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Kraus

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

GitHub(opens in a new tab)

Bases: QuantumChannel

Kraus representation of a quantum channel.

For a quantum channel E\mathcal{E}, the Kraus representation is given by a set of matrices [A0,...,AK1][A_0,...,A_{K-1}] such that the evolution of a DensityMatrix ρ\rho is given by

E(ρ)=i=0K1AiρAi\mathcal{E}(\rho) = \sum_{i=0}^{K-1} A_i \rho A_i^\dagger

A general operator map G\mathcal{G} can also be written using the generalized Kraus representation which is given by two sets of matrices [A0,...,AK1][A_0,...,A_{K-1}], [B0,...,AB1][B_0,...,A_{B-1}] such that

G(ρ)=i=0K1AiρBi\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](opens in a new tab)

Initialize a quantum channel Kraus operator.

Parameters

Raises

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 (2N,2N)(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

Default value: 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

Default value: 1e-05

settings

Return settings.


Methods

adjoint

adjoint()

Return the adjoint quantum channel.

Note

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

compose

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

Return the operator composition with another Kraus.

Parameters

  • other (Kraus) – a Kraus object.
  • qargs (list(opens in a new tab) or None) – Optional, a list of subsystem positions to apply other on. If None apply on all subsystems (default: None).
  • front (bool(opens in a new tab)) – If True compose using right operator multiplication, instead of left multiplication [default: False].

Returns

The composed Kraus.

Return type

Kraus

Raises

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

Note

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

conjugate()

Return the conjugate quantum channel.

Note

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

copy

copy()

Make a deep copy of current operator.

dot

dot(other, qargs=None)

Return the right multiplied operator self * other.

Parameters

  • other (Operator) – an operator object.
  • qargs (list(opens in a new tab) or None) – Optional, a list of subsystem positions to apply other on. If None apply on all subsystems (default: None).

Returns

The right matrix multiplied Operator.

Return type

Operator

Note

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

expand

expand(other)

Return the reverse-order tensor product with another Kraus.

Parameters

other (Kraus) – a Kraus object.

Returns

the tensor product bab \otimes a, where aa

is the current Kraus, and bb is the other Kraus.

Return type

Kraus

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)

Return type

bool(opens in a new tab)

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 trace-preserving (TP)

Return type

bool(opens in a new tab)

is_unitary

is_unitary(atol=None, rtol=None)

Return True if QuantumChannel is a unitary channel.

Return type

bool(opens in a new tab)

output_dims

output_dims(qargs=None)

Return tuple of output dimension for specified subsystems.

power

power(n)

Return the power of the quantum channel.

Parameters

n (float(opens in a new tab)) – the power exponent.

Returns

the channel En\mathcal{{E}} ^n.

Return type

SuperOp

Raises

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

Note

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

reshape

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

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

Parameters

  • input_dims (None or tuple(opens in a new tab)) – new subsystem input dimensions. If None the original input dims will be preserved [Default: None].
  • output_dims (None or tuple(opens in a new tab)) – new subsystem output dimensions. If None the original output dims will be preserved [Default: None].
  • num_qubits (None or int(opens in a new tab)) – reshape to an N-qubit operator [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.

tensor

tensor(other)

Return the tensor product with another Kraus.

Parameters

other (Kraus) – a Kraus object.

Returns

the tensor product aba \otimes b, where aa

is the current Kraus, and bb is the other Kraus.

Return type

Kraus

Note

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

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.

Return type

Operator

transpose

transpose()

Return the transpose quantum channel.

Note

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

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