Kraus

class qiskit.quantum_info.Kraus(data, input_dims=None, output_dims=None)

Bases: QuantumChannel

Kraus representation of a quantum channel.

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.

Parâmetros:

• data (QuantumCircuit | circuit.instruction.Instruction | BaseOperator | np.ndarray) – data to initialize superoperator.

• input_dims (tuple | None) – the input subsystem dimensions.

• output_dims (tuple | None) – the output subsystem dimensions.

Levanta:

QiskitError – if input data cannot be initialized as 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

atol = 1e-08

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 = 1e-05

settings

Return settings.

Methods

adjoint()

Return the adjoint quantum channel.

Nota

This is equivalent to the matrix Hermitian conjugate in the SuperOp representation ie. for a channel \(\mathcal{E}\), the SuperOp of the adjoint channel \(\mathcal{{E}}^\dagger\) is \(S_{\mathcal{E}^\dagger} = S_{\mathcal{E}}^\dagger\).

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

Return the operator composition with another Kraus.

Parâmetros:

• other (Kraus) – a Kraus object.

• qargs (list or None) – Optional, 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].

Retorno:

The composed Kraus.

Tipo de retorno:

Kraus

Levanta:

QiskitError – if other cannot be converted to an operator, or has incompatible dimensions for specified subsystems.

Nota

Composition (&) by default is defined as left matrix multiplication for matrix operators, while @ (equivalent to dot()) is defined as right matrix multiplication. That is that A & B == A.compose(B) is equivalent to B @ A == B.dot(A) when A and B are of the same type.

Setting the front=True kwarg changes this to right matrix multiplication and is equivalent to the dot() method A.dot(B) == A.compose(B, front=True).

conjugate()

Return the conjugate quantum channel.

Nota

This is equivalent to the matrix complex conjugate in the SuperOp representation ie. for a channel \(\mathcal{E}\), the SuperOp of the conjugate channel \(\overline{{\mathcal{{E}}}}\) is \(S_{\overline{\mathcal{E}^\dagger}} = \overline{S_{\mathcal{E}}}\).

copy()

Make a deep copy of current operator.

dot(other, qargs=None)

Return the right multiplied operator self * other.

Parâmetros:

• other (Operator) – an operator object.

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

Retorno:

The right matrix multiplied Operator.

Tipo de retorno:

Operator

Nota

The dot product can be obtained using the @ binary operator. Hence a.dot(b) is equivalent to a @ b.

expand(other)

Return the reverse-order tensor product with another Kraus.

Parâmetros:

other (Kraus) – a Kraus object.

Retorno:

the tensor product \(b \otimes a\), where \(a\)

is the current Kraus, and \(b\) is the other Kraus.

Tipo de retorno:

Kraus

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)

Tipo de retorno:

bool

is_cptp(atol=None, rtol=None)

Return True if completely-positive trace-preserving.

is_tp(atol=None, rtol=None)

Test if a channel is trace-preserving (TP)

Tipo de retorno:

bool

is_unitary(atol=None, rtol=None)

Return True if QuantumChannel is a unitary channel.

Tipo de retorno:

bool

output_dims(qargs=None)

Return tuple of output dimension for specified subsystems.

power(n)

Return the power of the quantum channel.

Parâmetros:

n (float) – the power exponent.

Retorno:

the channel \(\mathcal{{E}} ^n\).

Tipo de retorno:

SuperOp

Levanta:

QiskitError – if the input and output dimensions of the SuperOp are not equal.

Nota

For non-positive or non-integer exponents the power is defined as the matrix power of the SuperOp representation ie. for a channel \(\mathcal{{E}}\), the SuperOp of the powered channel \(\mathcal{{E}}^\n\) is \(S_{{\mathcal{{E}}^n}} = S_{{\mathcal{{E}}}}^n\).

reshape(input_dims=None, output_dims=None, num_qubits=None)

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

Parâmetros:

• 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].

• num_qubits (None or int) – reshape to an N-qubit operator [Default: None].

Retorno:

returns self with reshaped input and output dimensions.

Tipo de retorno:

BaseOperator

Levanta:

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

tensor(other)

Return the tensor product with another Kraus.

Parâmetros:

other (Kraus) – a Kraus object.

Retorno:

the tensor product \(a \otimes b\), where \(a\)

is the current Kraus, and \(b\) is the other Kraus.

Tipo de retorno:

Kraus

Nota

The tensor product can be obtained using the ^ binary operator. Hence a.tensor(b) is equivalent to a ^ b.

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.

Retorno:

A kraus instruction for the channel.

Tipo de retorno:

qiskit.circuit.Instruction

Levanta:

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

to_operator()

Try to convert channel to a unitary representation Operator.

Tipo de retorno:

Operator

transpose()

Return the transpose quantum channel.

Nota

This is equivalent to the matrix transpose in the SuperOp representation, ie. for a channel \(\mathcal{E}\), the SuperOp of the transpose channel \(\mathcal{{E}}^T\) is \(S_{mathcal{E}^T} = S_{\mathcal{E}}^T\).