angle solver qsp/qsvt

[KetpuntoG]

Context:

This PR appears in the context of the QSVT project, specifically focusing on the implementation of the angle solver, functionality to calculate the angles needed to apply a polynomial transformation.

Description of the Change:

Added qml.poly_to_angles, the main functionality of the PR that takes as input a polynomial and returns the angles.
Also included is the function qml.transform_angles, which transforms the angle format between qsp and qsvt.
Both these functions and other auxiliary functions have been added in qsvt.py.

Benefits:

Calculating angles is not a simple task and existing libraries have some limitations. With this new functionality we will facilitate the use of routines such as qsp and qsvt.

Possible Drawbacks:

Currently only one solver has been introduced. In the future it is expected to add more that will work for more extreme cases: larger polynomials (degree > 1000) or degenerate polynomials

[sc-72631]

[github-actions]

Hello. You may have forgotten to update the changelog!
Please edit doc/releases/changelog-dev.md with:

• A one-to-two sentence description of the change. You may include a small working example for new features.
• A link back to this PR.
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[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 99.46%. Comparing base (828c1ec) to head (91eb3a8).

Additional details and impacted files

@@           Coverage Diff           @@
##           master    #6483   +/-   ##
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  Coverage   99.46%   99.46%           
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  Files         454      454           
  Lines       42649    42716   +67     
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+ Hits        42420    42487   +67     
  Misses        229      229           

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[KetpuntoG]

I had to put this in because codefactor complained that I was using global_phase before giving it any value in line 148

[Jaybsoni]

Mostly small comments on the doc-strings and general questions. Great work! The PR looks good.

[Jaybsoni]

It doesn't seem like it matches the comment?

Suggested change

	R = Polynomial.basis(poly_degree) - Polynomial(P) * Polynomial(np.conj(P[::-1]))
	R = Polynomial.basis(poly_degree) * (1 - Polynomial(P) * Polynomial(np.conj(P[::-1])))

Also why do you need to do P[::-1]?

[KetpuntoG]

Yep, actually this is a bit unintuitive at first glance

The key here is that if you have $P(z) = a + bz^1 + cz^2$, then
$P(1/z) = a + bz^{-1} + cz^{-2}$
Which is equal to
$P(1/z)= z^{-2} (a z^2 + bz + c) $
Which is equal to
$P(1/z) = z^{-2}(P[::-1])$

and it can be simplified as the expression showed

[JerryChen97]

Basically, here we decomposed the reflection into reversal + trasnlation

[soranjh]

Please consider adding a brief description as comment to the code, specially if you have modified the equations of the paper.

[Jaybsoni]

This just tests that transform(transform(angles, "QSP", "QSVT"), "QSVT", "QSP") is the identity operation. We should check that for a particular polynomial's QSP angles should produce the same poly with transformed QSVT angles.

[KetpuntoG]

internally when I ask for the QSVT angles, I calculate the QSP angles and apply the function. So it is going always fulfilled (whether the function works correctly or not).

[soranjh]

Thanks @KetpuntoG, left my first round of comments.

[soranjh]

Not sure I understand this.

[KetpuntoG]

P should mee that |P(x)| ≤ 1
As it is not always easy to detect and the user may not be aware, I am checking that it is true at least in x = -1, x = 0, x = 1. I updated the comment 👍