Creation of quantum_info.quantum_networks submodule and QuantumNetwork class

[renatomello]

This closes #1077

Checklist:

• Reviewers confirm new code works as expected.
• Tests are passing.
• Coverage does not decrease.
• Documentation is updated.

[codecov]

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##            master     #1137    +/-   ##
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  Coverage   100.00%   100.00%            
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  Files           68        69     +1     
  Lines         9906     10084   +178     
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[Canoming]

@Canoming is the use of np.emath.sqrt necessary? i.e. can a quantum network be multiplied by a complex number?

Ah, sorry. I just rethought it. According to our construction of non-pure networks from pure networks, the phase of the complex will be omitted. So here is the expected behavior of the multiplication and division:

1. If the number multiplied or divided is a positive number, we'll keep a pure network to be pure, so we can save the memory. The pure network will be multiplied/divided by the square root of the number (which is equivalent to multiply/divide the non-pure network with the target number)
2. If the number is non-positive, we'll convert everything to non-pure networks. The multiplication and division of negative number will convert a semidefinite positive operator to semidefinite negative operator, which cannot be represented by pure networks.

I've updated __mul__ and __truediv__ accordingly. Please check if my code behaves as what

[Canoming]

I updated the test file to match the change of behavior of __mul__ and __truediv__.

[Canoming]

It looks like a basic functional class now.

[renatomello]

@Canoming could you help me beef up the documentation for QuantumNetworks in general? It would be nice to have detailed documentation for other people that are going to use this class. Maybe with some examples on how to use it too

[renatomello]

@Canoming is the use of np.emath.sqrt necessary? i.e. can a quantum network be multiplied by a complex number?

Ah, sorry. I just rethought it. According to our construction of non-pure networks from pure networks, the phase of the complex will be omitted. So here is the expected behavior of the multiplication and division:

1. If the `number` multiplied or divided is a positive number, we'll keep a pure network to be pure, so we can save the memory. The pure network will be multiplied/divided by the square root of the `number` (which is equivalent to multiply/divide the non-pure network with the target `number`)

2. If the `number` is non-positive, we'll convert everything to non-pure networks. The multiplication and division of negative number will convert a semidefinite positive operator to semidefinite negative operator, which cannot be represented by pure networks.

I've updated __mul__ and __truediv__ accordingly. Please check if my code behaves as what

Could you add that to the docstrings of these functions too? Thanks.

[Canoming]

@Canoming could you help me beef up the documentation for QuantumNetworks in general? It would be nice to have detailed documentation for other people that are going to use this class. Maybe with some examples on how to use it too

Ah, sorry for the late reply. I can write some documentation for this class. But should we restore the quantum_network branch? The current version is only a prototype; we should add more functionality to this class later. Are we supposed to create a new branch for the next stage of development?

[renatomello]

Ah, sorry for the late reply. I can write some documentation for this class. But should we restore the quantum_network branch? The current version is only a prototype; we should add more functionality to this class later. Are we supposed to create a new branch for the next stage of development?

As long as the branch is up-to-date with master, it makes no difference if you restore this one or create a new one

[Canoming]

Emm.. I just found an issue in the latest version of QuantumNetwork. The following code will crash the kernel:

import numpy as np

from qibo.quantum_info.quantum_networks import QuantumNetwork
from qibo.gates import DepolarizingChannel

# Create a quantum network from numpy array is fine
test = np.ones((2,2,2,2))
test_choi = QuantumNetwork(test, (2,2,2,2), pure=True)
print(test_choi.partition)
print(test_choi.system_output)

# But create a quantum network from the output of the old quantum channel will crash the kernal
test_ch = DepolarizingChannel(0,0.5)
N = len(test_ch.target_qubits)
partition = (2**N, 2**N)
depolar_choi = QuantumNetwork(test_ch.to_choi(), partition)
print(f'A quantum channel is a quantum netowrk of the form {depolar_choi}')

The creation of the QuantumNetwork, depolar_choi, will prompt the error and crash the kernel:

python: symbol lookup error: /home/ximing/miniconda3/envs/qibo_dev/lib/python3.11/site-packages/mkl/../../../libmkl_intel_thread.so.2: undefined symbol: omp_get_num_procs

I used these codes as the demo use case of QuantumNetwork, and they worked before. Not sure what causes the problem. Seems DepolarizingChannel hasn't changed in the recent commits. But as the kernel crashes, the debug functions also crash. Could you help to figure out what happened to these codes?

By the way, I'm also not sure where to put the documentation of QuantumNetwork. I think all the examples in qibo/examples are aiming for very specific algorithms. But QuantumNetwork is a very fundamental mathematical concept, and we haven't implemented the conversion of QuantumNetwork with other classes in qibo back and forth. Do you have any suggestions of where to put the documentation?

[BrunoLiegiBastonLiegi]

A couple of words to introduce this might be helpful

[BrunoLiegiBastonLiegi]

I believe this will fail if the QuantumNetwork was created with backend=None.
In the __init__ you should probably do:

self._backend = backend if backend is not None else GlobalBackend()

[BrunoLiegiBastonLiegi]

Suggested change

	"""Returns bool indicating if Choi operator :math:`\\mathcal{E}` of the networn is positive-semidefinite.
	"""Returns bool indicating if Choi operator :math:`\\mathcal{E}` of the network is positive-semidefinite.

[BrunoLiegiBastonLiegi]

Suggested change

	if self.pure() and number > 0.0:
	return QuantumNetwork(
	self.matrix(backend=self._backend) / np.sqrt(number),
	partition=self.partition,
	system_output=self.system_output,
	pure=True,
	backend=self._backend,
	)
	return QuantumNetwork(
	self.matrix(backend=self._backend) / number,
	partition=self.partition,
	system_output=self.system_output,
	pure=False,
	backend=self._backend,
	)
	number = np.sqrt(number) if self.pure() and number > 0.0 else number
	return QuantumNetwork(
	self.matrix(backend=self._backend) / number,
	partition=self.partition,
	system_output=self.system_output,
	pure=self.pure(),
	backend=self._backend,
	)