The regular lattice

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Indeed, by identifying the subspace of column-states |c<sup>`</sup>i, the quantum walk on the L - level glued trees graph starting from the left root is e↵ectively the same as a quantum walk on a line with 2L + 1 vertices, with all edge weights the same (see Fig. 4.2, bottom panel). In the limit of L ! 1, the walk on GL is nearly identical to a quantum walk on the infinite, translationally invariant line. The probability amplitude to go from column ` to column `⁰ for L ! 1 in a time t is then (cf. Eq. 4.14)

hc`<sup>0</sup>|e <sup>iH</sup>t|c`i = e ^{i3 t}i^{`0 `}J`<sup>0</sup> `(2p

2t), (4.23)

with J`<sup>0</sup> ` being a Bessel function of order `<sup>0</sup> `; this corresponds to propaga-tion with speed 2p

2 . To verify this, we numerically compute the probability

|hc^`|e ^iHt|c⁰i|² for a large system with L = 500 and = 1 at t = 100, 175, 250 and 354. The results are shown in Fig. 4.5. The wavefront of the distribution at t is located approximately 2p

2 t away from the left root.

4.3 Quantum walk on square lattices

The quantum walk on the glued trees graph that we considered in the previous section is e↵ectively a one-dimensional quantum walk problem. In this section, we focus on continuous-time quantum walks on structures topologically equivalent to two-dimensional (2D) square lattices.

4.3.1 The regular lattice

We first consider a 2D regular square lattice with eL = 2L + 1 sites per row or column. Each pair of nearest-neighbor vertices has an edge connecting them, and each vertex has degree four. Periodic boundary conditions (PBC) are imposed in two directions; the lattice can then be thought of as being mapped onto the surface of a three dimensional torus (or doughnut). The N = eL² vertices of the system give rise to N orthonormal basis vectors, denoted by|ni, or |nxi ⌦ |nyi ⌘ |nx, n_yi with n_x, n_y 2 [ L, L] being integer labels in x and y directions

Without loss of generality, we set the transition rate = 1 for every edge.

The continuous-time quantum walk is defined by the Hamiltonian given by the

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4.3 Quantum walk on square lattices

transition rate matrix. The Hamiltonian H acting on a state nx, ny

↵ reads to fulfill the PBC. Eq. 4.24 can be written in a tensor product form as

H |nxi ⌦ |nyi = 2 |nxi |nx 1i |nx+ 1i ⌦ |nyi +|n^xi ⌦ 2 |n^yi |n^y 1i |n^y+ 1i

⌘ Hx|nxi ⌦ |nyi + |nxi ⌦ Hy|nyi

(4.26)

The Hamiltonian H is then decomposed into two parts

H = Hx+ Hy, (4.27)

corresponding to the operators for x and y components Hx|nxi = 2 |nxi |nx 1i |nx+ 1i

Hy|n^yi = 2 |n^yi |n^y 1i |n^y+ 1i (4.28) The eigenvectors, | k_x(y)i, of Hx(y) are the so-called Bloch states [30] given by

| kxi = 1

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From Eq. (4.26), we find the eigenvalues Ek and the eigenvectors| ^ki for the full Hamiltonian H:

Using the solution of the eigenvalue problem for H, we obtain the exact expression for the probability amplitude of moving from state |ni = |nx, nyi to state |mi = |mx, myi in time t:

and the corresponding probability (denoted by Pn!m(t))

Pn!m(t)⌘ hm|e ^iHt|ni ². (4.34)

In the limit N ! 1, we may use a continuum approximation to rewrite the expression in Eq. (4.33) to [31]

hm|e ^iHt|ni !e ^i4t

where we have used an integral representation [29]

Jn(t) = 1 2⇡iⁿ

Z 2⇡

0

d✓ e^i✓ne^{it cos ✓}. (4.36) for the Bessel function Jn(t) of the first kind. The probability of moving from |ni

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4.3 Quantum walk on square lattices

to|mi in a lattice of infinite size is then

Nlim!1Pn!m(t) = Jmx nx(2t)Jmy ny(2t) ² (4.37) In Fig. 4.6 we show the probability distribution for quantum walk on a finite 2D lattice run for di↵erent time spans with the starting point at the origin (0, 0).

Since the time evolution of the walk is governed by a separable Hamiltonian H = Hx+ Hy with a separable initial condition, the di↵erent spatial dimensions behave independently; thus the probability at any t shows a symmetrical pattern along x and y directions.

To verify the results for an infinite lattice, we calculate the probabilities P0!n(t), as a function of time t, for various n for finite N = 513². As shown in Fig. 4.7, the data obtained from the Bloch ansatz are in excellent agreement with the exact solutions for N ! 1. Furthermore, the plots in Fig. 4.8 for a large lattice of size N = 1025² clearly show that the leading edge of the probability distribution P0!n(t) of arriving at vertices |ni = |n, ni along the diagonal path between the staring point|0, 0i and the corner vertex (L, L) moves approximately with speed 2; this is indeed implied in the exact expression (in Eq. (4.37)) for an infinitely large system.

Now we turn to the comparison with the classical random walk. Using the eigenvalues Ekand the eigenvectors| ki, the probability P_n!m^cl (t) for the classical case can be calculated via

P_n^cl_!m(t) = 1 N

X

k

e ^Ekthm| ^kih ^k|ni. (4.38)

We focus on the return probabilities Pr(t)⌘ Pn!n(t) (for the quantum case) and Pr(t)⌘ Pn^cl!n(t) (for the classical case), which are the probabilities to be still or again at the initial state at time t. A fast decay of the return probability implies a fast propagation through the graph since the probabilities (1 Pr(t)) to be at any but the initial state grow quickly. In Fig. (4.9), we consider Pr(t) for the lattice size 33⇥ 33. As discussed before, for the classical random walk the long time limit of the probabilities P_mn^cl (t) reaches the equipartition value 1/N . In the same way, the long time limit of the return probability P_r^cl(t) is given by 1/N . In

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Figure 4.6: Time evolution of the probability distribution over the square lattice of size 65⇥ 65 with the initial state |nx= 0i ⌦ |ny = 0i.

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4.3 Quantum walk on square lattices

30 40 50

Figure 4.7: The probability for a large 2D square lattice of size N = 513² of arriving on vertex (a)|64, 16i and (b) |32, 16i with initial state |0, 0i, plotted as a function of time t. The black line corresponds to data obtained using the Bloch ansatz; the orange dashed line indicates the exact solution for N ! 1, given in Eq. (4.37).

Figure 4.8: Propagation in a large lattice of size N = 1025²along a diagonal path starting at the middle vertex |0, 0i to the corner vertex (L, L). The vertex |n, ni located at 2t is indicated by the red dashed line, which is also the approximate location of the wavefront at time t.

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10^-1 10⁰ 10¹ 10²

t

10^-8 10^-6 10^-4 10^-2 10⁰

P

(t)

quantum classical

Figure 4.9: The return probabilities, P_r(t) and P_r^cl(t), for the quantum and classi-cal random walks on a 2D lattice of size 33². The blue dashed line, proportional to 1/t², indicates the decay behavior of the quantum probability in the intermediate range, where the classical probability decays as P_r^cl(t)⇠ 1/t before it converges to P_r^cl(1) = 1/N

contrast, the probability P_r(t) for the quantum case does not decay to a constant value at t ! 1, but oscillates over time. In the intermediate range (between t ⇡ 0.5 and t ⇡ 100) the classical probability decays algebraicly as Pr^cl(t)⇠ 1/t, while the quantum probability decays faster (Pr(t) ⇠ 1/t²) before it oscillates around the long time average.