qiskit.quantum_info.Kraus

class Kraus(data, input_dims=None, output_dims=None)

Kraus representation of a quantum channel.

The Kraus representation for a quantum channel \(\mathcal{E}\) is a set of matrices \([A_0,...,A_{K-1}]\) such that

For a quantum channel \(\mathcal{E}\), the Kraus representation is given by a set of matrices \([A_0,...,A_{K-1}]\) such that the evolution of a DensityMatrix \(\rho\) is given by

\[\mathcal{E}(\rho) = \sum_{i=0}^{K-1} A_i \rho A_i^\dagger\]

A general operator map \(\mathcal{G}\) can also be written using the generalized Kraus representation which is given by two sets of matrices \([A_0,...,A_{K-1}]\), \([B_0,...,A_{B-1}]\) such that

\[\mathcal{G}(\rho) = \sum_{i=0}^{K-1} A_i \rho B_i^\dagger\]

See reference [1] for further details.

References

1. 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]

Initialize a quantum channel Kraus operator.

Parameters

• (QuantumCircuit or (data) – Instruction or BaseOperator or matrix): data to initialize superoperator.

• input_dims (tuple) – the input subsystem dimensions. [Default: None]

• output_dims (tuple) – the output subsystem dimensions. [Default: None]

Raises

QiskitError – if input data cannot be initialized as a a list of Kraus matrices.

Additional Information:

If the input or output dimensions are None, they will be automatically determined from the input data. If the input data is a list of Numpy arrays of shape (2**N, 2**N) qubit systems will be used. If the input does not correspond to an N-qubit channel, it will assign a single subsystem with dimension specified by the shape of the input.

__init__(data, input_dims=None, output_dims=None)

Initialize a quantum channel Kraus operator.

Parameters

• (QuantumCircuit or (data) – Instruction or BaseOperator or matrix): data to initialize superoperator.

• input_dims (tuple) – the input subsystem dimensions. [Default: None]

• output_dims (tuple) – the output subsystem dimensions. [Default: None]

Raises

QiskitError – if input data cannot be initialized as a a list of Kraus matrices.

Additional Information:

If the input or output dimensions are None, they will be automatically determined from the input data. If the input data is a list of Numpy arrays of shape (2**N, 2**N) qubit systems will be used. If the input does not correspond to an N-qubit channel, it will assign a single subsystem with dimension specified by the shape of the input.

Methods

__init__(data[, input_dims, output_dims])	Initialize a quantum channel Kraus operator.
add(other)	Return the linear operator self + other.
adjoint()	Return the adjoint of the operator.
compose(other[, qargs, front])	Return the composed quantum channel self @ other.
conjugate()	Return the conjugate of the QuantumChannel.
copy()	Make a deep copy of current operator.
dot(other[, qargs])	Return the right multiplied quantum channel self * other.
expand(other)	Return the tensor product channel other ⊗ self.
input_dims([qargs])	Return tuple of input dimension for specified subsystems.
is_cp([atol, rtol])	Test if Choi-matrix is completely-positive (CP)
is_cptp([atol, rtol])	Return True if completely-positive trace-preserving.
is_tp([atol, rtol])	Test if a channel is completely-positive (CP)
is_unitary([atol, rtol])	Return True if QuantumChannel is a unitary channel.
multiply(other)	Return the linear operator other * self.
output_dims([qargs])	Return tuple of output dimension for specified subsystems.
power(n)	The matrix power of the channel.
reshape([input_dims, output_dims])	Return a shallow copy with reshaped input and output subsystem dimensions.
set_atol(value)	Set the class default absolute tolerance parameter for float comparisons.
set_rtol(value)	Set the class default relative tolerance parameter for float comparisons.
subtract(other)	Return the linear operator self - other.
tensor(other)	Return the tensor product channel self ⊗ other.
to_instruction()	Convert to a Kraus or UnitaryGate circuit instruction.
to_operator()	Try to convert channel to a unitary representation Operator.
transpose()	Return the transpose of the QuantumChannel.

Attributes

atol	The default absolute tolerance parameter for float comparisons.
data	Return list of Kraus matrices for channel.
dim	Return tuple (input_shape, output_shape).
num_qubits	Return the number of qubits if a N-qubit operator or None otherwise.
qargs	Return the qargs for the operator.
rtol	The relative tolerance parameter for float comparisons.

add(other)

Return the linear operator self + other.

DEPRECATED: use operator + other instead.

Parameters

other (BaseOperator) – an operator object.

Returns

the operator self + other.

Return type

BaseOperator

adjoint()

Return the adjoint of the operator.

property atol

The default absolute tolerance parameter for float comparisons.

compose(other, qargs=None, front=False)

Return the composed quantum channel self @ other.

Parameters

• other (QuantumChannel) – a quantum channel.

• qargs (list or None) – a list of subsystem positions to apply other on. If None apply on all subsystems [default: None].

• front (bool) – If True compose using right operator multiplication, instead of left multiplication [default: False].

Returns

The quantum channel self @ other.

Return type

Kraus

Raises

QiskitError – if other cannot be converted to a Kraus or has incompatible dimensions.

Additional Information:

Composition (@) is defined as left matrix multiplication for SuperOp matrices. That is that A @ B is equal to B * A. Setting front=True returns right matrix multiplication A * B and is equivalent to the dot() method.

conjugate()

Return the conjugate of the QuantumChannel.

copy()

Make a deep copy of current operator.

property data

Return list of Kraus matrices for channel.

property dim

Return tuple (input_shape, output_shape).

dot(other, qargs=None)

Return the right multiplied quantum channel self * other.

Parameters

• other (QuantumChannel) – a quantum channel.

• qargs (list or None) – a list of subsystem positions to apply other on. If None apply on all subsystems [default: None].

Returns

The quantum channel self * other.

Return type

Kraus

Raises

QiskitError – if other cannot be converted to a Kraus or has incompatible dimensions.

expand(other)

Return the tensor product channel other ⊗ self.

Parameters

other (QuantumChannel) – a quantum channel subclass.

Returns

the tensor product channel other ⊗ self as a Kraus object.

Return type

Kraus

Raises

QiskitError – if other cannot be converted to a channel.

input_dims(qargs=None)

Return tuple of input dimension for specified subsystems.

is_cp(atol=None, rtol=None)

Test if Choi-matrix is completely-positive (CP)

is_cptp(atol=None, rtol=None)

Return True if completely-positive trace-preserving.

is_tp(atol=None, rtol=None)

Test if a channel is completely-positive (CP)

is_unitary(atol=None, rtol=None)

Return True if QuantumChannel is a unitary channel.

multiply(other)

Return the linear operator other * self.

DEPRECATED: use other * operator instead.

Parameters

other (complex) – a complex number.

Returns

the linear operator other * self.

Return type

BaseOperator

Raises

NotImplementedError – if subclass does not support multiplication.

property num_qubits

Return the number of qubits if a N-qubit operator or None otherwise.

output_dims(qargs=None)

Return tuple of output dimension for specified subsystems.

power(n)

The matrix power of the channel.

Parameters

n (int) – compute the matrix power of the superoperator matrix.

Returns

the matrix power of the SuperOp converted to a Kraus channel.

Return type

Kraus

Raises

QiskitError – if the input and output dimensions of the QuantumChannel are not equal, or the power is not an integer.

property qargs

Return the qargs for the operator.

reshape(input_dims=None, output_dims=None)

Return a shallow copy with reshaped input and output subsystem dimensions.

Arg:

input_dims (None or tuple): new subsystem input dimensions.

If None the original input dims will be preserved [Default: None].

output_dims (None or tuple): new subsystem output dimensions.

If None the original output dims will be preserved [Default: None].

Returns

returns self with reshaped input and output dimensions.

Return type

BaseOperator

Raises

QiskitError – if combined size of all subsystem input dimension or subsystem output dimensions is not constant.

property rtol

The relative tolerance parameter for float comparisons.

classmethod set_atol(value)

Set the class default absolute tolerance parameter for float comparisons.

DEPRECATED: use operator.atol = value instead

classmethod set_rtol(value)

Set the class default relative tolerance parameter for float comparisons.

DEPRECATED: use operator.rtol = value instead

subtract(other)

Return the linear operator self - other.

DEPRECATED: use operator - other instead.

Parameters

other (BaseOperator) – an operator object.

Returns

the operator self - other.

Return type

BaseOperator

tensor(other)

Return the tensor product channel self ⊗ other.

Parameters

other (QuantumChannel) – a quantum channel subclass.

Returns

the tensor product channel self ⊗ other as a Kraus object.

Return type

Kraus

Raises

QiskitError – if other cannot be converted to a channel.

to_instruction()

Convert to a Kraus or UnitaryGate circuit instruction.

If the channel is unitary it will be added as a unitary gate, otherwise it will be added as a kraus simulator instruction.

Returns

A kraus instruction for the channel.

Return type

qiskit.circuit.Instruction

Raises

QiskitError – if input data is not an N-qubit CPTP quantum channel.

to_operator()

Try to convert channel to a unitary representation Operator.

transpose()

Return the transpose of the QuantumChannel.