feat: implement new optimal control example

tutorials-v5/optimal-control/01-cpo-GRAPE-cnot.md

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+---
+jupyter:
+  jupytext:
+    text_representation:
+      extension: .md
+      format_name: markdown
+      format_version: '1.3'
+      jupytext_version: 1.17.0
+  kernelspec:
+    display_name: qiskit-stable8
+    language: python
+    name: python3
+---
+
+### GRAPE calculation of control fields for cnot implementation
+
+[This is an updated implementation based on the deprecated notebook of GRAPE CNOT implementation by Robert Johansson](https://nbviewer.org/github/qutip/qutip-notebooks/blob/master/examples/control-grape-cnot.ipynb)
+
+```python
+import matplotlib.pyplot as plt
+import numpy as np
+import qutip as qt
+# the library for quantum control
+import qutip_qtrl.pulseoptim as cpo
+```
+
+```python
+# total duration
+T = 2 * np.pi
+# number of time steps
+times = np.linspace(0, T, 500)
+```
+
+```python
+U_0 = qt.operators.identity(4)
+U_target = qt.core.gates.cnot()
+```
+
+### Starting Point
+
+```python
+U_0
+```
+
+### Target Operator
+
+```python
+U_target
+```
+
+```python
+# Drift Hamiltonian
+g = 0
+H_drift = g * (
+    qt.tensor(qt.sigmax(), qt.sigmax()) + qt.tensor(qt.sigmay(), qt.sigmay())
+)
+```
+
+```python
+H_ctrl = [
+    qt.tensor(qt.sigmax(), qt.identity(2)),
+    qt.tensor(qt.sigmay(), qt.identity(2)),
+    qt.tensor(qt.sigmaz(), qt.identity(2)),
+    qt.tensor(qt.identity(2), qt.sigmax()),
+    qt.tensor(qt.identity(2), qt.sigmay()),
+    qt.tensor(qt.identity(2), qt.sigmaz()),
+    qt.tensor(qt.sigmax(), qt.sigmax()),
+    qt.tensor(qt.sigmay(), qt.sigmay()),
+    qt.tensor(qt.sigmaz(), qt.sigmaz()),
+]
+```
+
+```python
+H_labels = [
+    r"$u_{1x}$",
+    r"$u_{1y}$",
+    r"$u_{1z}$",
+    r"$u_{2x}$",
+    r"$u_{2y}$",
+    r"$u_{2z}$",
+    r"$u_{xx}$",
+    r"$u_{yy}$",
+    r"$u_{zz}$",
+]
+```
+
+## GRAPE
+
+```python
+result = cpo.optimize_pulse_unitary(
+    H_drift,
+    H_ctrl,
+    U_0,
+    U_target,
+    num_tslots=500,
+    evo_time=(2 * np.pi),
+    # this attribute is crucial for convergence!!
+    amp_lbound=-(2 * np.pi * 0.05),
+    amp_ubound=(2 * np.pi * 0.05),
+    fid_err_targ=1e-9,
+    max_iter=500,
+    max_wall_time=60,
+    alg="GRAPE",
+    optim_method="FMIN_L_BFGS_B",
+    method_params={
+        "disp": True,
+        "maxiter": 1000,
+    },
+)
+```
+
+```python
+for attr in dir(result):
+    if not attr.startswith("_"):
+        print(f"{attr}: {getattr(result, attr)}")
+
+print(np.shape(result.final_amps))
+```
+
+## Plot control fields for cnot gate in the presense of single-qubit tunnelling
+
+```python
+def plot_control_amplitudes(times, final_amps, labels):
+    num_controls = final_amps.shape[1]
+
+    y_max = 0.1  # Fixed y-axis scale
+    y_min = -0.1
+
+    for i in range(num_controls):
+        fig, ax = plt.subplots(figsize=(8, 3))
+
+        for j in range(num_controls):
+            # Highlight the current control
+            color = "black" if i == j else "gray"
+            alpha = 1.0 if i == j else 0.1
+            ax.plot(
+                times,
+                final_amps[:, j],
+                label=labels[j],
+                color=color,
+                alpha=alpha
+                )
+        ax.set_title(f"Control Fields Highlighting: {labels[i]}")
+        ax.set_xlabel("Time")
+        ax.set_ylabel(labels[i])
+        ax.set_ylim(y_min, y_max)  # Set fixed y-axis limits
+        ax.grid(True)
+        ax.legend()
+        plt.tight_layout()
+        plt.show()
+
+
+plot_control_amplitudes(times, result.final_amps / (2 * np.pi), H_labels)
+```
+
+## Fidelity/overlap
+
+```python
+U_target
+```
+
+```python
+U_f = result.evo_full_final
+U_f.dims = [[2, 2], [2, 2]]
+```
+
+```python
+U_f
+```
+
+```python
+print(f"Fidelity: {qt.process_fidelity(U_f, U_target)}")
+```
+
+## Proceess tomography
+
+
+Quantum Process Tomography (QPT) is a technique used to characterize an unknown quantum operation by reconstructing its process matrix (also called the χ (chi) matrix). This matrix describes how an input quantum state is transformed by the operation.
+
+
+Defines the basis operators 
+{
+𝐼
+,
+𝑋
+,
+𝑌
+,
+𝑍
+}
+for the two-qubit system.
+
+These operators form a complete basis to describe any quantum operation in the Pauli basis.
+
+
+### Ideal cnot gate
+
+```python
+op_basis = [[qt.qeye(2), qt.sigmax(), qt.sigmay(), qt.sigmaz()]] * 2
+op_label = [["i", "x", "y", "z"]] * 2
+```
+
+U_target is the ideal CNOT gate.
+
+qt.to_super(U_target) converts it into superoperator form, which is necessary for QPT.
+
+qt.qpt(U_i_s, op_basis) computes the χ matrix for the ideal gate.
+
+```python
+fig = plt.figure(figsize=(12, 6))
+
+U_i_s = qt.to_super(U_target)
+
+chi = qt.qpt(U_i_s, op_basis)
+
+fig = qt.qpt_plot_combined(chi, op_label, fig=fig, threshold=0.001)
+```
+
+```python
+op_basis = [[qt.qeye(2), qt.sigmax(), qt.sigmay(), qt.sigmaz()]] * 2
+op_label = [["i", "x", "y", "z"]] * 2
+```
+
+```python
+fig = plt.figure(figsize=(12, 6))
+
+U_f_s = qt.to_super(U_f)
+
+chi = qt.qpt(U_f_s, op_basis)
+
+fig = qt.qpt_plot_combined(chi, op_label, fig=fig, threshold=0.01)
+```
+
+## Versions
+
+
+```python
+qt.about()
+```