Simple quantum random walk

walker.py

"""
Requires matplotlib for plotting. Tested with python 27. If you want to try this
without plotting, remove the final two lines and the pylab import. The guts only
depends on math and will work with vanilla python.
"""
import math
import pylab


def probabilities(posn):
    """Returns a list of the probabilies for each place."""
    return [sum([abs(amp) ** 2 for amp in place]) for place in posn]


def normalise(posn):
    """Normalise function to normalise an input 1D line."""
    N = math.sqrt(sum(probabilities(posn)))
    return [[amp / N for amp in place] for place in posn]


def timestep(posn):
    """Defines action of a timestep, i.e. a Hadamard gate on each element."""
    return normalise([[x[0] + x[1], x[0] - x[1]] for x in posn])


def shift(coin):
    """Shift the up elements leftwards and the down elements rightwards."""
    newposn = [[0, 0] for i in range(len(coin))]
    for j in range(1, len(posn) - 1):
        newposn[j + 1][0] += coin[j][0]
        newposn[j - 1][1] += coin[j][1]
    return normalise(newposn)


# Initialise lists.
min, max = -500, 501
posn = [[0, 0] for i in range(min, max)]
posn[-min] = [1 / math.sqrt(2), 1j / math.sqrt(2)]

# Run for some steps...
for time in range(-min):
    posn = shift(timestep(posn))

# Plot.
pylab.plot(range(min, max), probabilities(posn))
pylab.show()

This is some old code I wrote to demonstrate some simple python to a physics student some years ago. The intention was to show them some neat syntax with a simple algorithm that they were already familiar with. I came across it today and was surprised to find that it still works.