QuantumChannel -- Chi, Choi, SuperOp, PTM, Kraus

Quantum channels describe the most general physical evolution of a quantum state. In pyqpanda3, a quantum channel can be represented in five mathematically equivalent forms. Each representation class can be freely converted to any other representation via its constructor.

All channel representations share a common base class QuantumChannel and provide methods for evolving quantum states and querying dimensions.

QuantumChannel

Marker class for all quantum channel representations. QuantumChannel cannot be constructed directly — use one of its concrete subclasses (Kraus, Chi, Choi, SuperOp, PTM) instead.

Signature

python

QuantumChannel(*args, **kwargs)

Note: The constructor accepts *args, **kwargs for internal use only. Create specific channel types via their named constructors (e.g., Kraus(), Choi()).

All channel subclasses implement the following methods:

Method	Return Type	Description
`evolve(state)`	`DensityMatrix`	Apply the channel to a quantum state
`get_input_dim()`	`int`	Return the input Hilbert-space dimension
`get_output_dim()`	`int`	Return the output Hilbert-space dimension

StateSystemType

Enumeration for qubit ordering conventions used in quantum state representations. Specifies how bit strings in state labels map to qubit indices.

Values

Value	Description
`Q2Q1Q0`	Standard ordering: in state label `'10'`, qubit 1 is `'1'` and qubit 0 is `'0'`

Example

python

from pyqpanda3.quantum_info import StateSystemType

ordering = StateSystemType.Q2Q1Q0
print(ordering)

Output:

StateSystemType.Q2Q1Q0

Mathematical Background

A quantum channel $E$ is a completely positive trace-preserving (CPTP) map. The five representations are:

Representation	Matrix dimension	Description
Kraus	--	Set of operators ${K_{i}}$ such that $E (ρ) = \sum_{i} K_{i} ρ K_{i}^{†}$
SuperOp	$d^{2} \times d^{2}$	The superoperator matrix $S$ in the column-stacking basis
Choi	$d^{2} \times d^{2}$	The Choi matrix $Λ = \sum_{i j} \| i ⟩ ⟨ j \| \otimes E (\| i ⟩ ⟨ j \|)$
Chi	$d^{2} \times d^{2}$	The Chi matrix in the Pauli basis
PTM	$d^{2} \times d^{2}$	The Pauli Transfer Matrix

Kraus

The Kraus representation of a quantum channel. A channel in Kraus form is defined by a set of operators ${K_{i}}$ such that:

E (ρ) = \sum_{i} K_{i} ρ K_{i}^{†}

with the trace-preservation condition $\sum_{i} K_{i}^{†} K_{i} = I$ .

The Kraus class also supports a two-sided (generalized) form with separate left and right operator lists:

E (ρ) = \sum_{i} L_{i} ρ R_{i}^{†}

Constructors

Default

python

Kraus()

Constructs an empty Kraus object with no operators.

From a single operator (Eigen matrix)

python

Kraus(matrix: numpy.ndarray)

Construct a Kraus object from a single 2-D matrix. The left operator list contains the given matrix; the right operator list is empty.

Warning: When passing a single numpy.ndarray, the call may dispatch to the list overload Kraus(left: list[numpy.ndarray]) instead of the single-matrix constructor. To ensure correct single-matrix construction, wrap the array in a list: Kraus([matrix]).

Parameter	Type	Description
matrix	`numpy.ndarray`	A single operator matrix.

From a single operator (Matrix)

python

Kraus(matrix: Matrix)

Same as above but accepts a Matrix object.

Parameter	Type	Description
matrix	`Matrix`	A single operator as a `Matrix`.

From a list of operators (left only)

python

Kraus(left: list[numpy.ndarray])
Kraus(left: list[Matrix])

Construct a Kraus object from a list of 2-D matrices. The left operator list is populated; the right operator list is empty.

Parameter	Type	Description
left	`list[numpy.ndarray]` or `list[Matrix]`	A list of Kraus operator matrices.

From two lists of operators (left and right)

python

Kraus(left: list[numpy.ndarray], right: list[numpy.ndarray])
Kraus(left: list[Matrix], right: list[Matrix])

Construct a two-sided Kraus object from separate left and right operator lists.

Parameter	Type	Description
left	`list[numpy.ndarray]` or `list[Matrix]`	The left operator list.
right	`list[numpy.ndarray]` or `list[Matrix]`	The right operator list.

From another channel representation

python

Kraus(other: Choi)
Kraus(other: Chi)
Kraus(other: SuperOp)
Kraus(other: PTM)
Kraus(other: Kraus)

Convert another channel representation to the Kraus form.

Methods

evolve

python

evolve(state: DensityMatrix) -> DensityMatrix
evolve(state: StateVector) -> DensityMatrix

Apply the channel to a quantum state and return the result as a DensityMatrix.

get_input_dim

python

get_input_dim() -> int

Return the input Hilbert-space dimension of the channel.

get_output_dim

python

get_output_dim() -> int

Return the output Hilbert-space dimension of the channel.

left

python

left() -> list[Matrix]

Return the list of left operators.

right

python

right() -> list[Matrix]

Return the list of right operators.

append

python

append(other: Kraus) -> bool

Append the operators from another Kraus object to the end of this one.

Parameter	Type	Description
other	`Kraus`	Another `Kraus` object whose operators will be appended.

Note: append() requires that the Kraus object already contains at least one operator. Calling append() on an empty Kraus object will raise an error.

left_push_back

python

left_push_back(val: Matrix)
left_push_back(val: numpy.ndarray)

Append a single matrix to the left operator list.

right_push_back

python

right_push_back(val: Matrix)
right_push_back(val: numpy.ndarray)

Append a single matrix to the right operator list.

clear

python

clear()

Remove all operators from both the left and right lists.

Operators

Equality

python

k_a == k_b -> bool

Return True if the two Kraus objects have identical internal data.

Chi

The Chi (process) matrix representation in the Pauli basis. For a channel $E$ , the Chi matrix $χ$ is defined by:

E (ρ) = \sum_{m, n} χ_{m n} P_{m} ρ P_{n}

where ${P_{m}}$ is the set of Pauli operators.

Constructors

python

Chi(other: Chi)
Chi(other: Choi)
Chi(other: PTM)
Chi(other: SuperOp)
Chi(other: Kraus)

Construct a Chi object by converting from any other channel representation.

Methods

evolve

python

evolve(state: DensityMatrix) -> DensityMatrix
evolve(state: StateVector) -> DensityMatrix

Apply the channel to a quantum state and return the result.

get_input_dim

python

get_input_dim() -> int

Return the input Hilbert-space dimension.

get_output_dim

python

get_output_dim() -> int

Return the output Hilbert-space dimension.

ndarray

python

ndarray() -> numpy.ndarray[numpy.complex128]

Return the internal Chi matrix as a numpy.ndarray.

Operators

Equality

python

chi_a == chi_b -> bool

Choi

The Choi matrix representation (also known as the Choi-Jamiolkowski isomorphism). For a channel $E$ acting on a $d$ -dimensional system, the Choi matrix is:

Λ = \sum_{i, j = 0}^{d - 1} | i ⟩ ⟨ j | \otimes E (| i ⟩ ⟨ j |)

The resulting matrix has dimension $d^{2} \times d^{2}$ .

Constructors

python

Choi(other: Kraus)
Choi(other: Choi)
Choi(other: PTM)
Choi(other: Chi)
Choi(other: SuperOp)

Construct a Choi object by converting from any other channel representation.

Methods

evolve

python

evolve(state: DensityMatrix) -> DensityMatrix
evolve(state: StateVector) -> DensityMatrix

Apply the channel to a quantum state and return the result.

get_input_dim

python

get_input_dim() -> int

Return the input Hilbert-space dimension.

get_output_dim

python

get_output_dim() -> int

Return the output Hilbert-space dimension.

ndarray

python

ndarray() -> numpy.ndarray[numpy.complex128]

Return the internal Choi matrix as a numpy.ndarray of shape $d^{2} \times d^{2}$ .

Operators

Equality

python

choi_a == choi_b -> bool

SuperOp

The superoperator (column-stacking-basis) matrix representation. The superoperator $S$ acts on the vectorized density matrix:

vec (E (ρ)) = S vec (ρ)

The resulting matrix has dimension $d^{2} \times d^{2}$ .

Constructors

python

SuperOp(other: Choi)
SuperOp(other: SuperOp)
SuperOp(other: Chi)
SuperOp(other: PTM)
SuperOp(other: Kraus)

Construct a SuperOp object by converting from any other channel representation.

Methods

evolve

python

evolve(state: DensityMatrix) -> DensityMatrix
evolve(state: StateVector) -> DensityMatrix

Apply the channel to a quantum state and return the result.

get_input_dim

python

get_input_dim() -> int

Return the input Hilbert-space dimension.

get_output_dim

python

get_output_dim() -> int

Return the output Hilbert-space dimension.

ndarray

python

ndarray() -> numpy.ndarray[numpy.complex128]

Return the internal superoperator matrix as a numpy.ndarray of shape $d^{2} \times d^{2}$ in the column-stacking basis.

Operators

Equality

python

s_a == s_b -> bool

PTM

The Pauli Transfer Matrix representation. The PTM $R$ encodes the channel in the Pauli basis:

R_{m n} = \frac{1}{d} Tr [P_{m} E (P_{n})]

where ${P_{m}}$ is the normalized Pauli basis. The PTM is a real-valued $d^{2} \times d^{2}$ matrix.

Constructors

python

PTM(other: Choi)
PTM(other: PTM)
PTM(other: Chi)
PTM(other: SuperOp)
PTM(other: Kraus)

Construct a PTM object by converting from any other channel representation.

Methods

evolve

python

evolve(state: DensityMatrix) -> DensityMatrix
evolve(state: StateVector) -> DensityMatrix

Apply the channel to a quantum state and return the result.

get_input_dim

python

get_input_dim() -> int

Return the input Hilbert-space dimension.

get_output_dim

python

get_output_dim() -> int

Return the output Hilbert-space dimension.

ndarray

python

ndarray() -> numpy.ndarray[numpy.complex128]

Return the internal Pauli transfer matrix as a numpy.ndarray of shape $d^{2} \times d^{2}$ in the Pauli basis.

Operators

Equality

python

ptm_a == ptm_b -> bool

Examples

python

import numpy as np
from pyqpanda3.quantum_info import Kraus, Chi, Choi, SuperOp, PTM, DensityMatrix

# Define a depolarizing channel for a single qubit
# E(rho) = (1-p)*rho + (p/3)*(X*rho*X + Y*rho*Y + Z*rho*Z)
p = 0.1
I = np.eye(2, dtype=complex)
X = np.array([[0, 1], [1, 0]], dtype=complex)
Y = np.array([[0, -1j], [1j, 0]], dtype=complex)
Z = np.array([[1, 0], [0, -1]], dtype=complex)

K0 = np.sqrt(1 - p) * I
K1 = np.sqrt(p / 3) * X
K2 = np.sqrt(p / 3) * Y
K3 = np.sqrt(p / 3) * Z

kraus = Kraus([K0, K1, K2, K3])

# Convert to other representations
choi = Choi(kraus)
chi = Chi(kraus)
superop = SuperOp(kraus)
ptm = PTM(kraus)

# Evolve a density matrix
rho = DensityMatrix([[1, 0], [0, 0]])
rho_out = kraus.evolve(rho)

# Query dimensions
print(kraus.get_input_dim())
print(kraus.get_output_dim())

Output:

2
2

QuantumChannel -- Chi, Choi, SuperOp, PTM, Kraus ​

QuantumChannel ​

Signature ​

StateSystemType ​

Values ​

Example ​

Mathematical Background ​

Kraus ​

Constructors ​

Default ​

From a single operator (Eigen matrix) ​

From a single operator (Matrix) ​

From a list of operators (left only) ​

From two lists of operators (left and right) ​

From another channel representation ​

Methods ​

evolve ​

get_input_dim ​

get_output_dim ​

left ​

right ​

append ​

left_push_back ​

right_push_back ​

clear ​

Operators ​

Equality ​

Chi ​

Constructors ​

Methods ​

evolve ​

get_input_dim ​

get_output_dim ​

ndarray ​

Operators ​

Equality ​

Choi ​

Constructors ​

Methods ​

evolve ​

get_input_dim ​

get_output_dim ​

ndarray ​

Operators ​

Equality ​

SuperOp ​

Constructors ​

Methods ​

evolve ​

get_input_dim ​

get_output_dim ​

ndarray ​

Operators ​

Equality ​

PTM ​

Constructors ​

Methods ​

evolve ​

get_input_dim ​

get_output_dim ​

ndarray ​

Operators ​

Equality ​

Examples ​

See Also ​

QuantumChannel -- Chi, Choi, SuperOp, PTM, Kraus

QuantumChannel

Signature

StateSystemType

Values

Example

Mathematical Background

Kraus

Constructors

Default

From a single operator (Eigen matrix)

From a single operator (Matrix)

From a list of operators (left only)

From two lists of operators (left and right)

From another channel representation

Methods

evolve

get_input_dim

get_output_dim

left

right

append

left_push_back

right_push_back

clear

Operators

Equality

Chi

Constructors

Methods

evolve

get_input_dim

get_output_dim

ndarray

Operators

Equality

Choi

Constructors

Methods

evolve

get_input_dim

get_output_dim

ndarray

Operators

Equality

SuperOp

Constructors

Methods

evolve

get_input_dim

get_output_dim

ndarray

Operators

Equality

PTM

Constructors

Methods

evolve

get_input_dim

get_output_dim

ndarray

Operators

Equality

Examples

See Also