Chi

目录

• 简介
• 在 QPanda3 量子信息模块中
  • 构造 Chi 对象
    • 从 Choi 构造
    • 从 Chi 构造
    • 从 SuperOp 构造
    • 从 Kraus 构造
    • 从 PTM 构造
  • 获取内部数据
    • 输入与输出维度
  • 量子态的演化
    • 对密度矩阵的演化
    • 对态矢量的演化
  • 布尔函数
    • 判等

上一章: SuperOp

下一章: PTM

---

简介

量子通道的 Chi 矩阵表示。

\(n\) 量子比特的量子通道 \(\mathcal{E}\) 的 Chi 矩阵表示为矩阵 χ，使得密度矩阵 \(\rho\) 的演化遵循以下公式：

\[\mathcal{E}(\rho)=\frac{1}{2^{n}} \sum_{i, j} \chi_{i, j} P_{i} \rho P_{j} \]

其中

\[\left[P_{0}, P_{1}, \ldots, P_{4^{n}-1}\right] \]

是按照字典序排列的 \(n\) 量子比特泡利基。Chi 矩阵与 Choi 表示之间通过 Choi 矩阵转换到泡利基得到。此外，定义中的

\[\frac{1}{2^{n}} \]

是一个归一化因子，用于缩放泡利基使其正交归一化。

请参考已有文献：C.J. Wood, J.D. Biamonte, D.G. Cory, Tensor networks and graphical calculus for open quantum systems, Quant. Inf. Comp. 15, 0579-0811 (2015).arXiv:1111.6950 [quant-ph]。

在 QPanda3 量子信息模块中

QPanda3 使用 Chi 类实现对 Chi 矩阵的抽象。

构造 Chi 对象

API文档链接

从 Choi 构造

从 Choi 对象生成一个 Chi 对象。请参考 Choi。

Python

 
from pyqpanda3.quantum_info import Kraus,Choi,Chi
import numpy as np
 
def test_construct_from_choi():
    # prepare Choi object
    # preapre Kraus object
    # Prepare two one-dimensional array consisting of a two-dimensional matrix
    K0 = np.array([[0, 1], [1+0.j, 0]])
    K1 = np.array([[1, 0], [0+0.j, -1]])
    K2 = np.array([[0+0.j, -1.j], [1.j, 0]])
 
    Ks1 = [K0, K1, K2]
    Ks2 = [K1]
    # Constructing a Kraus object
    Kra3 = Kraus(Ks1, Ks2)
    # Constructing a Choi object from a Kraus object
    choi = Choi(Kra3)
    
 
    # Constructing another SuperOp object from a Choi object
    chi = Chi(choi)
    # Pring the Chi object
    print("chi:",chi)
 
if __name__ == "__main__":
    test_construct_from_choi()
 

执行结果如下所示：

chi: [[0.+0.j 0.+0.j 0.+0.j 0.+0.j]  
 [0.+0.j 2.+0.j 0.+0.j 0.+0.j]  
 [0.+0.j 0.+0.j 2.+0.j 0.+0.j]  
 [0.+0.j 0.+0.j 0.+0.j 2.+0.j]]  

从 Chi 构造

从一个Chi对象生成另外一个 Chi 对象。

Python

 
from pyqpanda3.quantum_info import Kraus,Choi,Chi
import numpy as np
 
def test_construct_from_chi():
    # prepare Chi object
    # preapre Kraus object
    # Prepare two one-dimensional array consisting of a two-dimensional matrix
    K0 = np.array([[0, 1], [1+0.j, 0]])
    K1 = np.array([[1, 0], [0+0.j, -1]])
    K2 = np.array([[0+0.j, -1.j], [1.j, 0]])
 
    Ks1 = [K0, K1, K2]
    Ks2 = [K1]
    # Constructing a Kraus object
    Kra3 = Kraus(Ks1, Ks2)
    # Constructing a Chi object from a Kraus object
    chi = Chi(Kra3)
    
 
    # Constructing another Chi object from a Chi object
    chi2 = Chi(chi)
    # Pring the SuperOp object
    print("chi2:", chi2)
 
if __name__ == "__main__":
    test_construct_from_chi()
 

执行结果如下所示：

chi2: [[0.+0.j 0.+0.j 0.+0.j 0.+0.j]  
 [0.+0.j 2.+0.j 0.+0.j 0.+0.j]  
 [0.+0.j 0.+0.j 2.+0.j 0.+0.j]  
 [0.+0.j 0.+0.j 0.+0.j 2.+0.j]]  

从 SuperOp 构造

从 SuperOp 对象生成 Chi 对象。请参考 SuperOp。

Python

 
from pyqpanda3.quantum_info import Kraus, SuperOp, Chi
import numpy as np
 
 
def test_construct_from_superop():
    # prepare SuperOp object
    # preapre Kraus object
    # Prepare two one-dimensional array consisting of a two-dimensional matrix
    K0 = np.array([[0, 1], [1+0.j, 0]])
    K1 = np.array([[1, 0], [0+0.j, -1]])
    K2 = np.array([[0+0.j, -1.j], [1.j, 0]])
 
    Ks1 = [K0, K1, K2]
    Ks2 = [K1]
    # Constructing a Kraus object
    Kra3 = Kraus(Ks1, Ks2)
    # Constructing a SuperOp object from a Kraus object
    sop = SuperOp(Kra3)
    
 
    # Constructing another Chi object from a SuperOp object
    chi = Chi(sop)
    # Pring the Chi object
    print("chi:", chi)
 
if __name__ == "__main__":
    test_construct_from_superop()
 

执行结果如下所示：

chi: [[0.+0.j 0.+0.j 0.+0.j 0.+0.j]  
 [0.+0.j 2.+0.j 0.+0.j 0.+0.j]  
 [0.+0.j 0.+0.j 2.+0.j 0.+0.j]  
 [0.+0.j 0.+0.j 0.+0.j 2.+0.j]]  

从 Kraus 构造

从 Kraus 对象生成 Chi 对象。请参考 Kraus。

Python

 
from pyqpanda3.quantum_info import Kraus,Chi
import numpy as np
 
def test_construct_from_kraus():
    # prepare Kraus object
    # Prepare two one-dimensional array consisting of a two-dimensional matrix
    K0 = np.array([[0, 1], [1+0.j, 0]])
    K1 = np.array([[1, 0], [0+0.j, -1]])
    K2 = np.array([[0+0.j, -1.j], [1.j, 0]])
 
    Ks1 = [K0, K1, K2]
    Ks2 = [K1]
    # Constructing a Kraus object
    Kra3 = Kraus(Ks1, Ks2)
    
 
    # Constructing a Chi object from a Kraus object
    chi = Chi(Kra3)
    # Pring the Chi object
    print("chi:",chi)
 
if __name__ == "__main__":
    test_construct_from_kraus()
 

执行结果如下所示：

chi: [[0.+0.j 0.+0.j 0.+0.j 0.+0.j]  
 [0.+0.j 2.+0.j 0.+0.j 0.+0.j]  
 [0.+0.j 0.+0.j 2.+0.j 0.+0.j]  
 [0.+0.j 0.+0.j 0.+0.j 2.+0.j]]  

从 PTM 构造

从 PTM 对象生成 Chi 对象。请参考 PTM。

Python

 
from pyqpanda3.quantum_info import Kraus, Chi, PTM
import numpy as np
def test_construct_from_ptm():
    # prepare PTM object
    # preapre Kraus object
    # Prepare two one-dimensional array consisting of a two-dimensional matrix
    K0 = np.array([[0, 1], [1+0.j, 0]])
    K1 = np.array([[1, 0], [0+0.j, -1]])
    K2 = np.array([[0+0.j, -1.j], [1.j, 0]])
 
    Ks1 = [K0, K1, K2]
    Ks2 = [K1]
    # Constructing a Kraus object
    Kra3 = Kraus(Ks1, Ks2)
    # Constructing a PTM object from a Kraus object
    ptm = PTM(Kra3)
    
 
    # Constructing another Chi object from a PTM object
    chi = Chi(ptm)
    # Pring the Chi object
    print("chi:", chi)
if __name__ == "__main__":
    test_construct_from_ptm()
 

执行结果如下所示：

chi: [[0.+0.j 0.+0.j 0.+0.j 0.+0.j]  
 [0.+0.j 2.+0.j 0.+0.j 0.+0.j]  
 [0.+0.j 0.+0.j 2.+0.j 0.+0.j]  
 [0.+0.j 0.+0.j 0.+0.j 2.+0.j]]  

获取内部数据

输入与输出维度

获取量子通道的输入维度 input_dim 和输出维度 output_dim。

获取input_dim的API文档链接

获取output_dim的API文档链接

Python

 
from pyqpanda3.quantum_info import Kraus,Chi
import numpy as np
 
def test_dim():
    # prepare Kraus object
    # Prepare two one-dimensional array consisting of a two-dimensional matrix
    K0 = np.array([[0, 1], [1+0.j, 0]])
    K1 = np.array([[1, 0], [0+0.j, -1]])
    K2 = np.array([[0+0.j, -1.j], [1.j, 0]])
 
    Ks1 = [K0, K1, K2]
    Ks2 = [K1]
    # Constructing a Kraus object
    Kra3 = Kraus(Ks1, Ks2)
    
 
    # Constructing a Chi object from a Kraus object
    chi = Chi(Kra3)
 
    # Get the input dimension input_dim corresponding to the Chi object
    input_dim = chi.get_input_dim()
    # Get the output dimension output_dim corresponding to the Chi object
    output_dim = chi.get_output_dim()
    print("input_dim:",input_dim)
    print("output_dim:",output_dim)
if __name__ == "__main__":
    test_dim()
 

执行结果如下所示：

input_dim: 2  
output_dim: 2  

量子态的演化

Chi.evolve的API文档链接

对密度矩阵的演化

对 DensityMatrix 对象进行演化，并返回演化后的 DensityMatrix 对象。密度矩阵的维度由 dim 方法获取，并应与 Chi 对象的输入维度相等。请参考密度矩阵。

Python

 
from pyqpanda3.quantum_info import Kraus, Chi
from pyqpanda3.quantum_info import DensityMatrix
import numpy as np
 
def test_evolve():
    # Prepare DensityMatrix object
    # Prepare a complex array
 
    data = np.array(
        [[0.69272168 + 0.j,0.38338527 - 0.17772383j],
         [0.38338527 + 0.17772383j,0.30727832 + 0.j]]
    )
    # Construct a DensityMatrix object
    dm = DensityMatrix(data)
    
 
    # prepare Chi object
    # prepare Kraus object
    # Prepare two one-dimensional array consisting of a two-dimensional matrix
    K0 = np.array([[0, 1], [1+0.j, 0]])
    K1 = np.array([[1, 0], [0+0.j, -1]])
    K2 = np.array([[0+0.j, -1.j], [1.j, 0]])
 
    Ks1 = [K0, K1, K2]
    Ks2 = [K1]
    # Constructing a Kraus object
    Kra3 = Kraus(Ks1, Ks2)
    # Constructing a Chi object from a Kraus object
    chi = Chi(Kra3)
    
 
    # Evolution of the quantum state density matrix, and the evolution result is returned as a DensityMatrix object
    res = chi.evolve(dm)
    # Print result
    print("res:", res)
if __name__ == "__main__":
    test_evolve()
 

执行结果如下所示：

res: [[ 1.30727832+0.j         -0.38338527+0.17772383j]  
 [-0.38338527-0.17772383j  1.69272168+0.j        ]]  

对态矢量的演化

对 StateVector 对象进行演化，并返回演化后的 DensityMatrix 对象。StateVector 对象的维度由 dim() 方法获取，并应该与 Chi 对象的输入维度相等。请参考密度矩阵和态矢量。

Python

 
from pyqpanda3.quantum_info import Kraus,Chi
from pyqpanda3.quantum_info import StateVector
import numpy as np
 
def test_evolve():
    # Construct a StateVector object
    stv = StateVector()
 
    # prepare Chi object
    # prepare Kraus object
    # Prepare two one-dimensional array consisting of a two-dimensional matrix
    K0 = np.array([[0, 1], [1+0.j, 0]])
    K1 = np.array([[1, 0], [0+0.j, -1]])
    K2 = np.array([[0+0.j, -1.j], [1.j, 0]])
 
    Ks1 = [K0, K1, K2]
    Ks2 = [K1]
    # Constructing a Kraus object
    Kra3 = Kraus(Ks1, Ks2)
    # Constructing a Chi object from a Kraus object
    chi = Chi(Kra3)
    
 
    # Evolution of the StateVector object is performed, and the result is returned as a DensityMatrix object
    res = chi.evolve(stv)
    # Print result
    print("res:",res)
if __name__ == "__main__":
    test_evolve()
 

执行结果如下所示：

res: [[1.+0.j 0.+0.j]  
 [0.+0.j 2.+0.j]]  

布尔函数

判等

判断两个 Chi 对象的内部数据是否相等。

API文档链接

Python

 
from pyqpanda3.quantum_info import Kraus,Chi
import numpy as np
 
def test_equal():
    # prepare Chi object
    # prepare Kraus object
    # Prepare two one-dimensional array consisting of a two-dimensional matrix
    K0 = np.array([[0, 1], [1+0.j, 0]])
    K1 = np.array([[1, 0], [0+0.j, -1]])
    K2 = np.array([[0+0.j, -1.j], [1.j, 0]])
 
    Ks1 = [K0, K1, K2]
    Ks2 = [K1]
    # Constructing a Kraus object
    Kra3 = Kraus(Ks1, Ks2)
    # Constructing a Chi object from a Kraus object
    chi = Chi(Kra3)
    
 
    chi2 = chi
    # Determine whether the internal data of two Chi objects are equal
    print(chi2==chi)
if __name__ == "__main__":
    test_equal()
 

执行结果如下所示：

True