Tutorial covering the concept of geometric phase for a qubit, tutorial available under https://mareknarozniak.com/2021/01/09/qubit-berry-phase/

plotting.py

import matplotlib.pyplot as plt


def plotQubit(angles, refrel, refimag, resrel, resimag, title, x_axis, filename, url=''):
    fig, ax = plt.subplots(1, 1, constrained_layout=True)

    ax.set_title(title)
    x_axis += '\n' + url
    ax.set_xlabel(x_axis)
    ax.set_ylabel('phase')

    ax.plot(angles, refrel, color='tab:blue', label='ref real')
    ax.plot(angles, refimag, color='tab:orange', label='ref imaginary')

    ax.plot(angles, resrel, color='tab:blue', linestyle='--', label='res real')
    ax.plot(angles, resimag, color='tab:orange', linestyle='--', label='res imaginary')

    ax.grid()
    ax.legend()

    # fig.show()
    fig.savefig(filename)

qm.py

import numpy as np

from qutip import Options, mesolve, sigmax, sigmay, sigmaz


def sweepPhaseDown(psi0, theta_min, theta_max, phi, tau):
    def angle(t):
        return (1-(t/tau))*theta_max + (t/tau)*theta_min
    res = 300
    times = np.linspace(0., tau, res)
    opts = Options(store_final_state=True)
    H = [
        [sigmax(), (lambda t, args: np.cos(-phi)*np.sin(angle(t)))],
        [sigmay(), (lambda t, args: np.sin(-phi)*np.sin(angle(t)))],
        [sigmaz(), (lambda t, args: np.cos(angle(t)))]]
    return mesolve(H, psi0, times, options=opts)


def sweepPhaseUp(psi0, theta_min, theta_max, phi, tau):
    def angle(t):
        return (1-(t/tau))*theta_min + (t/tau)*theta_max
    res = 300
    times = np.linspace(0., tau, res)
    opts = Options(store_final_state=True)
    H = [
        [sigmax(), (lambda t, args: np.cos(-phi)*np.sin(angle(t)))],
        [sigmay(), (lambda t, args: np.sin(-phi)*np.sin(angle(t)))],
        [sigmaz(), (lambda t, args: np.cos(angle(t)))]]
    return mesolve(H, psi0, times, options=opts)


def sweepPhaseRound(psi0, theta, phi_min, phi_max, tau):
    def angle(t):
        return (1-(t/tau))*phi_min + (t/tau)*phi_max
    res = 300
    times = np.linspace(0., tau, res)
    opts = Options(store_final_state=True)
    H = [
        [sigmax(), (lambda t, args: np.cos(-angle(t))*np.sin(theta))],
        [sigmay(), (lambda t, args: np.sin(-angle(t))*np.sin(theta))],
        [sigmaz(), (lambda t, args: np.cos(theta))]]
    return mesolve(H, psi0, times, options=opts)

res_qubit.png

res_qubit_2_1.png

res_qubit_2_2.png

run.py

import numpy as np

from qutip import basis

from qm import sweepPhaseDown, sweepPhaseUp, sweepPhaseRound
from plotting import plotQubit


url = 'https://mareknarozniak.com/2021/01/09/qubit-berry-phase/'

tau = 16.*np.pi
res = 60
angles = np.linspace(0., 2., res)

refrels = np.zeros(res)
refimag = np.zeros(res)

resrels = np.zeros(res)
resimag = np.zeros(res)

for i, angle in enumerate(angles):
    theta = np.pi*angle

    psi0 = np.cos(theta/2.)*basis(2, 0)+np.sin(theta/2.)*basis(2, 1)
    result = sweepPhaseRound(psi0, theta, 0., 2.*np.pi, tau)
    psif = result.final_state

    gamma = -2.*np.pi*np.sin(theta/2.)**2.
    dyn = np.pi

    grefe = psif.overlap(psi0)

    resrels[i] = grefe.real
    resimag[i] = grefe.imag

    ref = np.exp(1j*gamma)

    refrels[i] = ref.real
    refimag[i] = ref.imag

title = '$\\phi_f \\in [0, 2\\pi]$ and $\\theta$ fixed'
x_axis = '$\\theta$ [$\\pi$]'
filename = 'res_qubit.png'
plotQubit(angles, refrels, refimag, resrels, resimag, title, x_axis, filename, url=url)

refrels = np.zeros(res)
refimag = np.zeros(res)

resrels = np.zeros(res)
resimag = np.zeros(res)

# fix theta_min, vary phi
for i, angle in enumerate(angles):
    theta_min = 3.*np.pi/4.
    theta_max = 0.

    phi_min = 0.
    phi_max = np.pi*angle

    psi0 = np.cos(theta_max/2.)*basis(2, 0)+np.sin(theta_max/2.)*basis(2, 1)

    result = sweepPhaseDown(psi0, theta_min, theta_max, phi_min, tau)
    psif = result.final_state

    result = sweepPhaseRound(psif, theta_min, phi_min, phi_max, tau)
    psif = result.final_state

    result = sweepPhaseUp(psif, theta_min, theta_max, phi_max, tau)
    psif = result.final_state

    gamma = -(phi_max-phi_min)*np.sin(theta_min/2.)**2.
    grefe = psif.overlap(psi0)

    ref = np.exp(1j*gamma)

    resrels[i] = grefe.real
    resimag[i] = grefe.imag

    refrels[i] = ref.real
    refimag[i] = ref.imag

title = '$\\phi_f \\in [0, 2\\pi]$ and $\\theta = \\frac{3\\pi}{4}$ fixed'
x_axis = '$\\phi_f$ [$\\pi$]'
filename = 'res_qubit_2_1.png'
plotQubit(angles, refrels, refimag, resrels, resimag, title, x_axis, filename, url=url)

refrels = np.zeros(res)
refimag = np.zeros(res)

resrels = np.zeros(res)
resimag = np.zeros(res)

# fix phi_max, vary theta_min
for i, angle in enumerate(angles):
    theta_min = np.pi*angle
    theta_max = 0.

    phi_min = 0.
    phi_max = 3.*np.pi/4.

    psi0 = np.cos(theta_max/2.)*basis(2, 0)+np.sin(theta_max/2.)*basis(2, 1)

    result = sweepPhaseDown(psi0, theta_min, theta_max, phi_min, tau)
    psif = result.final_state

    result = sweepPhaseRound(psif, theta_min, phi_min, phi_max, tau)
    psif = result.final_state

    result = sweepPhaseUp(psif, theta_min, theta_max, phi_max, tau)
    psif = result.final_state

    gamma = -(phi_max-phi_min)*np.sin(theta_min/2.)**2.
    grefe = psif.overlap(psi0)

    ref = np.exp(1j*gamma)

    resrels[i] = grefe.real
    resimag[i] = grefe.imag

    refrels[i] = ref.real
    refimag[i] = ref.imag

title = '$\\theta \\in [0, 2\\pi]$ and $\\phi_f = \\frac{3\\pi}{4}$ fixed'
x_axis = '$\\theta$ [$\\pi$]'
filename = 'res_qubit_2_2.png'
plotQubit(angles, refrels, refimag, resrels, resimag, title, x_axis, filename, url=url)

@marekyggdrasil Is this particular variable to remind the reader that the dynamical phase will always be an even multiple of pi for this quantity to vanish? It seems the simulation is coded to make it cancel before addressing the geometric phase factor gamma. In my case, I think I need to figure out how to account for this dynamic phase in simulation and am unsure how to model it in your code. I envision that the inherent precession of the qubit will contribute to this dynamic phase for all time because the integral for the dynamic phase is a function of energy and time E(t) dt. On average a transmon qubit might have a frequency of approx 4 Ghz given the relationship theta= 2pi/T and f =1/T, my naive assumption is this amount of theta might need to be included in the simulation and not taken out of the simulation until the end of running program I can subtract this quantity out and hopefully be left with the acquired phase as only the geometric phase.

@marekyggdrasil Asking this for clarity, but does the Python function SweepRound correspond to a sweep of the polar angle across the Bloch sphere, where the angle phi remains parked on the equator? I am using your illustration of the Bloch sphere from your website as a visual for describing how this part of the simulation numerically evolves the state. Also, since this evolution is adiabatic, how does one quantify how slow or adiabatic the evolution is? Is there a formulaic relationship for quantum gates in the adiabatic limit using time-dependent quantum mechanics that is assumed here in the simulation?