# This code is part of Qiskit.
#
# (C) Copyright IBM 2022.
#
# This code is licensed under the Apache License, Version 2.0. You may
# obtain a copy of this license in the LICENSE.txt file in the root directory
# of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.
#
# Any modifications or derivative works of this code must retain this
# copyright notice, and modified files need to carry a notice indicating
# that they have been altered from the originals.
"""Bloch trajectory analysis of a single qubit."""
from typing import List, Union, Optional
import lmfit
import numpy as np
import qiskit_experiments.curve_analysis as curve
import qiskit_experiments.data_processing as dp
[docs]
class BlochTrajectoryAnalysis(curve.CurveAnalysis):
r"""A class to analyze a trajectory of the Bloch vector of a single qubit.
# section: fit_model
The following equations are used to approximate the dynamics of the qubit Bloch vector.
.. math::
\begin{align}
F_x(t) &= \frac{1}{\Omega^2} \left(
- p_z p_x + p_z p_x \cos(\Omega t') +
\Omega p_y \sin(\Omega t') \right) + b \tag{1} \\
F_y(t) &= \frac{1}{\Omega^2} \left(
p_z p_y - p_z p_y \cos(\Omega t') -
\Omega p_x \sin(\Omega t') \right) + b \tag{2} \\
F_z(t) &= \frac{1}{\Omega^2} \left(
p_z^2 + (p_x^2 + p_y^2) \cos(\Omega t') \right) + b \tag{3}
\end{align}
where :math:`t' = t + t_{\rm offset}` with :math:`t` is pulse duration to scan
and :math:`t_{\rm offset}` is an extra fit parameter that may represent an edge effect.
Note that this analysis assumes a microwave drive with the flat top Gaussian envelope,
where the amplitude of edges, namely Hamiltonian coefficients of interest,
is time dependent and smaller than the flat top part.
The edge effect indicates the total net interaction that might be induced
by the comparable square drive pulse of duration :math:`t_{\rm offset}`,
which is usually smaller than the actual edge durations.
:math:`p_x, p_y, p_z, b` are fit parameters, and :math:`\Omega = \sqrt{p_x^2+p_y^2+p_z^2}`.
The fit functions :math:`F_x, F_y, F_z` approximate the Pauli expectation
values :math:`\langle \sigma_x (t) \rangle, \langle \sigma_y (t) \rangle,
\langle \sigma_z (t) \rangle` of the target qubit, respectively.
In this analysis, the initial guess is generated by the following equations.
.. math::
p_x &= \omega \cos(\theta) \cos(\phi) \\
p_y &= \omega \cos(\theta) \sin(\phi) \\
p_z &= \omega \sin(\theta)
where :math:`\omega` is the mean oscillation frequency of eigenvalues,
:math:`\theta = \cos^{-1}\sqrt{\frac{\max F_z - \min F_z}{2}}`
and :math:`\phi \in [-\pi, \pi]`.
# section: fit_parameters
defpar t_{\rm off}:
desc: Offset to the pulse duration. For example, if pulse envelope is
a flat-topped Gaussian, two Gaussian edges may become an offset duration.
init_guess: Computed as :math:`N \sqrt{2 \pi} \sigma` where the :math:`N` is number of
pulses and :math:`\sigma` is Gaussian sigma of rising and falling edges.
Note that this implicitly assumes the :class:`~qiskit.pulse.library\
.GaussianSquare` pulse envelope.
bounds: [0, None]
defpar p_x:
desc: Fit parameter of oscillations of the X observable.
init_guess: See fit model section.
bounds: None
defpar p_y:
desc: Fit parameter of oscillations of the Y observable.
init_guess: See fit model section.
bounds: None
defpar p_z:
desc: Fit parameter of oscillations of the Z observable.
init_guess: See fit model section.
bounds: None
defpar b:
desc: Vertical offset of oscillations. This may indicate the state preparation and
measurement error.
init_guess: 0
bounds: None
"""
def __init__(
self,
name: Optional[str] = None,
):
eqr_temps = {
"x": "(-pz * px + pz * px * cos(W * X) + W * py * sin(W * X)) / W**2 + b",
"y": "(pz * py - pz * py * cos(W * X) - W * px * sin(W * X)) / W**2 + b",
"z": "(pz**2 + (px**2 + py**2) * cos(W * X)) / W**2 + b",
}
models = []
for axis, temp_eq in eqr_temps.items():
eq = temp_eq
eq = eq.replace("W", "sqrt(px**2 + py**2 + pz**2)")
eq = eq.replace("X", "(x + t_off)")
models.append(
lmfit.models.ExpressionModel(
expr=eq,
name=axis,
)
)
super().__init__(
models=models,
name=name,
)
@classmethod
def _default_options(cls):
"""Return the default analysis options."""
default_options = super()._default_options()
default_options.data_processor = dp.DataProcessor(
input_key="counts",
data_actions=[dp.Probability("1"), dp.BasisExpectationValue()],
)
default_options.plotter.set_figure_options(
xlabel="Flat top width",
ylabel="Pauli expectation values",
xval_unit="s",
ylim=(-1, 1),
)
default_options.data_subfit_map = {
"x": {"meas_basis": "x"},
"y": {"meas_basis": "y"},
"z": {"meas_basis": "z"},
}
return default_options
def _generate_fit_guesses(
self,
user_opt: curve.FitOptions,
curve_data: curve.ScatterTable,
) -> Union[curve.FitOptions, List[curve.FitOptions]]:
"""Create algorithmic initial fit guess from analysis options and curve data.
Args:
user_opt: Fit options filled with user provided guess and bounds.
curve_data: Formatted data collection to fit.
Returns:
List of fit options that are passed to the fitter function.
"""
fit_options = []
user_opt.bounds.set_if_empty(t_off=(0, np.inf), b=(-1, 1))
user_opt.p0.set_if_empty(b=1e-9)
x_data = curve_data.filter(series="x")
y_data = curve_data.filter(series="y")
z_data = curve_data.filter(series="z")
omega_xyz = []
for data in (x_data, y_data, z_data):
ymin, ymax = np.percentile(data.y, [10, 90])
if ymax - ymin < 0.2:
# oscillation amplitude might be almost zero,
# then exclude from average because of lower SNR
continue
fft_freq = curve.guess.frequency(data.x, data.y)
omega_xyz.append(fft_freq)
if omega_xyz:
omega = 2 * np.pi * np.average(omega_xyz)
else:
omega = 1e-3
zmin, zmax = np.percentile(z_data.y, [10, 90])
theta = np.arccos(np.sqrt((zmax - zmin) / 2))
# The FFT might be up to 1/2 bin off
df = 2 * np.pi / ((z_data.x[1] - z_data.x[0]) * len(z_data.x))
for omega_shifted in [omega, omega - df / 2, omega + df / 2]:
for phi in np.linspace(-np.pi, np.pi, 5):
new_opt = user_opt.copy()
new_opt.p0.set_if_empty(
px=omega_shifted * np.cos(theta) * np.cos(phi),
py=omega_shifted * np.cos(theta) * np.sin(phi),
pz=omega_shifted * np.sin(theta),
)
fit_options.append(new_opt)
if omega < df:
# empirical guess for low frequency case
new_opt = user_opt.copy()
new_opt.p0.set_if_empty(px=omega, py=omega, pz=0)
fit_options.append(new_opt)
return fit_options