Linear function usability improvements

[alexanderivrii]

Summary

This PR adds a few usability improvements to LinearFunctions:

• Constructing linear functions from more general quantum circuits that may contain barriers and delays, permutations, other linear functions, cliffords (when these represent valid linear functions), and nested quantum circuits of this form. (Note that this is now similar to how Cliffords can be constructed).

• Adding the __eq__ method, so that two linear functions are considered equivalent when their defining binary matrices are.

• Required for the first bullet (and very similar to Clifford's _pad_with_identity method), there is also a new method extend_with_identity which allows to extend a linear function over k qubits to a linear function over n >= k qubits, specifying the new positions of the original qubits and padding with identities on the remaining qubits.

• Added methods for pretty-printing LinearFunctions, either as a binary matrix or as the corresponding linear transformation.

Additional possible improvements (not implemented):

• In principle we could also easily add that ability to construct a linear function from a general operator, by first attempting to construct a clifford out of it (as per Add Clifford.from_matrix #9475), and then attempting to construct a linear function from this clifford.

• When "n == k", the extend_with_identity method can be used to permute the qubits of the linear function, which could be useful in itself.

Details:

For pretty-printing (aka visualizing) linear functions, I have added the methods mat_str() and function_str(), please feel free to suggest better names. Personally, I find all of the three printing methods quite useful (with the last one specifically requested by @ShellyGarion).
For example, running the following code

qc = QuantumCircuit(4)
qc.cx(0, 1)
qc.swap(1, 2)
qc.cx(2, 3)
qc.swap(1, 3)
lf = LinearFunction(qc)
print(lf)
print(lf.mat_str())
print(lf.function_str())

produces the following output

Instruction(name='linear_function', num_qubits=4, num_clbits=0, params=[array([[ True, False, False, False],
       [ True,  True, False,  True],
       [ True,  True, False, False],
       [False, False,  True, False]]), <qiskit.circuit.quantumcircuit.QuantumCircuit object at 0x0000024C99EDD6F0>])
[[1 0 0 0]
 [1 1 0 1]
 [1 1 0 0]
 [0 0 1 0]]
(x_0, x_0 + x_1 + x_3, x_0 + x_1, x_2)

The default representation is a bit ugly, however serves to remind that the entries of the matrix are booleans and not integers (same as for Cliffords) and also shows that the linear matrix has the "original_circuit" field. The mat_str printing method is useful to think of the linear function as a binary matrix, while the function_str method is useful to think of the linear function as a linear transformation aka a function that maps (x_0, x_1, x_2, x_3) to (x_0, x_0 + x_1 + x_3, x_0 + x_1, x_2).

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[coveralls]

Pull Request Test Coverage Report for Build 5561081491

• 62 of 79 (78.48%) changed or added relevant lines in 1 file are covered.
• 419 unchanged lines in 26 files lost coverage.
• Overall coverage increased (+0.07%) to 86.057%

---

Changes Missing Coverage	Covered Lines	Changed/Added Lines	%
qiskit/circuit/library/generalized_gates/linear_function.py	62	79	78.48%

Files with Coverage Reduction	New Missed Lines	%
crates/qasm2/src/expr.rs	1	93.76%
crates/qasm2/src/lex.rs	1	91.9%
qiskit/qasm/init.py	1	80.0%
qiskit/transpiler/passes/layout/set_layout.py	1	95.0%
qiskit/transpiler/passes/routing/layout_transformation.py	1	76.09%
qiskit/circuit/controlflow/_builder_utils.py	2	97.94%
qiskit/circuit/quantumcircuitdata.py	2	86.36%
qiskit/transpiler/passes/routing/algorithms/token_swapper.py	2	92.31%
qiskit/qasm/pygments/init.py	3	0%
qiskit/circuit/controlflow/if_else.py	4	97.71%

Totals
Change from base Build 5490783039:	0.07%
Covered Lines:	72312
Relevant Lines:	84028

---

💛 - Coveralls

[ShellyGarion]

Thank you very much for this PR, it is very helpful!

I have a few comments and suggestions:

• could you add the pretty printing functions to the release notes?
• does it make sense to add a basic tests for the printing of function_str?
• perhaps it's worth to add some pseudo-random tests, to check that all these elements can go well inside the circuit:
  -- choose pseudo-randomly linear gates, barriers, delays, permutations, linear functions, linear cliffords and create a circuit qc
  -- calculate Clifford(qc), calculate LinearFunction(qc), and check that you get the proper linear matrix inside the clifford.

[alexanderivrii]

@ShellyGarion, thanks for the review!

could you add the pretty printing functions to the release notes?

This is done in 68c2aa9.

does it make sense to add a basic tests for the printing of function_str?

I don't know if it's worth checking the output of pretty-printing functions, so I didn't add such a test for now.

perhaps it's worth to add some pseudo-random tests

I agree that this is a good idea; this is done in 6190778 following your suggested approach. To support this I had to extend the random_linear_circuit function (from test_linear_function.py) to support barriers, delays, permutation gates, linear functions, cliffords (but only corresponding to linear functions), and nested linear subcircuits. Though the default behavior matches the old behavior of the function: only flat circuit with CX and SWAP gates are generated.

[ShellyGarion]

LGTM, thanks @alexanderivrii !

[mtreinish]

Overall this LGTM, I just had a couple of small comments and questions inline.

[mtreinish]

LGTM, thanks for the quick updates