Improving readout performance in superconducting qubits using machine learning

import time
from pathlib import Path

# Choose to run experiments or to use saved data
use_IBM = False
if use_IBM:
    timestr = time.strftime("%Y%m%d-%H%M%S")
    print("Time label for data saved throughout this experiment:" + timestr)
data_folder = Path(
    "resources/superconducting-qubits-improving-measurement-performance/"
)

import copy
import sys

import warnings

warnings.filterwarnings("ignore")

import pickle
from typing import Dict, List, Tuple

import matplotlib.gridspec as gridspec
import matplotlib.pyplot as plt
import numpy as np
from qctrlcore.cluster_metrics import get_best_measurement, intercluster_distance

# Q-CTRL imports
from qctrlcore.iq_discriminators import (
    create_gradient_boosting_discriminator,
    create_linear_discriminator,
    create_random_forest_discriminator,
)
from qctrlvisualizer import get_qctrl_style
from qctrlvisualizer.discriminators import plot_discriminator
from scipy import interpolate, stats
from scipy.optimize import curve_fit
from sklearn.model_selection import train_test_split

plt.style.use(get_qctrl_style())

# Q-CTRL auxiliary functions
def training_test_data(results, test_sample_size):
    """Randomly separate experiment results in a training and testing set for calibration and use of discriminators."""

    # Copy results object into a training object and a test object
    training_res = copy.deepcopy(results)
    test_res = copy.deepcopy(results)

    # Randomly select part of the results of for training/test of the discriminator
    (
        training_res.results[0].data.memory,
        test_res.results[0].data.memory,
    ) = train_test_split(results.results[0].data.memory, test_size=test_sample_size)
    (
        training_res.results[1].data.memory,
        test_res.results[1].data.memory,
    ) = train_test_split(results.results[1].data.memory, test_size=test_sample_size)

    return training_res, test_res


def calculate_avg_err(rabi_data):
    """Calculate average value and standard deviation of Rabi data."""
    rabi_data_avg = []
    rabi_data_err = []

    # Find average and standard deviation of Rabi data
    for i in range(len(rabi_data[0])):
        rabi_temp = []
        for j in range(len(rabi_data)):
            rabi_temp.append(rabi_data[j][i])

        # Calculate average of each data point
        rabi_data_avg.append(np.average(rabi_temp))

        # Calculate standard deviation
        rabi_data_err.append(
            np.sqrt(
                np.sum([(rabi_data_avg[i] - rabi) ** 2 for rabi in rabi_temp])
                / len(rabi_temp)
            )
        )

    return rabi_data_avg, rabi_data_err


def error_gamma(gamma, err_alpha, err_beta, err_f, alpha, beta):
    """Calculate error in the MLE estimation."""
    return (1 / (alpha - beta)) * np.sqrt(
        gamma ** 2 * err_alpha ** 2 + (gamma - 1) ** 2 * err_beta ** 2 + err_f ** 2
    )


def visibility(data):
    """Calculate the visibility of a sinusoidal signal."""
    I_max = np.max(data)
    I_min = np.min(data)
    return round((I_max - I_min) / (I_max + I_min), 2)


def ground_excited_experiment(qubit, backend, num_shots, measurement_schedule):
    """Run an experiment where the ground state is prepared and measured and the excited state is prepared and measured."""
    # Backend's default settings
    backend_defaults = backend.defaults()

    # Define default pi-pulse
    inst_sched_map = backend_defaults.instruction_schedule_map
    pi_default = inst_sched_map.get("x", qubits=backend_config.meas_map[meas_map_idx])

    # Create two schedules
    # Ground state schedule
    gnd_schedule = pulse.Schedule(name="cal_00")
    gnd_schedule += measurement_schedule

    # Excited state schedule
    exc_schedule = pulse.Schedule(name="cal_11")
    exc_schedule += pi_default
    exc_schedule += measurement_schedule << exc_schedule.duration

    # Assemble schedules into an experiment
    gnd_exc_program = assemble(
        [gnd_schedule, exc_schedule],
        backend=backend,
        meas_level=1,
        meas_return="single",
        shots=num_shots,
        schedule_los=[{drive_chan: backend_defaults.qubit_freq_est[qubit]}] * 2,
    )

    # Run
    job = backend.run(gnd_exc_program)
    job_monitor(job)
    gnd_exc_results = job.result(timeout=120)

    return gnd_exc_results


def train_discriminator(
    gnd_exc_results, test_sample, discriminator=create_gradient_boosting_discriminator
):
    """Train a given discriminator to recognize I/Q data corresponding to excited and ground state using the results obtained in a ground/excited state experiment."""

    # Calibrate discriminator on gnd/exc state experiment
    gnd_exc_training_results, gnd_exc_test_results = training_test_data(
        gnd_exc_results, test_sample
    )

    trained_discriminator = []
    trained_discriminator.append(
        discriminator(gnd_exc_training_results, [qubit], ["0", "1"])
    )

    # Collect ground state and excited state data to discriminate
    gnd_data = [data[0] for data in gnd_exc_test_results.results[0].data.memory]
    exc_data = [data[0] for data in gnd_exc_test_results.results[1].data.memory]

    # Store measurement result arrays for inter-cluster distance calculation in post-processing
    gnd_exc_data = np.concatenate((np.array(gnd_data), np.array(exc_data)))

    # Discriminate ground state data
    count_array = list(
        map(int, trained_discriminator[0].discriminate(gnd_data))
    )  # Turn the string of classified results into a list of integers 0 or 1
    probability_exc_gnd = np.sum(count_array) / len(
        count_array
    )  # To find probability of excited state, sum the elements of the count list (sum all the ones) and normalize by the number of elements
    probability_gnd_gnd = (
        1 - probability_exc_gnd
    )  # Find the probability of ground state

    # Discriminate excited state data
    count_array = list(map(int, trained_discriminator[0].discriminate(exc_data)))
    probability_exc_exc = np.sum(count_array) / len(count_array)
    probability_gnd_exc = 1 - probability_exc_exc

    return probability_exc_gnd, probability_exc_exc, gnd_exc_data, trained_discriminator


def rabi_experiment(
    qubit, backend, num_shots, measurement_schedule, measurement_setting
):
    """Run a Rabi experiment."""
    # Get qubit frequency
    qubit = 0
    backend.properties(refresh=True)
    qubit_frequency_updated = backend.properties().qubit_property(qubit, "frequency")[0]

    # Drive and measurement channels
    inst_sched_map = backend_defaults.instruction_schedule_map
    pi_default = inst_sched_map.get("x", qubits=backend_config.meas_map[qubit])

    # The minimum duration of a pulse in the armonk backend is 64*dt
    num_time_points = 50  # Number of points in the Rabi experiment
    pulse_amp = 0.4
    pulse_times = np.array(
        [
            64
            + np.arange(
                0,
                get_closest_multiple_of_16(16 * 2 * num_time_points),
                get_closest_multiple_of_16(16 * 2),
            )
        ]
    )

    rabi_schedules = []
    for integer_duration in pulse_times[0]:
        waveform = np.ones([integer_duration]) * pulse_amp
        this_schedule = pulse.Schedule(name=f"I pulse: duration = {integer_duration}")
        this_schedule += pulse.Play(pulse.Waveform(waveform), drive_chan)
        this_schedule += measurement_schedule << this_schedule.duration
        rabi_schedules.append(this_schedule)

    if measurement_setting == "calibrated":
        # Create two schedules
        # Excited state schedule
        exc_schedule = pulse.Schedule(name="cal_11")
        exc_schedule += pi_default
        exc_schedule += measurement_schedule

        # Ground state schedule
        gnd_schedule = pulse.Schedule(name="cal_00")
        gnd_schedule += measurement_schedule

        # Add calibration schedules to Rabi schedules
        rabi_schedules = [gnd_schedule] + [exc_schedule] + rabi_schedules

        # Assemble schedules in experiment
        rabi_experiment_program = assemble(
            rabi_schedules,
            backend=backend,
            meas_level=1,
            meas_return="single",
            shots=num_shots,
            schedule_los=[{drive_chan: qubit_frequency_updated}]
            * (len(pulse_times[0]) + 2),
        )
        # Run
        job = backend.run(rabi_experiment_program)
        job_monitor(job)
        rabi_results = job.result(timeout=120)
        return rabi_results

    elif measurement_setting == "default":
        # Assemble schedules in experiment
        rabi_experiment_program = assemble(
            rabi_schedules,
            backend=backend,
            meas_level=2,
            meas_return="avg",
            shots=num_shots,
            schedule_los=[{drive_chan: qubit_frequency_updated}]
            * (len(pulse_times[0])),
        )

        # Run
        job = backend.run(rabi_experiment_program)
        job_monitor(job)
        rabi_results = job.result(timeout=120)
        return rabi_results

    else:
        sys.exit("Select correct measurement setting: 'default' or 'calibrated' ")


# Parameters
GHz = 1.0e9  # Gigahertz
MHz = 1.0e6  # Megahertz
us = 1.0e-6  # Microseconds
ns = 1.0e-9  # Nanoseconds
runs = 16  # Number of runs for collecting statistics
num_shots = 1024  # Number of times each program is repeated
num_shots_with_training = 8 * num_shots
test_sample_size = 0.167  # Size of sample for testing of discriminators
num_time_points = 50  # Number of points in the Rabi experiment
colors = {"Calibrated": "#680CE9", "Default": "#000000"}

if use_IBM:
    # IBM-Q imports
    import qiskit.pulse as pulse
    import qiskit.pulse.pulse_lib as pulse_lib
    from qiskit import IBMQ
    from qiskit.compiler import assemble
    from qiskit.tools.jupyter import *
    from qiskit.tools.monitor import job_monitor

    # IBM credentials and backend selection
    IBMQ.enable_account("YOUR TOKEN HERE")
    provider = IBMQ.get_provider(hub="ibm-q", group="open", project="main")
    backend = provider.get_backend("ibmq_armonk")
    backend_defaults = backend.defaults()
    backend_config = backend.configuration()
    assert backend_config.open_pulse, "Backend doesn't support OpenPulse"

    # Backend properties
    qubit = 0
    qubit_freq_est = backend_defaults.qubit_freq_est[qubit]
    dt = backend_config.dt
    print(f"Qubit: {qubit}")
    print(f"Hardware sampling time: {dt/ns} ns")
    print(f"Qubit frequency estimate: {qubit_freq_est/GHz} GHz")

    # Set command channels
    drive_chan = pulse.DriveChannel(qubit)

    # Set measurement channels
    meas_chan = pulse.MeasureChannel(qubit)
    acq_chan = pulse.AcquireChannel(qubit)

    # Measurement map
    meas_map_idx = None
    for i, measure_group in enumerate(backend_config.meas_map):
        if qubit in measure_group:
            meas_map_idx = i
            break
    assert meas_map_idx is not None, f"Couldn't find qubit {qubit} in the meas_map!"

else:
    qubit = 0
    dt = 2 / 9 * ns

# IBM-Q auxiliary functions
def get_closest_multiple_of_16(num):
    return int(num + 8) - (int(num + 8) % 16)


def fit_function(x_values, y_values, function, init_params):
    fitparams, conv = curve_fit(function, x_values, y_values, init_params)
    y_fit = function(x_values, *fitparams)
    return fitparams, y_fit

if use_IBM:
    # Use default measurement settings
    inst_sched_map = backend_defaults.instruction_schedule_map
    default_measure_schedule = inst_sched_map.get(
        "measure", qubits=backend_config.meas_map[meas_map_idx]
    )

    # Run experiment
    gnd_exc_experiment_result = ground_excited_experiment(
        qubit, backend, num_shots, default_measure_schedule
    )

    # Save data
    filename = "gnd_exc_experiment_result" + timestr
    outfile = open(filename, "wb")
    pickle.dump(gnd_exc_experiment_result, outfile)
    outfile.close()
else:
    # Load data
    filename = data_folder / "gnd_exc_experiment_result"
    infile = open(filename, "rb")
    gnd_exc_experiment_result = pickle.load(infile)
    infile.close()

# Discriminator used to classify data
discriminator_list = [
    create_random_forest_discriminator,
    create_gradient_boosting_discriminator,
    create_linear_discriminator,
]

discriminators = []

# Classify data
for disc in discriminator_list:
    discriminators.append(disc(gnd_exc_experiment_result, [qubit], ["0", "1"]))

# Plot result of the classification
gs = gridspec.GridSpec(1, 3, wspace=0.1)
fig = plt.figure()
fig.set_figheight(5)
fig.set_figwidth(15)
fig.suptitle("Discriminators", fontsize=16, y=1.0)
titles = ["RandomForest", "Gradient boosting", "Linear"]
for idx, disc in enumerate(discriminators):
    ax = fig.add_subplot(gs[idx])
    ax.tick_params(labelsize=14)
    plot_discriminator(
        disc, ax, flag_misclassified=True, show_boundary=True, title=False
    )
    ax.xaxis.label.set_fontsize(14)
    if idx == 0:
        ax.yaxis.label.set_fontsize(14)
    else:
        ax.yaxis.set_ticklabels([])
        ax.set_ylabel("")
    ax.set_title(titles[idx], fontsize=14)
plt.show()

if use_IBM:
    # Parameters for amplitude scan
    measurement_amp_min = 0.05
    measurement_amp_max = 1
    measurement_amp_steps = 10
    measurement_amp_array = np.linspace(
        measurement_amp_min, measurement_amp_max, measurement_amp_steps
    )

    # Parameters for duration scan
    measurement_samples_min = 1
    measurement_samples_max = 4
    measurement_samples_steps = 15
    measurement_samples_array = np.linspace(
        measurement_samples_min, measurement_samples_max, measurement_samples_steps
    )

    measurement_data = []
    silhouette_list = []

    # Iterate over amplitude values
    for measurement_amp in measurement_amp_array:
        print("\nAMPLITUDE:", measurement_amp, " a.u.")

        silhouette_time_list = []
        # Iterate over duration values
        for measurement_samples_us in measurement_samples_array:
            ### Construct the measurement pulse
            # Measurement pulse parameters
            measurement_sigma_us = (
                0.5  # Width of the gaussian part of the rise and fall in us
            )
            measurement_risefall_us = 0.1  # Truncating parameter: how many samples to dedicate to the risefall

            # Convert to machine format
            measurement_sigma = get_closest_multiple_of_16(
                measurement_sigma_us * 1e-6 / dt
            )
            measurement_risefall = get_closest_multiple_of_16(
                measurement_risefall_us * 1e-6 / dt
            )
            measurement_samples = get_closest_multiple_of_16(
                measurement_samples_us * 1e-6 / dt
            )
            print("DURATION:", measurement_samples_us, " a.u.")

            # Define measurement pulse
            measurement_pulse = pulse_lib.gaussian_square(
                duration=measurement_samples,
                sigma=measurement_sigma,
                amp=measurement_amp,
                risefall=measurement_risefall,
                name="measurement_pulse",
            )

            # Import backend configurations
            backend_config = backend.configuration()

            # Set measurement channels
            meas_chan = pulse.MeasureChannel(qubit)
            acq_chan = pulse.AcquireChannel(qubit)

            # Add a measurement stimulus on the measure channel pulse to trigger readout
            measure_schedule = pulse.Play(measurement_pulse, meas_chan)

            # Trigger data acquisition, and store measured values into respective memory slots
            measure_schedule += pulse.Acquire(
                measurement_pulse.duration,
                pulse.AcquireChannel(backend_config.meas_map[meas_map_idx][0]),
                pulse.MemorySlot(backend_config.meas_map[meas_map_idx][0]),
            )

            # Run 0-1 state discrimination experiment
            gnd_exc_experiment_result = ground_excited_experiment(
                qubit, backend, num_shots_with_training, measure_schedule
            )

            # Train and classify data with Q-CTRL's discriminator
            gnd_exc_results = train_discriminator(
                gnd_exc_experiment_result, test_sample_size
            )[2]

            # Store results
            measurement_data.append(gnd_exc_results)
            silhouette_time_list.append(intercluster_distance(gnd_exc_results))

        silhouette_list.append(silhouette_time_list)

    # Save data
    filename = "measurement_amplitude_array" + timestr
    outfile = open(filename, "wb")
    pickle.dump(measurement_amp_array, outfile)
    outfile.close()
    filename = "measurement_samples_array" + timestr
    outfile = open(filename, "wb")
    pickle.dump(measurement_samples_array, outfile)
    outfile.close()
    filename = "measurements" + timestr
    outfile = open(filename, "wb")
    pickle.dump(measurement_data, outfile)
    outfile.close()
    filename = "silhouette_list" + timestr
    outfile = open(filename, "wb")
    pickle.dump(silhouette_list, outfile)
    outfile.close()
else:
    # Load data
    filename = data_folder / "measurement_amplitude_array"
    infile = open(filename, "rb")
    measurement_amp_array = pickle.load(infile)
    infile.close()
    filename = data_folder / "measurement_samples_array"
    infile = open(filename, "rb")
    measurement_samples_array = pickle.load(infile)
    infile.close()
    filename = data_folder / "measurements"
    infile = open(filename, "rb")
    measurement_data = pickle.load(infile)
    infile.close()
    filename = data_folder / "silhouette_list"
    infile = open(filename, "rb")
    silhouette_list = pickle.load(infile)
    infile.close()

silhouette_scores_example = []
discriminated_data = []
idx_measurements = [40, 70, 143]  # Example of measurement results

for idx_measurement in idx_measurements:
    # Find silhouette score of a given measurement data
    x_idx = int(idx_measurement % len(measurement_samples_array))
    y_idx = int(idx_measurement / len(measurement_samples_array))

    # Make Result object from measurement data
    silhouette_score_result = copy.deepcopy(gnd_exc_experiment_result)
    data_idx = int(len(gnd_exc_experiment_result.results[0].data.memory))
    exc_data_idx = int(len(measurement_data[idx_measurement]) / 2)
    silhouette_score_result.results[0].data.memory = [
        [sample] for sample in (measurement_data[idx_measurement][:data_idx]).tolist()
    ]
    silhouette_score_result.results[1].data.memory = [
        [sample]
        for sample in (
            measurement_data[idx_measurement][exc_data_idx : exc_data_idx + data_idx]
        ).tolist()
    ]

    # Classify data
    discriminated_data.append(
        create_gradient_boosting_discriminator(
            silhouette_score_result, [qubit], ["0", "1"]
        )
    )

    # Get silhouette score
    silhouette_scores_example.append(round(silhouette_list[y_idx][x_idx], 3))

# Plot result of the classification
gs = gridspec.GridSpec(1, 3, wspace=0.1)
fig = plt.figure()
fig.set_figheight(5)
fig.set_figwidth(15)
fig.suptitle("Discriminator performance", fontsize=16, y=1.0)
for idx, data in enumerate(discriminated_data):
    ax = fig.add_subplot(gs[idx])
    ax.tick_params(labelsize=14)
    plot_discriminator(
        data, ax, flag_misclassified=True, show_boundary=True, title=False
    )
    ax.xaxis.label.set_fontsize(14)
    if idx == 0:
        ax.yaxis.label.set_fontsize(14)
    else:
        ax.yaxis.set_ticklabels([])
        ax.set_ylabel("")
    ax.set_title("Silhouette score:%s" % silhouette_scores_example[idx], fontsize=14)
plt.show()

# Evaluate all measurements to determine which gives the best silhoutte score
gnd_exc_data, idx = (
    get_best_measurement(measurement_data)["measurement"],
    get_best_measurement(measurement_data)["index"],
)

# Find duration and amplitude of best measurement results
dur_idx = int(idx % len(measurement_samples_array))
amp_idx = int(idx / len(measurement_samples_array))

measurement_duration = measurement_samples_array[dur_idx]
measurement_amp = measurement_amp_array[amp_idx]

print("Best duration: ", measurement_duration)
print("Best amplitude: ", measurement_amp)

# Plot landscape of probability of excited state
fig, ax = plt.subplots(1, constrained_layout=True)
fig.set_figheight(5)
fig.set_figwidth(14)

Y = measurement_amp_array
X = measurement_samples_array
X, Y = np.meshgrid(X, Y)
Z0 = silhouette_list

cs = ax.contourf(
    X,
    Y,
    Z0,
    len(measurement_amp_array) * len(measurement_samples_array),
    cmap="coolwarm",
    vmin=0.5,
    vmax=0.8,
)

ax.set_ylabel("Amplitude", fontsize=14, rotation=90)
ax.set_xlabel(r"Time [$\mu$s]", fontsize=14, rotation=0)
ax.set_title("Silhouette score", fontsize=16)
ax.tick_params(axis="both", which="major", labelsize=14)

cb_ax = fig.add_axes([1.02, 0.12, 0.02, 0.8])
cbar1 = fig.colorbar(cs, cax=cb_ax, orientation="vertical")
cbar1.ax.tick_params(labelsize=14)

# Mark the value of the default setting and the best value from the scan
if use_IBM:
    backend_defaults = backend.defaults()
    backend_config = backend.configuration()
    inst_sched_map = backend_defaults.instruction_schedule_map
    default_measure_schedule = inst_sched_map.get(
        "measure", qubits=backend_config.meas_map[meas_map_idx]
    )  # measurement pulse parameters for default measurement
    default_amplitude = np.real(default_measure_schedule.instructions[1][1].pulse.amp)
    default_duration = default_measure_schedule.duration * dt / us
else:
    default_amplitude = 0.605
    default_duration = 3.556

ax.autoscale(False)  # Avoid scatterplot changing the plot limits
ax.scatter(
    default_duration,
    default_amplitude,
    color=colors["Default"],
    marker="X",
    s=200,
    zorder=1,
    label="Default",
)
ax.scatter(
    measurement_duration,
    measurement_amp,
    color=colors["Calibrated"],
    marker="X",
    s=200,
    zorder=1,
    label="Best Silhouette",
)
ax.legend(loc="lower left", fontsize=14)

plt.show()

if use_IBM:
    # Measurement pulse parameters
    measurement_duration_us = measurement_samples_array[dur_idx]
    measurement_amp = measurement_amp_array[amp_idx]
    measurement_sigma_us = 0.5  # Width of the gaussian part of the rise and fall in us
    measurement_risefall_us = (
        0.1  # Truncating parameter: how many samples to dedicate to the risefall
    )

    # Convert to machine format
    measurement_sigma = get_closest_multiple_of_16(measurement_sigma_us * 1e-6 / dt)
    measurement_risefall = get_closest_multiple_of_16(
        measurement_risefall_us * 1e-6 / dt
    )
    measurement_duration = get_closest_multiple_of_16(
        measurement_duration_us * 1e-6 / dt
    )

    # Define measurement pulse
    calibrated_measurement_pulse = pulse_lib.gaussian_square(
        duration=measurement_duration,
        sigma=measurement_sigma,
        amp=measurement_amp,
        risefall=measurement_risefall,
        name="measurement_pulse",
    )

    # Add a measurement stimulus on the measure channel pulse to trigger readout
    calibrated_measure_schedule = pulse.Play(calibrated_measurement_pulse, meas_chan)

    # Trigger data acquisition, and store measured values into respective memory slots
    calibrated_measure_schedule += pulse.Acquire(
        calibrated_measurement_pulse.duration,
        pulse.AcquireChannel(backend_config.meas_map[meas_map_idx][0]),
        pulse.MemorySlot(backend_config.meas_map[meas_map_idx][0]),
    )

    # Run Rabi experiment with calibrated measurement pulse
    rabi_calibrated_result_list = [
        rabi_experiment(
            qubit, backend, num_shots, calibrated_measure_schedule, "calibrated"
        )
        for i in range(runs)
    ]

    # Discriminate Rabi data
    rabi_discriminated_list = []
    calibrated_probability_gnd = []
    calibrated_probability_exc = []
    for k in range(len(rabi_calibrated_result_list)):
        # Train the discriminator with a gnd/exc experiment done before the Rabi experiment and collect probabilities
        prob_exc_gnd, prob_exc_exc, data, trained_disc = train_discriminator(
            rabi_calibrated_result_list[k], test_sample_size
        )
        calibrated_probability_gnd.append(1 - prob_exc_gnd)
        calibrated_probability_exc.append(prob_exc_exc)

        # Discriminate each experiment
        rabi_discriminated = []
        for j in range(2, len(rabi_calibrated_result_list[0].results)):
            rabi = [
                data[0]
                for data in rabi_calibrated_result_list[k].results[j].data.memory
            ]
            count_array = list(
                map(int, trained_disc[0].discriminate(rabi))
            )  # Turn the string of classified results into a list of integers 0 or 1
            probability_exc = np.sum(count_array) / len(
                count_array
            )  # To find probability of excited state, sum the elements of the count list (sum all the ones) and normalize by the number of elements
            probability_gnd = (
                1 - probability_exc
            )  # Find the probability of ground state

            rabi_discriminated.append(probability_gnd)
        rabi_discriminated_list.append(rabi_discriminated)

    # Run Rabi experiment with default measurement pulse
    rabi_default_result_list = [
        rabi_experiment(qubit, backend, num_shots, default_measure_schedule, "default")
        for i in range(runs)
    ]

    # Discriminate Rabi data
    rabi_default_list = []
    for i in range(len(rabi_default_result_list)):
        rabi_discriminated = []
        for j in range(len(rabi_default_result_list[0].results)):
            rabi_discriminated.append(
                rabi_default_result_list[i].to_dict()["results"][j]["data"]["counts"][
                    "0x0"
                ]
                / num_shots
            )
        rabi_default_list.append(rabi_discriminated)

    # Save data
    filename = "rabi_discriminated_list" + timestr
    outfile = open(filename, "wb")
    pickle.dump(rabi_discriminated_list, outfile)
    outfile.close()
    filename = "probability_ground_calibrated" + timestr
    outfile = open(filename, "wb")
    pickle.dump(calibrated_probability_gnd, outfile)
    outfile.close()
    filename = "probability_excited_calibrated" + timestr
    outfile = open(filename, "wb")
    pickle.dump(calibrated_probability_exc, outfile)
    outfile.close()
    filename = "rabi_default_list" + timestr
    outfile = open(filename, "wb")
    pickle.dump(rabi_default_list, outfile)
    outfile.close()
else:
    # Load data
    filename = data_folder / "rabi_discriminated_list"
    infile = open(filename, "rb")
    rabi_discriminated_list = pickle.load(infile)
    infile.close()
    filename = data_folder / "probability_ground_calibrated"
    infile = open(filename, "rb")
    calibrated_probability_gnd = pickle.load(infile)
    infile.close()
    filename = data_folder / "probability_excited_calibrated"
    infile = open(filename, "rb")
    calibrated_probability_exc = pickle.load(infile)
    infile.close()
    filename = data_folder / "rabi_default_list"
    infile = open(filename, "rb")
    rabi_default_list = pickle.load(infile)
    infile.close()

# Calculate average Rabi data and deviation, excluding outliers
rabi_cal_avg, rabi_cal_err = calculate_avg_err(rabi_discriminated_list)
rabi_def_avg, rabi_def_err = calculate_avg_err(rabi_default_list)

# Drive duration values
pulse_times = np.array(
    [
        64
        + np.arange(
            0,
            get_closest_multiple_of_16(16 * 2 * num_time_points),
            get_closest_multiple_of_16(16 * 2),
        )
    ]
)
time_array = pulse_times[0] * dt / us


# Fit cosine function to the data
fit_params_def, y_fit_def = fit_function(
    time_array,
    rabi_def_avg,
    lambda x, A, B, drive_period, phi: (
        A * np.cos(2 * np.pi * x / drive_period - phi) + B
    ),
    [0.5, 0.5, 3 * 10 ** -1, 0],
)

# Fit a sinusoind to the data
fit_params_cal, y_fit_cal = fit_function(
    time_array,
    rabi_cal_avg,
    lambda x, A, B, drive_period, phi: (
        A * np.cos(2 * np.pi * x / drive_period - phi) + B
    ),
    [0.5, 0.5, 3 * 10 ** -1, 0],
)
# Plot
fig, ax = plt.subplots()
fig.set_figheight(6)
fig.set_figwidth(10)
ax.plot(time_array, y_fit_def, color=colors["Default"], linewidth=2)
ax.errorbar(
    time_array,
    rabi_def_avg,
    yerr=rabi_def_err,
    color=colors["Default"],
    fmt=".",
    label="Default",
    markersize=6,
)

ax.plot(time_array, y_fit_cal, color=colors["Calibrated"], linewidth=2)
ax.errorbar(
    time_array,
    rabi_cal_avg,
    yerr=rabi_cal_err,
    color=colors["Calibrated"],
    fmt=".",
    label="Calibrated",
    markersize=6,
)

plt.xlabel("Time [$\mu$s]", fontsize=14)
plt.ylabel(r"Probability of $\vert 0 \rangle$ ", fontsize=14)
plt.tick_params(axis="both", which="major", labelsize=14)
plt.legend(fontsize=14)
plt.show()

print(
    "The visibility of the Rabi oscillations obtained with the default setting is: ",
    visibility(y_fit_def),
)
print(
    "The visibility of the Rabi oscillations obtained with the optimal setting is: ",
    visibility(y_fit_cal),
)

# Find average values of \alpha and \beta and their deviation
avg_probability_gnd = np.average(calibrated_probability_gnd)
deviation_probability_gnd = np.sqrt(
    np.sum([(avg_probability_gnd - prob) ** 2 for prob in calibrated_probability_gnd])
    / len(calibrated_probability_gnd)
)
avg_probability_exc = np.average(calibrated_probability_exc)
deviation_probability_exc = np.sqrt(
    np.sum([(avg_probability_exc - prob) ** 2 for prob in calibrated_probability_exc])
    / len(calibrated_probability_exc)
)

# Written in terms of the variable of the theoretical model
alpha = avg_probability_gnd
beta = 1 - avg_probability_exc

# Calculate estimate of probability of ground state
gamma_list = []
for i in range(num_time_points):
    gamma_list.append(((1 - beta) - (1 - rabi_cal_avg[i])) / (alpha - beta))

# Calculate the error on the MLE estimate
err_gamma = []
for i, gamma in enumerate(gamma_list):
    err_gamma.append(
        error_gamma(
            gamma,
            deviation_probability_gnd,
            deviation_probability_exc,
            np.sqrt((rabi_cal_avg[i] * (1 - rabi_cal_avg[i])) / (runs * num_shots)),
            alpha,
            beta,
        )
    )

# Fit a sinusoind to the MLE data
fit_params_gamma, y_fit_gamma = fit_function(
    time_array,
    gamma_list,
    lambda x, A, B, drive_period, phi: (
        A * np.cos(2 * np.pi * x / drive_period - phi) + B
    ),
    [0.5, 0.5, 3 * 10 ** -1, 0],
)
# Plot
fig, ax = plt.subplots()
fig.set_figheight(6)
fig.set_figwidth(10)
ax.plot(time_array, y_fit_def, color=colors["Default"], linewidth=2)
ax.errorbar(
    time_array,
    rabi_def_avg,
    yerr=rabi_def_err,
    color=colors["Default"],
    fmt=".",
    label="Default",
    markersize=6,
)

ax.plot(time_array, y_fit_gamma, color=colors["Calibrated"], linewidth=2)
ax.errorbar(
    time_array,
    gamma_list,
    yerr=err_gamma,
    color=colors["Calibrated"],
    fmt=".",
    label="Calibrated",
    markersize=6,
)

plt.xlabel("Time [$\mu$s]", fontsize=14)
plt.ylabel(r"Probability of $\vert 0 \rangle$ ", fontsize=14)
plt.tick_params(axis="both", which="major", labelsize=14)
plt.legend(fontsize=14)
plt.show()

print(
    "The visibility of the Rabi oscillations obtained with the default settings is: ",
    visibility(y_fit_def),
)
print(
    "The visibility of the Rabi oscillations obtained with the optimal settings and MLE is: ",
    visibility(y_fit_gamma),
)