Quantum walks over a square lattice

Quantum random walk finds application in efficient quantum algorithms as well as in quantum network theory. Here we study the mixing time of a discrete quantum walk over a square lattice in the presence of dynamic percolation and decoherence. We consider bit-flip and phase damping noise, and evaluate the instantaneous mixing time for both the cases. Using numerical analysis we show that, in the case of phase damping noise, the probability distribution of walker's position is sufficiently close to the uniform distribution after infinite time. However, during the action of bit-flip noise, even after infinite time the total variational distance between the two probability distributions is large enough.

Figure 1

Plot of $f(M,{\mathrm{\Gamma }}_{\text{dep}})$ against ${\mathrm{\Gamma }}_{\text{dep}}$ for three different lattice size. From each of the curves one can numerically find the value of the function $f(M,{\mathrm{\Gamma }}_{\text{dep}})$ for a particular dephasing rate ${\mathrm{\Gamma }}_{\text{dep}}$, and calculate the trace norm in the range $0\le t\le \infty$. It is clear from the plots that with increasing dephasing rate the value of the function decreases, which means that the dephasing noise helps in faster convergence of the walk.

Figure 2

Time evolution of average trace distance ($\overline{D}(t,\delta )$) under continuous dephasing. Each curve represents the time evolution of $\overline{D}(t,\delta )$ for a particular value of coin parameter $a$ and dephasing rate ${\mathrm{\Gamma }}_{\text{dep}}$. Notice that for higher values of coin parameter and dephasing rate, convergence towards uniform distribution becomes faster.

Figure 3

Plot of $f(M,{\mathrm{\Gamma }}_{\text{bit}})$ against ${\mathrm{\Gamma }}_{\text{bit}}$ for three different lattice size. From each of the curves one can numerically find the value of the function $f(M,{\mathrm{\Gamma }}_{\text{bit}})$ for a particular bit-flip rate ${\mathrm{\Gamma }}_{\text{bit}}$, and calculate the trace norm in the range $0\le t\le \infty$. It is clear from the plots that with increasing bit-flip rate the value of the function decreases, which means that the bit-flip noise helps in faster convergence of the walk. However, the rate of convergence is smaller than the previous case.

Figure 4

Time evolution of average trace distance ($\overline{D}(t,\delta )$) under continuous bit-flip noise. Each curve represents the time evolution of $\overline{D}(t,\delta )$ for a particular value of coin parameter $a$ and bit-flip rate ${\mathrm{\Gamma }}_{\text{bit}}$. For higher value of bit-flip rate convergence towards uniform distribution becomes faster. However, the convergence towards uniform distribution stops after a certain value of average trace distance.