditions. New J. Phys. 12, 123022 (2010).
Konno, N. Quantum Random Walks in One Dimension. Quant. Inf. Proc. 1, 345–354 (2002); A new type of limit theorems for the onedimensional quantum random walk. J. Math. Soc. Jpn. 57, 935–1234 (2005).
Vazquez, J. L. The Porous Medium Equation, Mathematical Theory (Oxford University Press, Oxford, 2006).
Barenblatt, G. I. Scaling, SelfSimilarity and Intermediate Asymptotics (Cambridge University Press, Cambridge, 1996).
Ohara, A. & Wada, T. Information geometry of qGaussian densities and behaviors of solutions to related diffusion equations. J. Phys. A 43, 035002 (2010).
Tsallis, C. Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World (Springer, New York, NY, 2009).
Anteneodo, C. Nonextensive random walks. Physica A 358, 289–298 (2005).
Schwämmle, V., Nobre, F. D. & Tsallis, C. qGaussians in the porousmedium equation: stability and time evolution. Eur. J. Phys. B 66, 537–546 (2008).
Romanelli, A. Distribution of chirality in the quantum walk: Markovian process and entanglement. Phys. Rev. A 81, 062349 (2010).
NavarreteBenlloch, C., Pérez, A. & Roldán, E. Nonlinear optical Galton board. Phys. Rev. A 75, 062333 (2007).
Ribeiro, P., Milman, P. & Mosseri, R. Aperiodic quantum random walks. Phys. Rev. Lett. 93, 190503 (2004).
Rohde, P. P., Brennen, G. K. & Gilchrist, A. G. Quantum walks with memory provided by recycled coins and a memory of the coinflip history. Phys. Rev. A 87, 052302 (2013).
Yariv, A. Optical Electronics in Modern Communications (Oxford University Press, Oxford, 1997).
Romanelli, A. The Fibonacci quantum walk and its classical trace map. Physica A 388, 3985–3990 (2009).
McGettrick, M. One Dimensional Quantum Walks with Memory. Quantum Inf. Comp. 10, 0509–0524 (2010).
Joye, A. & Merkli, M. Dynamical Localization of Quantum Walks in Random Environments. J. Stat. Phys. 140, 1–29 (2010).
Shikano, Y. & Katsura, H. Localization and fractality in inhomogeneous quantum walks with selfduality. Phys. Rev. E 82, 031122 (2010); Notes on Inhomogeneous Quantum Walks. AIP Conf. Proc. 1363, 151–154 (2011).
Romanelli, A. et al. Quantum random walk on the line as a Markovian process. Physica A 338, 395–405 (2004).
Godoy, S. & GarcíaColín, L. S. From the quantum random walk to classical mesoscopic diffusion in crystalline solids. Phys. Rev. E 53, 5779–5785 (1996).
Gaeta, G. Asymptotic symmetries in an optical lattice. Phys. Rev. A 72, 033419 (2005).
Mori, H. Transport, Collective Motion and Brownian Motion. Prog. Theor. Phys. 33, 423–455 (1965); A ContinuedFraction Representation of the TimeCorrelation Functions. Prog. Theor. Phys. 34, 399–416 (1965).
Olver, P. J. Applications of Lie Groups to Differential Equations (SpringerVerlag, New York, NY, 1986).
Acknowledgements
Y.S. thanks Masao Hirokawa for valuable discussions. This work was partially supported by the Joint Studies Program of the Institute for Molecular Science.
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Y.S. provided the theoretical model of the discretetime quantum walk; T.W. and J.H. provided the numerical analysis; and T.W. provided the analytical solution of the associated Markovian model with assistance from Y.S. Y.S. conducted this project. All authors contributed to writing the manuscript.
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Supplementary Information
Probability Distribution of Feedforward DTQW
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Shikano, Y., Wada, T. & Horikawa, J. Discretetime quantum walk with feedforward quantum coin. Sci Rep 4, 4427 (2014). https://doi.org/10.1038/srep04427
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DOI: https://doi.org/10.1038/srep04427
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