1. Code, python3.7:

import numpy as np
import math
import matplotlib.pyplot as plt
import time
import colorsys
import cmath
size=1000

def run_classical_randwalk(itersteps,initsim_mat):
    simmat=initsim_mat
    for iterstep in range(itersteps):
        newsimmat=np.zeros((2*size+1,2), dtype=complex)
        for matindex in range(2*size+1):
            to_right=0.5*simmat[matindex][0]
            to_left=0.5*simmat[matindex][0]
            if(matindex-1>=0):
                newsimmat[matindex-1][0]+=to_left
            if(matindex+1<=2*size):
                newsimmat[matindex+1][0]+=to_right
        simmat=newsimmat

    psisquared=np.zeros(2*size+1)
    for matindex in range(2*size+1):
        psisquared[matindex]+=abs(newsimmat[matindex][0])

    average_x=0
    min_x=0
    max_x=0
    datastartflag=0
    for matindex in range(2*size+1):
        if(datastartflag==0):
            min_x=matindex
        if(psisquared[matindex]>0):
            datastartflag=1
            max_x=matindex
        average_x+=psisquared[matindex]*(matindex-(size+1))
    print(f"validdatarange {max_x-min_x}")
    return(range(min_x-(size+1),max_x-size,2),psisquared[min_x:max_x+1:2],average_x)

def run_quantum_randwalk(itersteps,initsim_mat):
    simmat=initsim_mat
    for iterstep in range(itersteps):
        newsimmat=np.zeros((2*size+1,2), dtype=complex)
        for matindex in range(2*size+1):
            hadamard_spinup=1/math.sqrt(2)*(simmat[matindex][0]+simmat[matindex][1])
            hadamard_spindown=1/math.sqrt(2)*(simmat[matindex][0]-simmat[matindex][1])
            if(matindex-1>=0):
                newsimmat[matindex-1][1]+=hadamard_spindown
            if(matindex+1<=2*size):
                newsimmat[matindex+1][0]+=hadamard_spinup
        simmat=newsimmat

    psisquared=np.zeros(2*size+1)
    for matindex in range(2*size+1):
        psisquared[matindex]+=abs(newsimmat[matindex][0])**2+abs(newsimmat[matindex][1])**2

    average_x=0
    min_x=0
    max_x=0
    datastartflag=0
    for matindex in range(2*size+1):
        if(datastartflag==0):
            min_x=matindex
        if(psisquared[matindex]>0):
            datastartflag=1
            max_x=matindex
        average_x+=psisquared[matindex]*(matindex-(size+1))
    print(f"validdatarange {max_x-min_x}")
    return(range(min_x-(size+1),max_x-size,2),psisquared[min_x:max_x+1:2],average_x)

simmat=np.zeros((2*size+1,2), dtype=complex)
#(-size, ....,-1,0,1, size)
#first index spin up, second index spin down
simmat[size+1][0]=1.0
simmat[size+1][1]=0.0#1.0#1.0j

fig=plt.figure()
plt.xlabel("position")
plt.ylabel("probability of occurence")
q_list_return_50=run_quantum_randwalk(50,simmat)
c_list_return_50=run_classical_randwalk(50,simmat)
plt.plot(q_list_return_50[0],q_list_return_50[1],color="#e67300")
plt.plot(c_list_return_50[0],c_list_return_50[1],color="#0000a0")
xlim=plt.gca().get_xlim()
ylim=plt.gca().get_ylim()
plt.vlines(q_list_return_50[2],*ylim,color="#e67300",alpha=.7)
plt.vlines(c_list_return_50[2],*ylim,color="#0000a0",alpha=.7)
print(fig.axes)
plt.gca().spines['top'].set_visible(False)
plt.gca().spines['right'].set_visible(False)
plt.gca().grid(color='grey', linestyle='-', linewidth=0.25, alpha=0.5)
plt.show()
fig.savefig("One_dimensional_quantum_random_walk.svg")

This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.

You are free:

• to share – to copy, distribute and transmit the work
• to remix – to adapt the work

Under the following conditions:

• attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
• share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.

https://creativecommons.org/licenses/by-sa/3.0CC BY-SA 3.0 Creative Commons Attribution-Share Alike 3.0 truetrue