^{1,a)}, H. Vogts

^{1,b)}, A. H. Werner

^{1,c)}and R. F. Werner

^{1,d)}

### Abstract

We study the asymptotic position distribution of general quantum walks on a lattice, including walks with a random coin, which is chosen from step to step by a general Markov chain. In the unitary (i.e., nonrandom) case, we allow any unitary operator which commutes with translations and couples only sites at a finite distance from each other. For example, a single step of the walk could be composed of any finite succession of different shift and coin operations in the usual sense, with any lattice dimension and coin dimension. We find ballistic scaling and establish a direct method for computing the asymptotic distribution of position divided by time, namely as the distribution of the discrete time analog of the group velocity. In the random case, we let a Markov chain (control process) pick in each step one of finitely many unitary walks, in the sense described above. In ballistic order, we find a nonrandom drift which depends only on the mean of the control process and not on the initial state. In diffusive scaling, the limiting distribution is asymptotically Gaussian, with a covariance matrix (diffusion matrix) depending on momentum. The diffusion matrix depends not only on the mean but also on the transition rates of the control process. In the nonrandom limit, i.e., when the coins chosen are all very close or the transition rates of the control process are small, leading to long intervals of ballistic evolution, the diffusion matrix diverges. Our method is based on spatial Fourier transforms, and the first and second order perturbation theory of the eigenvalue 1 of the transition operator for each value of the momentum.

We gratefully acknowledge the support of the DFG (Forschergruppe 635) and the EU projects CORNER, QUICS and CoQuit.

I. INTRODUCTION

II. THE SYSTEMS

A. General form of decoherent, translation invariant quantum walks

III. ASYMPTOTIC POSITION BY THE PERTURBATION METHOD

IV. UNITARY QUANTUM WALKS

A. Ballistic order

B. Higher orders

V. TIME DECOHERENT WALKS

A. Ballistic determinism for decoherent processes

B. Markovcontrolled walks in diffusive scaling

C. Higher orders without ballistic determinism

D. Memoryless decoherence

VI. EXAMPLES

A. The Hadamard walk with reflections

B. One dimensional quantum walks with equal position distribution

C. Nonunitary Kraus operators

D. Quantum walks with momentum shifts

E. Commuting Kraus operators

### Key Topics

- Eigenvalues
- 63.0
- Ballistics
- 42.0
- Process monitoring and control
- 30.0
- Probability theory
- 23.0
- Bernoulli's principle
- 19.0

## Figures

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.

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.

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 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.

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 ψ_{0} = (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.

Correction of order 1/*t* to the asymptotic position distribution for the Hadamard walk from the initial state ψ_{0} = (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.

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).

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.

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.

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.

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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