#!/usr/bin/env python
# coding: utf-8

# # Bloch Sphere
# 
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# 
# ## Single-qubit State
# 
# Unlike classical bits, qubits can be in both the computational basis vector $\left|0\right>$ state and the $\left|1\right>$ state, usually expressed as
# 
# $$\left|\psi\right> = a\left|0\right> + b\left|1\right>$$
# 
# Here $a$ and $b$ are complex numbers. Due to the normalization condition of the quantum state $\left<\psi\middle|\psi\right> = 1$, therefore
# 
# $$\left|a\right|^2 + \left|b\right|^2 = 1$$
# 
# For a two-dimensional Hilbert space, we can make the following mapping of the computational basis vector
# 
# $$
# \left|0\right> =
#     \begin{pmatrix}
#     1 \\
#     0
#     \end{pmatrix},
# \left|1\right> =
#     \begin{pmatrix}
#     0 \\
#     1
#     \end{pmatrix}
# $$
# 
# Thus, any single-qubit state can be expressed as
# 
# $$\left|\psi\right> =
#     \begin{pmatrix}
#     a \\
#     b
#     \end{pmatrix}
# $$
# 
# In the general case, we do not care about the global phase, so we can assume $a=\cos\left(\theta/2\right), b=e^{i\phi}\sin\left(\theta/2\right)$：
# 
# $$\left|\psi\right> = \cos\left(\theta/2\right) \left|0\right> + e^{i\phi}\sin\left(\theta/2\right)\left|1\right>$$
# 
# Here we might as well represent that arbitrary quantum state in the unit ball, as follows, taking $\theta$ and $\phi$ as the elevation and azimuth angles, respectively.
# 
# ![bloch-sphere](https://mindspore-website.obs.cn-north-4.myhuaweicloud.com/website-images/r2.2/docs/mindquantum/docs/source_zh_cn/images/bloch_sphere.png)
# 
# Next we will show how to demonstrate a single-qubit state in MindSpore Quantum and the evolution of the single-qubit state in the form of an animation.
# 
# ## Importing Related Modules

# In[9]:


from mindquantum.core.circuit import Circuit
from mindquantum.core.gates import RX, RZ
from mindquantum.io.display import BlochScene


# In order to dynamically display quantum states in Jupyter Notebook, we need to run the following command.

# In[10]:


# get_ipython().run_line_magic('matplotlib', 'ipympl')


# ## Building Quantum Circuit
# 
# From the above bloch sphere, we can control the elevation angle $\theta$ by the revolving gate `RX` and the azimuth angle $\phi$ by `RZ`. Therefore, we can build the following quantum circuit.

# In[11]:


circ = Circuit()               # Build circuit for preparing an arbitrary single-qubit quantum state
circ += RX('theta').on(0)      # Control elevation via RX gate
circ += RZ('phi').on(0)        # Control azimuth via RZ gate
circ.svg()


# Here we might as well take $\theta=\pi/4, \phi=\pi/4$ and calculate the corresponding quantum state.

# In[12]:


import numpy as np

state1 = circ.get_qs(pr={'theta': np.pi/4, 'phi': np.pi/4})
print(state1)


# ## Displaying the Quantum State
# 
# In MindSpore Quantum, `BlochScene` is the module used to display the Bloch sphere. We can add as many single-qubit states as we want to the `BlochScene` and also animate the evolution of the single-qubit quantum states.

# In[13]:


scene = BlochScene()                       # Create Bloch drawing scene
fig, ax = scene.create_scene()             # Initialize the scene
state_obj1 = scene.add_state(ax, state1)   # Add a quantum state to the scene


# In addition, we can also show the Bloch sphere in dark mode, as follows.

# In[14]:


scene = BlochScene('dark')                       # Create Bloch drawing scene
fig, ax = scene.create_scene()                   # Initialize the scene
state_obj1 = scene.add_state(ax, state1)         # Add a quantum state to the scene


# ## Displaying the Evolution of the Quantum State
# 
# We can also create animations in the Bloch scene when the quantum state is a time-dependent quantum state. Here we may assume that the elevation $\theta$ and azimuthal $\phi$ are time-dependent and we can obtain the quantum states for all time.

# In[20]:


t = np.linspace(0, 10, 500)
all_theta = 4 * np.sin(2 * t)
all_phi = 5 * np.cos(3 * t)
states = []
for theta, phi in zip(all_theta, all_phi):
    states.append(circ.get_qs(pr={'theta': theta, 'phi': phi}))
states = np.array(states)


# In the following, we create a dark Bloch scene and initialize the scene with the first one of the evolved quantum states.

# In[22]:


scene = BlochScene('dark')                          # Create Bloch drawing scene
fig, ax = scene.create_scene()                      # Initialize the scene
state_obj = scene.add_state(ax, states[0])          # Add a quantum state to the scene


# To be able to display dynamically the evolution of quantum states, we create an animated object from the Bloch scene.

# In[23]:


anim = scene.animation(fig, ax, state_obj, states)


# ![bloch-sphere-anim](https://mindspore-website.obs.cn-north-4.myhuaweicloud.com/website-images/r2.2/docs/mindquantum/docs/source_zh_cn/images/bloch_sphere.gif)
# 
# From this, we can see that the quantum state of a single qubit has moved in the Bloch sphere.

# In[1]:


from mindquantum.utils.show_info import InfoTable

InfoTable('mindquantum', 'scipy', 'numpy')