The figure shows a caustic of a unitary quantum walk on a two-dimensional lattice. Closed curves correspond to the image of the map p↦∇ω(p) applied to a discrete set of coordinate lines in momentum space [0, 2π)2. The caustic, i.e., the region where the line density diverges is exactly the region where the asymptotic probability distribution of the quantum walk exhibits peaks.
Figures (a)–(c) show the time evolution of a quantum walk with index three and three internal states for 50, 150, and 300 time steps. The green/dot-dashed curve corresponds to the undisturbed walk with a coin that is constant in time, whereas the red/solid and blue/dashed curves correspond to a bernoulli process where in every time step instead of the coin of the green/dot-dashed quantum walk a different coin is applied with probability 0.01 or 0.2, respectively. Graph (d) shows the standard deviation of the position distributions with a visible crossover from ballistic to diffusive spreading behavior for the perturbed quantum walks.
For the quantum walk with Hadamard coin, the figures show (a) the dispersion relations ω±, (b) the velocities v ±, and (c) the asymptotic probability distribution. The initial state is chosen to be . The extremal points of the functions v ±(p), hence the inflection points of ω±, are responsible for the peaks in (c) at .
For the two-dimensional walk (26) the plots show (a) the dispersion relation ω±, (b) a contour plot, and (c) a 3D plot of the contribution of the ω+-branch to the asymptotic position density, for a particle starting at the origin. The curves in (a) indicate points of vanishing curvature. At these points the velocity density (i.e., the inverse of the Jacobian of the transformation p↦v +(p)) diverges. This produces the enclosing curve in (b), and infinitely high values in (c). For more complicated walks such lines also appear in the interior of the velocity region.
Correction of order 1/t to the asymptotic position distribution for the Hadamard walk from the initial state ψ0 = (1, 0) located at the origin. The polygon connects the exact values for n = 10. The asymptotic distribution (red/dashed) overestimates the left peak and underestimates the right peak. The correction after (31) for the same t is shown by the green/solid curve.
For the quantum walk according to (67) the plot shows the 1/t correction to the asymptotic position distribution in ballistic scaling after t = 10 depending on the parameter ε.
The plot shows the variance s(p) for a dephased Hadamard Walk (69) with . The undisturbed Hadamard Walk is applied with probability ε ∈ [0, 1] and the walk W π/3 is applied with probability (1 − ε).
Position distributions of the walk defined by (72) after (a) 50 and (b) 200 time steps (green/dotted lines) in ballistic scaling. The red/dashed line shows the asymptotic position distribution for comparison.
Plot (a) shows the correction of order 1/t (green/solid curve) to the asymptotic distribution (red/dashed curve) for the walk (72). The exact values (connected by a polygon) are shown for the same value (t = 10) as the correction. In (b) the 1/t correction C 1(λ, t) to the characteristic function C t (λ) is shown for t = 10.
Asymptotic position distribution for a walk (75) with momentum shift q = π/16 (green/dotted line) after (a) 20 steps and (b) 100 steps. The red/dashed line is the asymptotic distribution.
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