Adding Kraus to channel

[Shivansh20128]

Description

This PR is a work I progress for adding the function kraus_to_channel.

#Resolves #14

Changes

• Added code for kraus_to_channel function.
• Added tests for the function.

Checklist

Before marking your PR ready for review, make sure you checked the following locally. If this is your first PR, you might be notified of some workflow failures after a maintainer has approved the workflow jobs to be run on your PR.

Additional information is available in the documentation.

• Use ruff for errors related to code style and formatting.
• Verify all previous and newly added unit tests pass in pytest.
• Check the documentation build does not lead to any failures. Sphinx build can be checked locally for any failures related to your PR
• Use linkcheck to check for broken links in the documentation
• Use doctest to verify the examples in the function docstrings work as expected.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 98.0%. Comparing base (1379480) to head (5eef4c6).

Additional details and impacted files

@@          Coverage Diff           @@
##           master   #1071   +/-   ##
======================================
  Coverage    98.0%   98.0%           
======================================
  Files         181     182    +1     
  Lines        3520    3525    +5     
  Branches      773     773           
======================================
+ Hits         3452    3457    +5     
  Misses         44      44           
  Partials       24      24           

☔ View full report in Codecov by Sentry.
📢 Have feedback on the report? Share it here.

🚀 New features to boost your workflow:

• ❄️ Test Analytics: Detect flaky tests, report on failures, and find test suite problems.

[purva-thakre]

Can you add a unit test where the superoperator acts on a state?

See Section 2.2 of https://arxiv.org/pdf/1509.02921 for more info.

It would also be pretty easy to construct a superoperator from the channels in Section 2.1.1

[Shivansh20128]

Before going into that, can you tell me if the function has been implemented correctly. Because I was not very sure about it. If you read the previous discussion on the issue, you will see that the two results, one from using the function on a vector, and other from directly applying a series of kraus operators were coming out to be transpose of each other.
To tackle that, I modified the implementation of the function a little by changing the order of A and B ( from (A, B.conj()) to (B, A.conj())) But I am not sure about it.

super_op = sum(tensor(B, A.conj()) for A, B in kraus_list)

[vprusso]

Hi @Shivansh20128 ,

Thanks for your response. From what I can tell (and what I recall) I believe your approach here is sensible.

Perhaps one way to sanity check your approach would be to convert a set of Kraus operators to a quantum channel. Then, if you can use some of the existing functionality in toqito to check that you can either go from the channel back to the operators, or apply the channel in a way that gives the same result as a quantum channel that was arrived at through another means, perhaps that would be serve as some good sanity checks?

I understand that some of the above was a bit vague, but in general, do you think that's a sensible approach?

[Shivansh20128]

apply the channel in a way that gives the same result as a quantum channel that was arrived at through another means, perhaps that would be serve as some good sanity checks?

I tried this approach here:

import numpy as np

from toqito.matrix_ops import tensor


def kraus_to_channel(
    kraus_list: list[tuple[np.ndarray, np.ndarray]]
) -> np.ndarray:
    """Convert a collection of Kraus operators into the corresponding quantum channel (superoperator).

    :param kraus_list: List of tuples (A, B) where A and B are Kraus operators as numpy arrays
    :return: The superoperator as a numpy array
    """
    super_op = sum(tensor(B, A.conj()) for A, B in kraus_list)
    return super_op

from toqito.channel_ops import apply_channel

kraus_operators = [
    (np.array([[1, 0], [0, 0]]), np.array([[1, 0], [0, 0]])),  # Projection onto |0⟩
    (np.array([[0, 1], [0, 0]]), np.array([[0, 1], [0, 0]])),  # Projection onto |1⟩
]

# Generate the quantum channel using your function
quantum_channel = kraus_to_channel(kraus_operators)

# Define a test quantum state (density matrix)
rho = np.array([[0.5, 0.5], [0.5, 0.5]])

# Apply the quantum channel using Toqito's apply_channel function
rho_after_channel = apply_channel(rho, kraus_operators)

# Apply the superoperator to the vectorized form of rho
rho_vec = rho.flatten("F")  # Column-major order
rho_after_super_op = quantum_channel @ rho_vec
rho_after_super_op = rho_after_super_op.reshape(2, 2, order="F")  # Reshape back

# Compare both methods
print("Using apply_channel:\n", rho_after_channel)
print("Using superoperator:\n", rho_after_super_op)
print("Difference:\n", rho_after_channel - rho_after_super_op)

# The difference should be close to zero
assert np.allclose(rho_after_channel, rho_after_super_op), "Mismatch in quantum channel application!"

And I am getting a match on the results:

Using apply_channel:
 [[1. 0.]
 [0. 0.]]
Using superoperator:
 [[1. 0.]
 [0. 0.]]
Difference:
 [[0. 0.]
 [0. 0.]]

If you also think this is good enough, we can go ahead.

[vprusso]

You might want to try other values of rho, but I think this is definitely the correct sanity check approach for what you are working on, and it seems right to me! Nice approach, @Shivansh20128 !

[Shivansh20128]

Okay. I will add more values of rho in the test cases.

[purva-thakre]

We have commonly predefined channels in toqito/channels.

[Shivansh20128]

I will work on this later.
Thanks

[purva-thakre]

We have quite a few functions to generate random things. See if those are useful, instead of manually defining your own randomly generated input.

https://toqito--1071.org.readthedocs.build/en/1071/autoapi/rand/index.html

[Shivansh20128]

I will work on this later.
Thanks

[vprusso]

Yes, I think specifically in this case the random_density_matrix function would be useful!

[purva-thakre]

@Shivansh20128 something has gone wrong with the formatting for this. I was trying to make these clickable similar to what I did in #1091

Each function should be :func:.name_function where .name_function is sandwiched between `

My apologies!