Limit distributions of two-dimensional quantum walks

One-parameter family of discrete-time quantum-walk models on the square lattice, which includes the Grover-walk model as a special case, is analytically studied. Convergence in the long-time limit $t\to \infty$ of all joint moments of two components of walker’s pseudovelocity, $X_t/t$ and $Y_t/t$, is proved and the probability density of limit distribution is derived. Dependence of the two-dimensional limit density function on the parameter of quantum coin and initial four-component qudit of quantum walker is determined. Symmetry of limit distribution on a plane and localization around the origin are completely controlled. Comparison with numerical results of direct computer simulations is also shown.

Figure 1

(Color online) The two-dimensional fundamental density-function ${\mu }_p\left(v_x,v_y\right)$ of limit distribution of pseudovelocities, when $p=1/4$.

Figure 2

(Color online) Dependence of localization probability around the origin $\Delta$ on the parameter $p\in \left(0,1\right)$. (a) The case ${{}^T\varphi }_0=\left(1,1,-1,-1\right)/2$. When $p=1/2$ (the Grover-walk model), $\Delta =0$. (b) The case ${{}^T\varphi }_0=\left(1,1,1,1\right)/2$. When $p=1/2$ (the Grover-walk model), $\Delta =2\left(\pi -2\right)/\pi =0.726\cdots$.

Figure 3

(Color online) The case $p=1/4$ and ${{}^T\varphi }_0=\left(1,-1,1,1\right)/2$. Since ${\mathcal{M}}_3={\mathcal{M}}_6=0$ in this case, the limit distribution has the reflection symmetry for the $v_x$ axis; $\nu \left(v_x,-v_y\right)=\nu \left(v_x,v_y\right)$. (a) Distribution of pseudovelocity ${\mathbf{V}}_t=\left(X_t/t,Y_t/t\right)$ at time step $t=30$ numerically obtained by computer simulation. (b) Probability density of limit distribution.

Figure 4

(Color online) The case $p=1/4$ and ${{}^T\varphi }_0=\left(1,1,1,-1\right)/2$. Since ${\mathcal{M}}_2={\mathcal{M}}_6=0$ in this case, the limit distribution has the reflection symmetry for the $v_y$ axis; $\nu \left(-v_x,v_y\right)=\nu \left(v_x,v_y\right)$. (a) Distribution of pseudovelocity ${\mathbf{V}}_t=\left(X_t/t,Y_t/t\right)$ at time step $t=30$ numerically obtained by computer simulation. (b) Probability density of limit distribution.

Figure 5

(Color online) The case $p=1/4$ and ${{}^T\varphi }_0=\left(1,1,0,0\right)/\sqrt{2}$. Since ${\mathcal{M}}_2={\mathcal{M}}_3={\mathcal{M}}_6=0$ in this case, the limit distribution has the reflection symmetries both for the $v_x$ axis and the $v_y$ axis; $\nu \left(v_x,-v_y\right)=\nu \left(-v_x,v_y\right)=\nu \left(v_x,v_y\right)$. (a) Distribution of pseudovelocity ${\mathbf{V}}_t=\left(X_t/t,Y_t/t\right)$ at time step $t=30$ numerically obtained by computer simulation. (b) Probability density of limit distribution.

Figure 6

(Color online) The case $p=1/4$ and ${{}^T\varphi }_0=\left(1,-1,-1,1\right)/2$. Since ${\mathcal{M}}_2={\mathcal{M}}_3=0$ in this case, the limit distribution has the birotational symmetry for the $v_z$ axis, which is perpendicular both to $v_x$ and $v_y$ axes; $\nu \left(-v_x,-v_y\right)=\nu \left(v_x,v_y\right)$ (a) Distribution of pseudovelocity ${\mathbf{V}}_t=\left(X_t/t,Y_t/t\right)$ at time step $t=30$ numerically obtained by computer simulation. (b) Probability density of limit distribution.