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

• or (data (QuantumCircuit) – 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.

Attributes

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

Methods

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