channel_metrics.diamond_norm

Compute the diamond norm between two quantum channels.

Module Contents

Functions

diamond_norm(choi_1, choi_2)

Return the diamond norm distance between two quantum channels.
channel_metrics.diamond_norm.diamond_norm(choi_1, choi_2)

Return the diamond norm distance between two quantum channels.

The calculation uses the simplified semidefinite program of Watrous in [1].

This function has been adapted from [1].

Note

This calculation becomes very slow for 4 or more qubits.

Examples

Consider the depolarizing and identity channels in a 2-dimensional space. The depolarizing channel parameter is set to 0.2:

>>> import numpy as np
>>> from toqito.channels import depolarizing
>>> from toqito.channel_metrics import diamond_norm
>>> choi_depolarizing = depolarizing(dim=2, param_p=0.2)
>>> choi_identity = np.identity(2**2)
>>> dn = diamond_norm(choi_depolarizing, choi_identity)
>>> print("Diamond norm between depolarizing and identity channels: ", '%.2f' % dn)
Diamond norm between depolarizing and identity channels:  -0.00

Note

You do not need to use ‘%.2f’ % when you use this function.

We use this to format our output such that doctest compares the calculated output to the expected output upto two decimal points only. The accuracy of the solvers can calculate the float output to a certain amount of precision such that the value deviates after a few digits of accuracy.

Similarly, we can compute the diamond norm between the dephasing channel (with parameter 0.3) and the identity channel:

>>> import numpy as np
>>> from toqito.channels import dephasing
>>> from toqito.channel_metrics import diamond_norm
>>> choi_dephasing = dephasing(dim=2)
>>> choi_identity = np.identity(2**2)
>>> dn = diamond_norm(choi_dephasing, choi_identity)
>>> print("Diamond norm between dephasing and identity channels: ", '%.2f' % dn)
Diamond norm between dephasing and identity channels:  -0.00

Note

You do not need to use ‘%.2f’ % when you use this function.

We use this to format our output such that doctest compares the calculated output to the expected output upto two decimal points only. The accuracy of the solvers can calculate the float output to a certain amount of precision such that the value deviates after a few digits of accuracy.

References

[1]

John Watrous. Semidefinite programs for completely bounded norms. 2009. arXiv:0901.4709.

[2]

Rigetti. Forest benchmarking. https://github.com/rigetti/forest-benchmarking.

Raises:

  • ValueError – If matrices are not of equal dimension.

  • ValueError – If matrices are not square.

Parameters:

  • choi_1 (numpy.ndarray) – A 4**N by 4**N matrix (where N is the number of qubits).

  • choi_2 (numpy.ndarray) – A 4**N by 4**N matrix (where N is the number of qubits).

Return type:

float