Measurement¶

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Measuring position¶

Real quantum system are measurable. This property and physical effect is complicated to model with just pen and paper, since it relies on a component of 'dice-throwing'. Computationally, using a Metropolis-Hastings random walk algorithm, the process of measurement becomes instead rather straightforward.

BraketLab models measurable kets, meaning you may do as follows:

In [1]:

import braketlab as bk
import matplotlib.pyplot as plt
import numpy as np


p = bk.basisbank.get_harmonic_oscillator_function(5)

m = p.measure(repetitions = 10000) # measure position, repeat experiment 10000 times with reset

# visualize results
population, bins = np.histogram(m, bins = 61)
plt.figure()
plt.plot(.5*(bins[1:]+bins[:-1]), population, ".")
plt.xlabel("Position")
plt.ylabel("Population")
plt.show()

import braketlab as bk import matplotlib.pyplot as plt import numpy as np p = bk.basisbank.get_harmonic_oscillator_function(5) m = p.measure(repetitions = 10000) # measure position, repeat experiment 10000 times with reset # visualize results population, bins = np.histogram(m, bins = 61) plt.figure() plt.plot(.5*(bins[1:]+bins[:-1]), population, ".") plt.xlabel("Position") plt.ylabel("Population") plt.show()

Arbitrary measurements¶

Disclaimer: this feature is still a on the experimental stage, and will be more tested in the future.

If you provide the measurement-method with a Hermitian operator with it's eigenstates, the outputs will distribute correspondngly with a projection onto the eigenstates of the operator. (This feature presupposes, however, that the operator has a set of eigenfunctions, which currently no operators in BraketLab posess by default. But feel free to experiment.

In the following example, we create the Hamiltonian of the Harmonic Oscillator with a truncated set of eigenfunctions. We thereafter measure the energy.

In [2]:

x = bk.get_default_variables(0, 1)[0]

x = bk.get_default_variables(0, 1)[0]

In [3]:

n = 2

psi = bk.basisbank.get_harmonic_oscillator_function(2)

Hpsi = (bk.get_kinetic_operator() *psi + .5*x**2*psi)

n = 2 psi = bk.basisbank.get_harmonic_oscillator_function(2) Hpsi = (bk.get_kinetic_operator() *psi + .5*x**2*psi)

In [4]:

E_n = psi.bra@Hpsi
print("Energy %i = %f" % (n, E_n))

E_n = psi.bra@Hpsi print("Energy %i = %f" % (n, E_n))

Energy 2 = 2.500000

In [15]:

H = bk.get_kinetic_operator()  #making a dummy operator 

eigenstates = []
eigenvalues = []
for n in range(6):
    eigenstates.append(bk.basisbank.get_harmonic_oscillator_function(n))
    eigenvalues.append(eigenstates[-1].bra@(bk.get_kinetic_operator() *eigenstates[-1] + .5*x**2*eigenstates[-1]))

H.eigenstates = np.array(eigenstates)
H.eigenvalues = np.array(eigenvalues)
#psi.measure(observable = H, repetitions = 100 )

print("Eigenvalues:", H.eigenvalues)
print("Measurement outcome:", psi.measure(observable = H, repetitions = 100 ))

H = bk.get_kinetic_operator() #making a dummy operator eigenstates = [] eigenvalues = [] for n in range(6): eigenstates.append(bk.basisbank.get_harmonic_oscillator_function(n)) eigenvalues.append(eigenstates[-1].bra@(bk.get_kinetic_operator() *eigenstates[-1] + .5*x**2*eigenstates[-1])) H.eigenstates = np.array(eigenstates) H.eigenvalues = np.array(eigenvalues) #psi.measure(observable = H, repetitions = 100 ) print("Eigenvalues:", H.eigenvalues) print("Measurement outcome:", psi.measure(observable = H, repetitions = 100 ))

Eigenvalues: [0.5 1.5 2.5 3.5 4.5 5.5]
[3.15544362e-30 0.00000000e+00 1.00000000e+00 0.00000000e+00
 5.89816045e-34 0.00000000e+00]
stepsize: 0.08333333333333333
Measurement outcome: [2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5
 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5
 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5
 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5
 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5
 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5]

Let's also measure the energy of another distribution in the harmonic oscillator potential:

In [24]:

import sympy as sp

psi_d = bk.ket( sp.exp(-.1*(x-1.5)**2) ) 

psi_d = psi_d*(psi_d.bra@psi_d)**-.5

print("Measurement outcome:", psi_d.measure(observable = H, repetitions = 100 ))

import sympy as sp psi_d = bk.ket( sp.exp(-.1*(x-1.5)**2) ) psi_d = psi_d*(psi_d.bra@psi_d)**-.5 print("Measurement outcome:", psi_d.measure(observable = H, repetitions = 100 ))

[0.51227518 0.0640344  0.16053068 0.04819255 0.07297033 0.03014132]
stepsize: 0.08333333333333333
Measurement outcome: [0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5]