Quantum Ultra-Walks: Walks on a Line with Hierarchical Spatial Heterogeneity

We discuss the model of a one-dimensional, discrete-time walk on a line with spatial disorder in the form of a variable set of ultrametric barriers. We develop a formalism by which the classical random walk as well as a quantum walk can be treated in parallel by using a"coined"walk with internal degrees of freedom. For the random walk, this amounts to a 2nd-order Markov process with a stochastic coin, better know as an (anti-)persistent walk. When this coin varies spatially in the hierarchcial manner of"ultradiffusion,"it reproduces the well-known results of that model. The exact analysis employed for obtaining the walk dimension $d_w$, based on the real-space renormalization group (RG), proceeds virtually identical for the corresponding quantum walk with a unitary coin. However, unitarity provides for a quite distinct result for the quantum walk dimension, which can be tuned to any value between ballistic spreading ($d_w=1$) and confinement ($d_w=\infty$). Yet, for any $d_w<\infty$ the quantum ultra-walk never appears to localize.

The above-mentioned studies of solvable QW on the line are all based on spatially homogeneous coins. Only few examples exist concerning heterogeneous but unitary environments, i.e., using coins that vary extensively with location quasi-periodically [33] or are drawn from some random ensemble [21], each yielding analytical insights only into localization properties in some limits. Here, we discuss a 1d -walk in which coins possess a strong spatial variation but with a hierarchical repetition of coins, which we shall call the quantum ultra-walk. It is inspired by classical models of diffusion over an ultrametric arrangement of barriers [34][35][36][37][38][39] that was meant to describe ultra-slow relaxation. Using a real-space renormalization group (RG), we can analytically determine the walk dimension d w [40], characterizing the asymptotic scaling variable x/t 1/dw (or pseudo-velocity [18]) for the walk, in closed form for a parameter that determines the relative strength of barriers. For example, d w describes the anomalous spread of the wavefunction with time in terms of the mean-square displacement, x 2 ∼ t 2/dw . The RG has been previously employed to obtain d w for quantum walks with a homogeneous coin in various fractal geometries [16,41] and to elucidate the complexity of Grover's search algorithm on the spectral dimension d s of the search-space [42]. We demonstrate our procedure first by rederiving the classical result in a novel manner by using a 2 nd -order Markov process [43,44]. It mimics the coined quantum walk in all but the final step of the analysis whilst using a stochastic instead of a unitary coin [45]. Despite of these parallels, quantum effects clearly assert themselves in the final analysis and, thus, in the behavior obtained for d w .
This paper is organized as follows: In Sec. II, we introduce the dynamic equation that describes the evolution of the discrete-time walk, classical or quantum, on a 1d -line with position-dependent coins. In Sec. III, we develop the RG for the case of a hierarchical dependence of such coins. In Sec. IV, we choose a hierarchy of stochastic coins to derive the familiar classical result, Eq. (19). In Sec. V, we then derive the solution for a corresponding hierarchy of unitary coins. We conclude with a discussion of our results in Sec. VI.

II. MASTER EQUATION FOR COINED WALKS
The time evolution of walks are governed by the discrete-time master equation [45] where U is a stochastic propagator for a random walk or a unitary propagator for a quantum walk. In the discrete site-basis |x , the probability density function is given by ρ (x, t) = ψ x,t for a random walk, while it is ρ (x, t) = |ψ x,t | 2 for quantum walks, where ψ x,t = x|Ψ (t) . For nearest-neighbor transitions on a line, it is where A x , B x , and M x specify the (possibly positiondependent) hopping operators for transitions to the left, right, or same site, on leaving from a site x. While a random walk would merely require local conservation of probability, A x + B x + M x = 1, the form of ρ (x, t) = |ψ x,t | 2 for quantum walks demands unitary propagation, I = U † U, which imposes on "coin-space" the conditions x B x+2 = 0, this algebra requires at least r = 2dimensional hopping matrices. It is common [1,7] to construct the hopping operators as a combination of a unitary coin-matrix C that entangles the components of ψ x,t locally while shift matrices S (j) provide for a subsequent transfer in either direction, i.e., For r = 2, the degree of each site on the line, we allow no self-loops (S M = 0, M x = 0) and we shift upper (lower) components of each ψ x,t to the right (left) using the projectors Then, it is easy to show that the unitarity conditions above are satisfied for arbitrary unitary coins C x . For a random walk, we could simply choose scalar Bernoulli coins p x such that A x = p x , B x = 1 − p x , and M x = 0, say, to satisfy the local conservation of probability, A x + B x + M x = 1. However, as the somewhat complicated construction of Maritan and Stella [36] illustrates, it is not possible to obtain a walk model with ultrametric barriers merely with such plain scalar coins. Here, we present an alternative version of that ultradiffusion model that is intuitive and has greater conceptual simplicity at the expense of adding an internal degree of freedom. Such a construction [45] is easily recognized as a second-order Markov process [43] or persistent random walk (PRW) [44]. It has the added benefit of being completely analogous to the above construction of the quantum walk: We merely replace the unitary coin by a stochastic coin C x , then the sum A x + B x + M x is also.

III. RENORMALIZATION OF WALKS WITH AN ULTRAMETRIC SET OF BARRIERS
In our study here, we consider position-dependent coins in such a way that all sites of odd index x share the same coin, and so do all sites that are once-, twice-, trice-, ..., divisible by 2. We define the binary decomposition x(i, j) = 2 i (2j+1) with a hierarchy-index, i ≥ 0, and running index, −∞ < j < ∞, providing a unique, oneto-one relation between x( = 0) and the pair (i, j). Then, all sites x ( = 0) that share the same value of i have an identical coin for all j, i.e., C x(i,j) = C i . When the sequence of such coins then becomes ever more reflective for a walker trying to transition through the respective site, the walker gets confined in a tree-like ultrametric set of domains with vastly varying timescales for exit. Figure 1. Depiction of the hierarchical set of barriers (red) of relative reflectivity −i for 0 < < 1 and hierarchical index i on a 1d -line, as implemented in Eq. (14) for the classical walk and Eq. (14) for the quantum walk, generating an ultrametrically arranged set of domains (tree) with a hierarchy of characteristic timescales ti for escape.
(Similar models have been proposed for slow relaxation and aging [34][35][36][37][38][39].) Two neighboring domains at level i form a larger domain at level i+ 1, and so on, from which an ultrametric hierarchy emerges. The barriers between such domains are depicted in Fig. 1.
To obtain the RG-recursions for such a hierarchical, 1d quantum walk, we employ generating functions [46], Then, the master equation (1) with U in Eq. (2) becomes For simplicity, we merely consider initial conditions (IC) localized at the origin, ψ x,t=0 = δ x,0 ψ IC and define the coin at x = 0 simply to be the identity matrix. Now, we recursively eliminate ψ x for all sites for which x is an odd number (i = 0), then set x → x/2 (i → i − 1) for the remaining sites, and repeat, step-by-step for k = 0, 1, 2, . . .. In each step, we successively eliminate all sites within an entire hierarchy i, each with an identical coin C i , starting at k = 0 with the "raw" hopping operator A After each step, the master equation becomes self-similar in form by identifying the renormalized hopping operators A i . Consider, for example, a site x with i + ≥ 2 (or x = 0), which pertains to every fourth site. Note that all sites ±1 or ±3 hops removed from x are of odd index (i = 0), while those ±2 hops removed must have i = 1. At RG-step k we have: [45] Solving this linear system for the even sites x, x±2, x±4, etc., yields and so on. Matching the solutions for those even sites to Eq. (6), we can read off the RG-recursions for all i > 0: Those RG-steps of decimation are illustrated in Fig. 2. Amazingly, we can entirely eliminate the hierarchyindex i: If we define the k-th renormalized shift matrices S which matches the definitions above for the unrenormalized systems at k = 0, at which point the shift matrices are independent of i. Then, the S {A,B,M } k remain i-independent for all k ≥ 0 when inserted into Eqs. (7). These satisfy the recursions: which instead now have an explicit k-dependence via the inverse coins C −1 k of the k-th hierarchy. (We assume that also the stochastic coin has an inverse, see below.)

A. Absorbing Walls
As a specific situation for such a setting, we will consider a walk between two absorbing walls of separation N = 2 l + 1, equidistant from the starting site x = 2 l−1 , as discussed in Fig. 2. As the wall-sites x = 0 and x = 2 l are fully absorbing, there is no flow out of those sites and at the end of l − 1 RG-steps the Eqs. (5) turn into Thus, for either wall it is We will discuss below the RG-prediction for the absorption for both, classical and quantum walks. Figure 2. Last three RG-steps k = l − 3, l − 2, and l − 1, in an ultra-walk with absorbing walls (yellow squares). Note that for each RG-step k → k + 1, the remaining hierarchy indices i for sites x go from i → i − 1, as reflected in Eq. (7). Then, for all sites x (k) → x (k+1) = x (k) 2 . The final step k = l − 1 → l is described by Eqs. (9)(10), assuming that the walk started at the central site, x = x (0) = 2 l−1 = N −1 2 in a finite system of size N = 2 l + 1, originally, i.e., at x (l−1) = 1.

IV. SOLUTION OF THE CLASSICAL ULTRA-WALK
Like for the quantum walk, the state variable describing a PRW is now a 2-component vector ψ x,t , which here expresses a memory of the previous step. Based on its prior behavior, the upper component ψ + x,t refers to a walker with the preference to step to the right in the next time-step, and the lower component ψ − x,t indicates a left-hop preference. The value of each component describes the probability of finding a walker at that site x and time t in state "±", and the total probability of finding a walker there, irrespective of preference, is simply the sum of the two, ρ x,t = ψ + x,t + ψ − x,t . Ignoring a po-tential left-right bias here, we consider a walker coming from the left (right) to have a probability η x to continue to move right (left), and a probability 1−η x to reverse direction in the next step. For η x > 1 2 (η x < 1 2 ) the walker exhibits (anti-)persistence, and for η x = 1 2 reduces again to an ordinary unbiased RW without memory. The master equations then reads: − IC , (11) whereψ IC (withψ + IC +ψ − IC = 1) represents the IC of the PRW, which we place again at the origin, x = 0. Thus,ψ + x (ψ − x ) only depends on hops from its left (right) neighbor; it is that inflow which induces the "+" ("−") state. We can then write Eq. (11) conveniently in matrix notation as a propagator like Eq. (2) with and M x = 0. As for the QW, we can decompose these matrices further to write them as a combination of a shift and a coin matrix, {A, B, M } x = S {A,B,M } C x , with the same shift matrices as in Eq. (3). However, here we introduce the stochastic coin matrix in which each row sums to unity.

A. Ultradiffusion as a hierarchically anti-persistent Walk
With the same choice of a hierarchically defined coin as in Sec. III, decomposing x = x(i, j), the classical PRW is renormalized exactly the same way such as to obtain Eq. (8).
[Note that C x in Eq. (13) also has an inverse, except for η x = 1 2 , the case of an ordianry (non-presistent) walk, which we can safely exclude in the following.] Then, it is easy to formulate a simple PRW that is in the same universality class as the ultradiffusion model solved in Ref. [36], by choosing for 0 ≤ ≤ 1 and some η 0 (< 1 2 , to ensure invertibility of all coins): For = 1, we expect to recover the ordinary PRW on a homogeneous 1d lattice. Since η i < 1 2 and, indeed, rapidly approaches zero, the walk is increasingly antipersistent, i.e., ever-larger domains form that are bordered by sites x of high hierarchical index i, frustrating the walk attempting to leave the domain with an exponentially smaller probability (or higher barriers) with i.
In this case, the renormalization recursion for the shift matrices in Eq. (8) can be parametrized as [45] A single iteration of Eq. (8) provides the RG-flow with intitial conditions a 0 = z and m 0 = 0. Note that this recursion is non-autonomous due to the explicit kdependence via η k . We could supplement η k as a dynamical variable via its recursion η k+1 = η k [36] and study fixed points of those three recursions. However, there is a more elegant approach using the transformations which turn Eqs. (16) into which is justified for η k in Eq. (14) at large k for < 1 but trivally holds for the homogeneous walk at = 1 also. Now, the RG-flow in Eq. (18) is purely autonomous but depends nontrivially on the parameter that characterizes the strength of the ultrametric barriers. This flow has two obvious fixed points, at α ∞ = 1 − 2 and µ ∞ = 1 − 1, and at α ∞ = 1 − and µ ∞ = − 1 . The second one can not be reached by any physical initial condition of the flow, as µ ∞ < 0. The first fixed point is physical for 0 ≤ ≤ 1 2 , where the largest eigenvalue of the Jacobian of the flow in Eq. (18) is λ = 2 , which reproduces the anomalous walk dimension, found in Ref. [36]. When → 1 2 , d w → 2 and the effect of the barriers becomes irrelevant such that ordinary diffusion ensues for all 1 2 ≤ ≤ 1.
[In fact, for = 1 only, we can find a closed form solution for all k of the RG-flow in Eq. (18), as in Refs. [41,47].] However, to find the fixed point for the diffusive solutions for 1 2 ≤ ≤ 1 requires a scaling ansatz with α k = (2 ) −k x k and µ k = 1+(2 ) −k y k such that now independent of , with fixed point x ∞ = y ∞ and a Jacobian eigenvalue of λ = 4, i.e., d w = 2. These results are summarized in Fig. 3. Finally, we note that replacing the anti-persistent hierarchy of coins with a persistent one, easily achieved by replacing η x → 1 − η x in Eq. (13), leads again to a purely diffusive walk for all . The asymptotic analysis for large k with k → 0 now yields the → 1 limit of Eq. (18) even for < 1 that then results in Eq. (20) with d w = 2. It is easy to see that the physical situation of a persistent hierarchy differs dramatically from the anti-persistent one: For large k, all coins now become transmissive (i.e., the identity matrix) instead of reflective, leaving mostly the non-diagonal coins at all odd sites (i = 0) to institute a simple persistent walk that is in the same universality class as ordinary diffusion [44].

B. Classical Walk with Absorbing Walls
Evaluation of Eq. (10) for the geometry of Fig. 2 using the coin in Eq. (13) and the RG parametrization in Eq. (15) for l → ∞, we readily obtain with S {A,B} given in Eq. (3). In the classical walk, the norm of a state-vecor is simply the sum of its (certainly non-negative) components, ψ x,t def = 1 1 •ψ x,t . Then, for an absorbing site x, the absorption there is F x = ∞ t=0 |ψ x,t | = ψ x (z = 1) . With S A + S B = I, we finally get for the total (combined) absorption: since ψ IC is normed to unity, of course. Thus, for any size barrier and any system size, any walk started in the middle will eventually get absorbed with certainty! For a classical walk on the 1d-line, we would have expected this, due to Polya, even for a heterogeneous environment.

V. SOLUTION OF THE QUANTUM ULTRA-WALK
To design a quantum analogue to the classical PRW on an ultrametric set of barriers, specified by Eq. (13), we consider the real, unitary quantum coins although many other interesting choices may exist [21]. Note that for = 1, this reproduces a homogeneous 1d QW [45]. However, for < 1, the coins become increasingly resistant to transit with η i → 0 for i → ∞, blocking the transition through sites x of higher index i, no matter from which direction those sites are approached.
Evolving the recursions in Eq. (8) with this coin for a few iterations from the unrenormalized values, already after one iteration a recurring pattern emerges that suggest the Ansatz which is amazingly similar to the classical case in Eq. (15). When iterated, the RG-flow in Eqs. (8) for a k and m k closes after each iteration for with a k=1 = z 2 sin η 0 , m k=1 = z 2 cos η 0 , and η 1 = η 0 as initial conditions. [Only the first step, from k = 0 to k = 1, does not fit this pattern.] Note the striking similarity of these recursions to those for the PRW in Eq. (16). Like those, Eq. (25) is non-autonomous. Analogous to the PRW in Eq. (17), we can find an elegant transformation, that turns Eq. (25) into where we have set sin η k+1 sin η k ∼ and (cos η k − cos η k+1 ) / sin η k+1 ∼ 1 2 η k ( − 1/ ). If we were to consider neglecting the last term in Eq. (27), that is small for ≤ 1, we find a fixed point at Having α ∞ < 0 imposes no restriction, since the definition of a k in Eq. (24) is invariant under a k → −a k . Having imaginary µ ∞ could be expected as a small correction off the real axis to m ∞ = 1 in a quantum problem. The associated eigenvalues are very interesting, with of which only λ + > 1 for 0 < < 1. In fact, it reproduces the eigenvalues found for a corresponding tight-binding spectrum considered in Ref. [38]. As shown in Fig. 3, it meets up with the classical result for d w really well for → 0. However, it generally predicts a slower spread than even the classical walk throughout. Closer inspection of the boundary layer [48] at → 1 shows that it is actually a very subtle extension of the (subdominant) diffusive solution found for the ordinary 1d QW discussed in Ref. [45]. While λ + → 5 + √ 17 /2 ≈ 4.56... for → 1, this fixed point is actually not valid for = 1, since µ ∞ → ±i and α ∞ → 0, for which the recursions in Eq. (27) are singular. Resolving that singularity with a scaling ansatz reproduces the diffusive solution with λ = 4 to which λ + discontinuously connects.
To reveal the physically relevant scaling of the quantum walk, we argue as follows: Without the last term in Eq. (27), there is also a fixed point values of α ∞ = and µ ∞ = 0, which in turn is inconsistent with dropping even an exponentially small term, however. Rather, we retain the term and apply the Ansatz µ k = η k ν k 1 with ν k finite to turn all of Eq. (27) autonomous: The flow in Eq. (29) has a fixed point at α ∞ = and ν ∞ = 1 2 1 − 2 . Its Jacobian eigenvalues are λ 1 = 1 + −2 and λ 2 = 2, i.e., λ 1 ≥ λ 2 > 1 for 0 ≤ ≤ 1. For a classical walk with a stochastic master equation, the leading eigenvalue λ 1 suffices to describe the walk dimension d w = log 2 λ 1 [46]. However, it has been shown [41,49] that the unitarity constraint imposed on the master equation for the quantum walk requires both leading eigenvalues such that [50] This walk dimension, also shown in Fig. 3, has all the characteristics of being physical, as it is d w → 1 for the homogeneous 1d QW at → 1, and it diverges for diverging barrier heights, → 0, where to leading order matches up with the classical result, d w ∼ − log 2 . It also predicts the fastest spread for the QW compared to RW or the fixed point leading to Eq. (28).

A. Quantum Walk with Absorbing Walls
Unlike for the classical case discussed in Sec. IV B, where the total absorption amounts conveniently to taking a local limit for z → 1 on the Laplace transform of the site amplitude, the adsorption in the QW is more involved. Due to unitarity, the adsorption sums up the square modulus of site amplitudes, which correspond to contour integrals over the entire unit circle in the complex z−plane for the modulus of their Laplace transforms, While such an integral is readily evaluated for the simple case of homogeneous QW ( = 1) [8], for < 1 only the local asymptotic evaluation of the RG recursions in Eq. (25) is available. While, e.g., ψ 0 (z) in Eq. (31) is a functional of the hopping operators, inserting their asymptotic form for a local expansion of the integral does not appear to be sufficient to obtain the absorption. RW: λ 1 λ + Figure 3. Plot of the inverse walk dimension 1/dw for the classical (RW) and the quantum walk (QW) with ultrametric barriers as a function of coin-parameter , where smaller represents higher barriers between an ultrametrically arrange hierarchy of domains that confine the walk. The black and the blue lines are the respective RG-predictions in Eq. (19) and in Eq. (30). Also, the unphysical prediction from λ+ in Eq. (28) is shown as red-dashed line.
Resorting instead to a direct simulation of QW shows that it is difficult to extract the scaling for moments of the walk to, say, verify the walk dimension in Eq. (30) with any reasonable accuracy. The irregular pattern of reflecting barriers leads to very noisy probability densities. However, it is quite easy to convince oneself, starting from very small systems and progressively doubling their size, that the total absorption remains exactly unity throughout for even the smallest values of . Thus, there appears to be no localization in ther interior of the system; all of the QW eventually reachs a wall!

VI. DISCUSSION
The ultra-walk provides an exactly solvable model of a walk, both classical or quantum, with tunable spatial heterogeneity. For the classical case, we reproduce previous results by alternative means, using a 2 nd −order Markov process that closely resembles the coined QW in form. For the discrete-time QW, we obtain entirely new results over a whole range of heterogeneity, with walk dimensions ranging from d QW w = 1 to infinity. However, numerical verification of these results for QW is difficult to obtain, due to the strong hierarchical nature. Focusing on the absorbtion problem at walls that recede from the starting site with increasing system size, we can at least ascertain, as in the classical case, that there is no localization, even in the most extreme heterogeneity. Future work will focus on the asymptotic evaluation of observables, like the absorption in Eq. (31), that are defined via a complex integration for the QW.