Adds Fourier coefficient calculation module

[glassnotes]

Context: The Fourier spectrum of a quantum model can be used to gauge its expressivity, as demonstrated in Schuld, Sweke and Meyer (2020) and Vidal and Theis, 2019.

Description of the Change: A new module, fourier, and associated documentation and visualization methods. Note: the visualization tools are being developed in #1198 to keep the review process more manageable.

Benefits: Users now have built-in tools and examples for computing the Fourier spectra and coefficients of quantum circuits.

Possible Drawbacks: Spectra are currently only calculated for a subset of possible operations. Further work is required to extend this.

Related GitHub Issues: N/A

[github-actions]

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[mariaschuld]

The module docstring is now like this:

Everything is clickable. Maybe we can use the format also for kernels and groupings.

[josh146]

The module docstring is now like this:

It looks fantastic!

[mariaschuld]

@glassnotes only one failing test, which is not even related to the module:

Traceback (most recent call last):
  File "/home/runner/work/pennylane/pennylane/tests/interfaces/test_qnode_jax.py", line 72, in test_simple_jacobian
    np.testing.assert_allclose(grads, np.array([-0.09784342, -0.19767685]))
  File "/opt/hostedtoolcache/Python/3.8.10/x64/lib/python3.8/site-packages/numpy/testing/_private/utils.py", line 1527, in assert_allclose
    assert_array_compare(compare, actual, desired, err_msg=str(err_msg),
  File "/opt/hostedtoolcache/Python/3.8.10/x64/lib/python3.8/site-packages/numpy/testing/_private/utils.py", line 840, in assert_array_compare
    raise AssertionError(msg)
AssertionError: 
Not equal to tolerance rtol=1e-07, atol=0

Mismatched elements: 2 / 2 (100%)
Max absolute difference: 3.83459161e-08
Max relative difference: 2.55436125e-07
 x: array([-0.097843, -0.197677])
 y: array([-0.097843, -0.197677])

Maybe this is caused by config.update("jax_enable_x64", True), and we have to disable it again. Let me try.

[glassnotes]

The module docstring is now like this:

Everything is clickable. Maybe we can use the format also for kernels and groupings.

Agreed! I will make an issue for this.

[glassnotes]

Maybe this is caused by config.update("jax_enable_x64", True), and we have to disable it again. Let me try.

Yes I think that's the reason. A temporary fix would be to not enable 64-bit support, and just set the equality threshold at something higher, like 1e-6. I'm not a big fan of this, but to get everything to work nicely with JAX requires modifying stuff outside the scope of the PR (see #1363 )

[mariaschuld]

Maybe this is caused by config.update("jax_enable_x64", True), and we have to disable it again. Let me try.

I can locally run the tests :) Let's see if CI is happy.

[mariaschuld]

@josh146 sorry for the delay. Happy if we merge? The codecov should only be one import line that is not tested (plus the strange omission of seeing that coefficients is covered).

[josh146]

yep go for it!