SuperOp

目录

• 简介
• 在 QPanda3 量子信息模块中
  • 构造 SuperOp 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
    • 获取输入和输出维度
  • 量子态的演化
    • 密度矩阵
    • 态矢量
  • 布尔函数
    • 判等

上一章: Choi

下一章: Chi

---

简介

量子通道的超算子表示。

量子通道 \(\mathcal{E}\) 的超算子表示 \(\mathcal{S}\) 是一个矩阵，使得密度矩阵 \(\rho\) 的演化可以表示为：

\[|\mathcal{E}(\rho)\rangle\rangle=S|\rho\rangle\rangle \]

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

构造 SuperOp 对象

API文档链接

从 Choi 构造

从一个 Choi 对象生成一个 SuperOp 对象。请参考 Choi。

Python

 
from pyqpanda3.quantum_info import Kraus,Choi,SuperOp
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
    sop = SuperOp(choi)
    # Pring the SuperOp object
    print("sop:",sop)
if __name__ == "__main__":
    test_construct_from_choi()
 

执行结果如下所示：

sop: [[ 1.+0.j  0.+0.j  0.+0.j  2.+0.j]
 [ 0.+0.j -1.+0.j  0.+0.j  0.+0.j]
 [ 0.+0.j  0.+0.j -1.+0.j  0.+0.j]
 [ 2.+0.j  0.+0.j  0.+0.j  1.+0.j]]

从 Chi 构造

从一个 Chi 对象生成一个 SuperOp 对象。请参考 Chi。

Python

 
from pyqpanda3.quantum_info import Kraus, SuperOp, 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 SuperOp object from a Chi object
    sop = SuperOp(chi)
    # Pring the SuperOp object
    print("sop:", sop)
if __name__ == "__main__":
    test_construct_from_chi()
 

执行结果如下所示：

sop: [[ 1.+0.j  0.+0.j  0.+0.j  2.+0.j]
 [ 0.+0.j -1.+0.j  0.+0.j  0.+0.j]
 [ 0.+0.j  0.+0.j -1.+0.j  0.+0.j]
 [ 2.+0.j  0.+0.j  0.+0.j  1.+0.j]]

从 SuperOp 构造

从一个 SuperOp 对象生成另一个 SuperOp 对象。

Python

 
from pyqpanda3.quantum_info import Kraus, SuperOp
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 SuperOp object from a SuperOp object
    sop2 = SuperOp(sop)
    # Pring the SuperOp object
    print("sop2:", sop2)
if __name__ == "__main__":
    test_construct_from_superop()
 

执行结果如下所示：

sop2: [[ 1.+0.j  0.+0.j  0.+0.j  2.+0.j]
 [ 0.+0.j -1.+0.j  0.+0.j  0.+0.j]
 [ 0.+0.j  0.+0.j -1.+0.j  0.+0.j]
 [ 2.+0.j  0.+0.j  0.+0.j  1.+0.j]]

从 Kraus 构造

从一个 Kraus 对象生成一个 SuperOp 对象。请参考 Kraus。

Python

 
from pyqpanda3.quantum_info import Kraus,SuperOp
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 SuperOp object from a Kraus object
    sop = SuperOp(Kra3)
    # Pring the SuperOp object
    print("sop:",sop)
if __name__ == "__main__":
    test_construct_from_kraus()
 

执行结果如下所示：

sop: [[ 1.+0.j  0.+0.j  0.+0.j  2.+0.j]
 [ 0.+0.j -1.+0.j  0.+0.j  0.+0.j]
 [ 0.+0.j  0.+0.j -1.+0.j  0.+0.j]
 [ 2.+0.j  0.+0.j  0.+0.j  1.+0.j]]

从 PTM 构造

从一个 PTM 对象生成一个 SuperOp 对象。请参考 PTM。

Python

 
from pyqpanda3.quantum_info import Kraus, SuperOp, 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 SuperOp object from a PTM object
    sop = SuperOp(ptm)
    # Pring the SuperOp object
    print("sop:", sop)
if __name__ == "__main__":
    test_construct_from_ptm()
 

执行结果如下所示：

sop: [[ 1.+0.j  0.+0.j  0.+0.j  2.+0.j]
 [ 0.+0.j -1.+0.j  0.+0.j  0.+0.j]
 [ 0.+0.j  0.+0.j -1.+0.j  0.+0.j]
 [ 2.+0.j  0.+0.j  0.+0.j  1.+0.j]]

获取内部数据

获取输入和输出维度

获取量子通道的输入维度 input_dim 和输出维度 output_dim。SuperOp 矩阵的形状为 (output_dim * output_dim, input_dim * input_dim)。

SuperOp.get_input_dim的API文档链接

SuperOp.get_output_dim的API文档链接

Python

 
from pyqpanda3.quantum_info import Kraus,SuperOp
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 SuperOp object from a Kraus object
    sop = SuperOp(Kra3)
 
    # Get the input dimension input_dim corresponding to the SuperOp object
    input_dim = sop.get_input_dim()
    # Get the output dimension output_dim corresponding to the SuperOp object
    output_dim = sop.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

SuperOp.evolve的API文档链接

密度矩阵

DensityMatrix 对象的演化，演化结果返回为 DensityMatrix 对象。DensityMatrix 的维度通过成员方法 dim 获取，并且应当与 SuperOp 对象的输入维度相等。请参考密度矩阵。

Python

 
from pyqpanda3.quantum_info import Kraus, SuperOp
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 SuperOp 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 SuperOp object from a Kraus object
    sop = SuperOp(Kra3)
    
 
    # Evolution of the quantum state density matrix, and the evolution result is returned as a DensityMatrix object
    res = sop.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        ]]

态矢量

StateVector 对象的演化，并且结果返回为 DensityMatrix 对象。StateVector 对象的维度通过成员方法 dim 获取，并且应当与 Choi 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,SuperOp
from pyqpanda3.quantum_info import StateVector
import numpy as np
 
def test_evolve():
    # Construct a StateVector object
    stv = StateVector()
 
    # prepare SuperOp 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 SuperOp object from a Kraus object
    sop = SuperOp(Kra3)
    
 
    # Evolution of the StateVector object is performed, and the result is returned as a DensityMatrix object
    res = sop.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 SuperOp 对象的内部数据是否相等。

API文档链接

Python

 
from pyqpanda3.quantum_info import Kraus,SuperOp
import numpy as np
 
def test_equal():
    # prepare SuperOp 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 SuperOp object from a Kraus object
    sop = SuperOp(Kra3)
    
 
    sop2 = sop
    # Determine whether the internal data of two SuperOp objects are equal
    print(sop2==sop)
if __name__ == "__main__":
    test_equal()
 

执行结果如下所示：

True