Continuous-time quantum walks with defects and disorder

With the advent of physical implementations of quantum walks, a general theoretical and efficient numerical framework is required for the study of their interactions with defects and disorder. In this paper, we derive analytic expressions for the eigenstates of a one-dimensional continuous-time quantum walk interacting with a single defect, before investigating the effects of multiple diagonal defects and disorder, with emphasis on its transmission and reflection properties. Complex resonance behavior is demonstrated, showing alternating bands of zero and perfect transmission for various defect parameters. Furthermore, we provide an efficient numerical method to characterize quantum walks in the presence of diagonal disorder, paving the way for selective control of quantum walks via the optimization of position-dependent defects. The numerical method can be readily extended to higher dimensions and multiple interacting walkers.

Figure 1

The numeric (orange solid line) and analytic (black dashed line) single-point-defect eigenstates are plotted for the set of (a) odd continuous eigenstates and (b) even continuous eigenstates. (c) The value of the even states at the defect location $|0\rangle$ for defect strength $\alpha =1/\sqrt{200},1,2$.

Figure 2

Left: The numeric and analytic single-point defect bound states are plotted and compared for the cases (i) $\alpha <0$ with the analytic result in long-dashed black lines and the numerical data points in red open circles, and (ii) $\alpha >0$ with the analytic result in short-dashed black lines and the numerical data points in green dots. Right: The complete eigenvalue spectrum of the Hamiltonian with a single-point defect.

Figure 3

(a) The time evolution of the initial state $|1\rangle$ for time $t^{\prime }=20$, with a defect placed at $d=5$ with strength $\alpha =1/\sqrt{2}$, using the matrix exponential definition (blue line) and the integral approach (18) (red data points). (b) A plot of the absolute error between both methods.

Figure 4

Transmission coefficient vs momentum for a CTQW momentum eigenstate incident on an $N$-point defect, where $N=2$ (blue dot-dashed line), $N=3$ (red dashed line), $N=5$ (black dotted line), and $N=8$ (green solid line). The defect-induced barriers are constructed with unity amplitude ($\alpha =1$) and equal separation $L$.

Figure 5

Transmission coefficient vs momentum for a CTQW momentum eigenstate incident on an evenly spaced 8-point defect, with barrier amplitudes $\alpha =0.1$ (blue dot-dashed line), $\alpha =0.5$ (red dashed line), $\alpha =1$ (black dotted line), and $\alpha =2$ (green solid line).

Figure 6

(a) Transmission coefficient vs momentum of a CTQW momentum eigenstate incident on two defects of unity amplitude, separated by distances $L=0$ (solid blue line), $L=1$ (dashed red line), $L=2$ (dotted black line), and $L=5$ (dot-dashed green line), and calculated via numerical Fourier methods. (b) Absolute difference between numeric and analytic results for the case $L=1$.