PTM

Table of Contents

• Introduction
• In QPanda3 Quantum Information
  • Constructing a PTM 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: Chi
Next Tutorial: Hellinger Distance

---

Introduction

Pauli Transfer Matrix (PTM) representation of a Quantum Channel.

The PTM representation of an n-qubit quantum channel \(\mathcal{E}\) is an n-qubit SuperOp \(\mathcal{R}\) defined with respect to vectorization in the Pauli basis instead of column-vectorization. The elements of the PTM

\[ \mathcal{R} \]

are given by

\[ R_{i, j}=\frac{1}{2^{n}} \operatorname{Tr}\left[P_{i} \mathcal{E}\left(P_{j}\right)\right] \]

where

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

is the \(\mathcal{n}\) -qubit Pauli basis in lexicographic order.

Evolution of a DensityMatrix \(\rho\)

ρ with respect to the PTM is given by

\[ \left.|\mathcal{E}(\rho)\rangle_{P}=S_{P}|\rho\rangle\right\rangle_{P} \]

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 PTM object

Here is API doc

From Choi

Generate another PTM object from a Choi object

Please refer to Choi

Python

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

Output

ptm: {
{
{(3,0)},{(0,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(-1,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(-1,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(0,0)},{(-1,0)},
}
}

From Chi

Generate another PTM object from a Chi object

Please refer to Chi

Python

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

Output

ptm: {
{
{(3,0)},{(0,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(-1,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(-1,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(0,0)},{(-1,0)},
}
}

From SuperOp

Generate another PTM object from a SuperOp object

Please refer to SuperOp

Python

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

Output

ptm: {
{
{(3,0)},{(0,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(-1,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(-1,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(0,0)},{(-1,0)},
}
}

From Kraus

Generate another PTM object from a Kraus object

Please refer to Kraus

Python

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

Output

ptm: {
{
{(3,0)},{(0,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(-1,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(-1,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(0,0)},{(-1,0)},
}
}

From PTM

Generate another PTM object from a PTM object

Python

 
from pyqpanda3.quantum_info import Kraus,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 PTM object from a PTM object
    ptm2 = PTM(ptm)
    # Pring the PTM object
    print("ptm2:", ptm2)
if __name__ == "__main__":
    test_construct_from_ptm()
 

Output

ptm2: {
{
{(3,0)},{(0,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(-1,0)},{(0,0)},{(0,0)},
}
{
{(0,0)},{(0,0)},{(-1,0)},{(0,0)},
}
{
{(0,0)},{(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

Here is API doc for PTM.get_input_dim

Here is API doc for PTM.get_output_dim

Python

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

Please refer to DensityMatrix.

Python

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

Please refer to DensityMatrix

Please refer to StateVector

Python

 
from pyqpanda3.quantum_info import Kraus,PTM
from pyqpanda3.quantum_info import StateVector
import numpy as np
 
def test_evolve():
    # Construct a StateVector object
    stv = StateVector()
 
    # prepare PTM 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 PTM object from a Kraus object
    ptm = PTM(Kra3)
    
 
    # Evolution of the StateVector object is performed, and the result is returned as a DensityMatrix object
    res = ptm.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 PTM objects are equal

Here is API doc

Python

from pyqpanda3.quantum_info import Kraus,PTM
import numpy as np
 
def test_equal():
    # prepare PTM 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 PTM object from a Kraus object
    ptm = PTM(Kra3)
    
 
    ptm2 = ptm
    # Determine whether the internal data of two PTM objects are equal
    print(ptm2==ptm)
if __name__ == "__main__":
    test_equal()
 

Output

True