Kraus

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

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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](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.

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

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

Return settings.

Methods

adjoint()

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Note

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

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

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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()

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Return the conjugate quantum channel.

Note

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

copy()

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Make a deep copy of current operator.

dot

dot(other, qargs=None)

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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)

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Return the reverse-order tensor product with another Kraus.

Parameters

other (Kraus) – a Kraus object.

Returns

the tensor product $b \otimes a$, where $a$

is the current Kraus, and $b$ is the other Kraus.

Return type

Kraus

input_dims

input_dims(qargs=None)

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Return tuple of input dimension for specified subsystems.

is_cp

is_cp(atol=None, rtol=None)

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

Return type

bool(opens in a new tab)

is_cptp

is_cptp(atol=None, rtol=None)

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Return True if completely-positive trace-preserving.

is_tp

is_tp(atol=None, rtol=None)

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Test if a channel is trace-preserving (TP)

Return type

bool(opens in a new tab)

is_unitary

is_unitary(atol=None, rtol=None)

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Return True if QuantumChannel is a unitary channel.

Return type

bool(opens in a new tab)

output_dims

output_dims(qargs=None)

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Return tuple of output dimension for specified subsystems.

power

power(n)

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Return the power of the quantum channel.

Parameters

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

Returns

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

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 $\mathcal{{E}}$, the SuperOp of the powered channel $\mathcal{{E}}^n$ is $S_{{\mathcal{{E}}^n}} = S_{{\mathcal{{E}}}}^n$.

reshape

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

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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)

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Return the tensor product with another Kraus.

Parameters

other (Kraus) – a Kraus object.

Returns

the tensor product $a \otimes b$, where $a$

is the current Kraus, and $b$ 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()

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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()

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Try to convert channel to a unitary representation Operator.

Return type

Operator

transpose

transpose()

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Return the transpose quantum channel.

Note

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 $S_{mathcal{E}^T} = S_{\mathcal{E}}^T$.