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

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

__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:

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

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

adjoint()

Return the adjoint of the operator.

atol

The default absolute tolerance parameter for float comparisons.

compose

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

conjugate()

Return the conjugate of the QuantumChannel.

copy

copy()

Make a deep copy of current operator.

data

Return list of Kraus matrices for channel.

dim

Return tuple (input_shape, output_shape).

dot

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

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

input_dims(qargs=None)

Return tuple of input dimension for specified subsystems.

is_cp

is_cp(atol=None, rtol=None)

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

is_cptp

is_cptp(atol=None, rtol=None)

Return True if completely-positive trace-preserving.

is_tp

is_tp(atol=None, rtol=None)

Test if a channel is completely-positive (CP)

is_unitary

is_unitary(atol=None, rtol=None)

Return True if QuantumChannel is a unitary channel.

multiply

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.

num_qubits

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

output_dims

output_dims(qargs=None)

Return tuple of output dimension for specified subsystems.

power

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.

qargs

Return the qargs for the operator.

reshape

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.

rtol

The relative tolerance parameter for float comparisons.

set_atol

classmethod set_atol(value)

Set the class default absolute tolerance parameter for float comparisons.

DEPRECATED: use operator.atol = value instead

set_rtol

classmethod set_rtol(value)

Set the class default relative tolerance parameter for float comparisons.

DEPRECATED: use operator.rtol = value instead

subtract

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

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

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

to_operator()

Try to convert channel to a unitary representation Operator.

transpose

transpose()

Return the transpose of the QuantumChannel.

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