mimiqcircuits.operations.krauschannel¶

Kraus channel definitions.

Classes

krauschannel()

Supertype for all N-qubit Kraus channels.

class mimiqcircuits.operations.krauschannel.krauschannel[source]¶

Bases: Operation

Supertype for all N-qubit Kraus channels.

A Kraus channel is a quantum operation on a density matrix \(\\rho\) of the form:

\[\mathcal{E}(\rho) = \sum_k E_k \rho E_k^\dagger,\]

where \(E_k\) are Kraus operators that need to fulfill \(\sum_k E_k^\dagger E_k \leq I\).

Special Properties:

isCPTP(): A Kraus channel is a completely positive and trace-preserving (CPTP) operation when \(\\sum_k E_k^\dagger E_k = I\). Currently, all noise channels are CPTP.
ismixedunitary(): A Kraus channel is called a mixed unitary channel when the Kraus operators \(E_k\) are each proportional to a unitary matrix \(U_k\), i.e., when \(\\mathcal{E}(\\rho) = \\sum_k p_k U_k \\rho U_k^\dagger\) with some probabilities \(0 \leq p_k \leq 1\) that add up to 1 and \(U_k^\dagger U_k = I\).

krausmatrices()[source]¶

Returns the Kraus matrices associated with the given Kraus channel.

A mixed unitary channel is written as:

\[\mathcal{E}(\rho) = \sum_k p_k U_k \rho U_k^\dagger,\]

where \(U_k\) are the unitary matrices returned by this function.

If the Kraus channel is parametric, the matrix elements are wrapped in a symengine or sympy object.

Returns:: A list of symengine matrices representing the Kraus operators.
Return type:: list

unwrappedkrausmatrices()[source]¶

Returns the unitary Kraus matrices associated to the mixed unitary Kraus channel without symbolic wrapper.

See also

unitarymatrices()

Example

>>> from mimiqcircuits import *
>>> op = PauliX(0.2)
>>> op.unwrappedkrausmatrices()
[[0.894427190999916, 0]
[0, 0.894427190999916]
, [0, 0.447213595499958]
[0.447213595499958, 0]
]

classmethod isCPTP()[source]¶

Determine whether the Kraus channel is Completely Positive and Trace Preserving (CPTP).

A quantum operation is CPTP if the Kraus operators fulfill the condition:

\[\sum_k E_k^\dagger E_k = I\]

If the condition is:

\[\sum_k E_k^\dagger E_k < I\]

then the operation is not CPTP.

Note

Currently, all noise channels are considered CPTP.

Parameters:: krauschannel – The Kraus channel to check.
Returns:: True if the channel is CPTP, False otherwise.
Return type:: bool

classmethod ismixedunitary()[source]¶

Determine whether the quantum operation is a mixed unitary channel.

A channel is considered mixed unitary if all the Kraus operators \(E_k\) are proportional to a unitary matrix \(U_k\), i.e.,

\[\mathcal{E}(\rho) = \sum_k p_k U_k \rho U_k^\dagger,\]

with some probabilities \(0 \leq p_k \leq 1\) that add up to 1, and \(U_k^\dagger U_k = I\).

Parameters:: krauschannel – The Kraus channel to check.
Returns:: True if the channel is a mixed unitary channel, False otherwise.
Return type:: bool

Examples

>>> from mimiqcircuits import *
>>> PauliX(0.1).ismixedunitary()
True

>>> AmplitudeDamping(0.1).ismixedunitary()
False

inverse()[source]¶

power(n)[source]¶

iswrapper()[source]¶

probabilities()[source]¶

Returns the probabilities for each Kraus operator in a mixed unitary channel.

A mixed unitary channel is written as:

\[\mathcal{E}(\rho) = \sum_k p_k U_k \rho U_k^\dagger,\]

where \(p_k\) are the probabilities.

This method is valid only for mixed unitary channels.

Returns:: A list of probabilities for each Kraus operator.
Return type:: list

numparams()[source]¶

cumprobabilities()[source]¶

Returns the cumulative sum of probabilities for each Kraus operator in a mixed unitary channel.

A mixed unitary channel is written as:

\[\mathcal{E}(\rho) = \sum_k p_k U_k \rho U_k^\dagger,\]

where \(p_k\) are the probabilities.

This method is valid only for mixed unitary channels.

Returns:: Cumulative probabilities for the Kraus operators.
Return type:: list

Examples

>>> from mimiqcircuits import *
>>> Depolarizing1(0.1).cumprobabilities()
[0.9, 0.933333333333333, 0.966666666666667, 1.0]

unwrappedcumprobabilities()[source]¶

Returns the cumulative sum of probabilities for the mixed unitary channel without symbolic wrapping.

See also

cumprobabilities()

Returns:: A list of cumulative probabilities as float values.
Return type:: list

krausoperators()[source]¶

Returns the Kraus operators associated with the given Kraus channel.

This should be implemented for each specific channel.

Returns:: A list of matrices representing the Kraus operators.
Return type:: list

haslossybranch()[source]¶

Whether any branch of the channel loses qubits when drawn.

That is whether any of its Kraus operators is a LossyOperator. It is False for channels without loss branches, such as PauliX(0.1).

Examples

>>> from mimiqcircuits import *
>>> PauliX(0.1).haslossybranch()
False
>>> import numpy as np
>>> survival = Operator(np.array([[1, 0], [0, np.sqrt(0.8)]]))
>>> lossy = LossyOperator(np.array([[0, 0], [0, np.sqrt(0.2)]]))
>>> Kraus([survival, lossy]).haslossybranch()
True

squaredkrausoperators()[source]¶

Returns the square of the Kraus operators (\(E_k^\dagger E_k\)). This computes the Hermitian conjugate (dagger) of each Kraus operator and multiplies it by the operator itself.

Returns:: A list of squared Kraus operators.
Return type:: list

unitarymatrices()[source]¶

Unitary matrices associated with the given mixed unitary Kraus channel.

A mixed unitary channel is written as:

\[\mathcal{E}(\rho) = \sum_k p_k U_k \rho U_k^\dagger,\]

where \(U_k\) are the unitary matrices.

An error is raised if the channel is not mixed unitary (i.e., ismixedunitary(self)==False).

Note

If the Kraus channel is parametric, the matrix elements are wrapped in a symbolic object (e.g., from sympy or symengine). To manipulate expressions, use the appropriate symbolic manipulation libraries.

Examples

>>> from mimiqcircuits import *
>>> PauliX(0.2).unitarymatrices()
[[1.0, 0]
[0, 1.0]
, [0, 1.0]
[1.0, 0]
]

unwrappedunitarymatrices()[source]¶

Returns the unitary matrices associated with the mixed unitary channel without symbolic wrapping.

Returns:: A list of unitary matrices as numerical values.
Return type:: list

unitarygates()[source]¶

Returns the unitary gates associated with the given mixed unitary Kraus channel.

A mixed unitary channel is written as:

\[\sum_k p_k U_k \rho U_k^\dagger,\]

where \(U_k\) are the unitary operators.

This method is valid only for mixed unitary channels.