SuperOp

Table of Contents

• Introduction
• In QPanda3 Quantum Information
  • Constructing a SuperOp object
    • From Choi
    • From Chi
    • From SuperOp
    • From Kraus
    • From PTM
  • Obtain internal data
    • Input and output dim
  • Evolution of quantum states
    • DensityMatrix
    • StateVector
  • Boolean function
    • Equal

Prev Tutorial: Choi
Next Tutorial: Chi

---

Introduction

Superoperator representation of a quantum channel.

The Superoperator representation of a quantum channel \(\mathcal{E}\) is a matrix \(\mathcal{S}\) such that the evolution of a DensityMatrix \(\rho\) is given by

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

Please refer: 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]

In QPanda3 Quantum Information

Constructing a SuperOp object

Here is API doc

From Choi

Generate another SuperOp object from a Choi object

Please refer to 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()
 

Output

sop: {
{
{(1,0)},{(0,0)},{(0,0)},{(2,0)},
}
{
{(0,0)},{(-1,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(-1,0)},{(0,0)},
}
{
{(2,0)},{(0,0)},{(0,0)},{(1,0)},
}
}

From Chi

Generate another SuperOp object from a Chi object

Please refer to 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()
 

Output

sop: {
{
{(1,0)},{(0,0)},{(0,0)},{(2,0)},
}
{
{(0,0)},{(-1,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(-1,0)},{(0,0)},
}
{
{(2,0)},{(0,0)},{(0,0)},{(1,0)},
}
}

From SuperOp

Generate another SuperOp object from a SuperOp object

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()
 

Output

sop2: {
{
{(1,0)},{(0,0)},{(0,0)},{(2,0)},
}
{
{(0,0)},{(-1,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(-1,0)},{(0,0)},
}
{
{(2,0)},{(0,0)},{(0,0)},{(1,0)},
}
}

From Kraus

Generate another SuperOp object from a Kraus object

Please refer to 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()
 

Output

sop: {
{
{(1,0)},{(0,0)},{(0,0)},{(2,0)},
}
{
{(0,0)},{(-1,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(-1,0)},{(0,0)},
}
{
{(2,0)},{(0,0)},{(0,0)},{(1,0)},
}
}

From PTM

Generate another SuperOp object from a PTM object

Please refer to 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()
 

Output

sop: {
{
{(1,0)},{(0,0)},{(0,0)},{(2,0)},
}
{
{(0,0)},{(-1,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(-1,0)},{(0,0)},
}
{
{(2,0)},{(0,0)},{(0,0)},{(1,0)},
}
}

Obtain internal data

Input and output dim

Obtain the input dimension input_dim and output dimension output_dim of the quantum channel

The shape of the SuperOp matrix is（output_dim * output_dim,input_dim * input_dim）

Here is API doc for SuperOp.get_input_dim

Here is API doc for SuperOp.get_output_dim

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()
 

Output

input_dim: 2
output_dim: 2

Evolution of quantum states

Here is API doc for SuperOp.evolve

DensityMatrix

Evolution of a DensityMatrix object, and the evolution result is returned as a DensityMatrix object

The dimension of the density matrix is obtained by the member method dim() and should be equal to the input dimension of the SuperOp object

Please refer to DensityMatrix.

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()
 

Output

res: {
{
{(1.30728,0)},{(-0.383385,0.177724)},
}
{
{(-0.383385,-0.177724)},{(1.69272,0)},
}
}

StateVector

Evolution of the StateVector object is performed, and the result is returned as a DensityMatrix object

The dimension of the StateVector object is obtained by the member method dim() and should be equal to the input dimension of the Choi object

Please refer to DensityMatrix.

Please refer to StateVector.

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()
 

Output

res: {
{
{(1,0)},{(0,0)},
}
{
{(0,0)},{(2,0)},
}
}

Boolean function

Equal

Determine whether the internal data of two SuperOp objects are equal

Here is API doc

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()
 

Output

True