Chi

Table of Contents

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

Prev Tutorial: SuperOp
Next Tutorial: PTM

---

Introduction

Pauli basis Chi-matrix representation of a quantum channel.

The Chi-matrix representation of an n-qubit quantum channel \(\mathcal{E}\) is a matrix χ such that the evolution of a DensityMatrix \(\rho\) is given by

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

where

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

is the n-qubit Pauli basis in lexicographic order. It is related to the Choi representation by a change of basis of the Choi-matrix into the Pauli basis. The

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

in the definition above is a normalization factor that arises from scaling the Pauli basis to make it orthonormal.

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 Chi object

Here is API doc

From Choi

Generate another Chi object from a Choi object

Please refer to 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()
 

Output

chi: {
{
{(0,0)},{(0,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(2,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(2,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(0,0)},{(2,0)},
}
}

From Chi

Generate another Chi object from a Chi object

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()
 

Output

chi2: {
{
{(0,0)},{(0,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(2,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(2,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(0,0)},{(2,0)},
}
}

From SuperOp

Generate another Chi object from a SuperOp object

Please refer to 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()
 

Output

chi: {
{
{(0,0)},{(0,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(2,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(2,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(0,0)},{(2,0)},
}
}

From Kraus

Generate another Chi object from a Kraus object

Please refer to 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()
 

Output

chi: {
{
{(0,0)},{(0,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(2,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(2,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(0,0)},{(2,0)},
}
}

From PTM

Generate another SuperOp object from a PTM object

Please refer to 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()
 

Output

chi: {
{
{(0,0)},{(0,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(2,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(2,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(0,0)},{(2,0)},
}
}

Obtain internal data

Input and outut dim

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

Here is API doc for Chi.get_input_dim

Here is API doc for Chi.get_output_dim

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()
 

Output

input_dim: 2
output_dim: 2

Evolution of quantum states

Here is API doc for Chi.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 Chi object

Please refer to DensityMatrix.

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()
 

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 Chi object

Please refer to DensityMatrix

Please refer to StateVector

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()
 

output

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

Boolean function

Equal

Determine whether the internal data of two Chi objects are equal

Here is API doc

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()
 

Output

True