Align definition of scale factor with Zero-Noise Extrapolation literature

[cosenal]

The mitigate_with_zne API in Catalyst accepts a scale_factors argument, which represents a range of scalar factors by which the circuit is scaled.

In the literature of Zero-Noise Extrapolation ¹², a scale factor value $\lambda$ is the ratio between the number of gates of the scaled circuit and the number of gates of the original circuit. However, in the Catalyst implementation, the term scale factor is used to denote the number $n$ of foldings of the circuit.

The relation between the two definitions is $\lambda = 1 + 2n$.

For example, scale_factor = 1 (i.e., $\lambda=1$, $n=0$) means that the circuit is not folded.
For scale_factor = 3 ($\lambda=3$, $n=1$), we have that the circuit is folded once

$$C \rightarrow C(C^{\dagger}C)$$

and the number of gates in the scaled circuit is three times the number of gates in the original circuit.
(See ² for a discussion on odd/even/non-integer scale factors.)

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This issue is about aligning the definition of scale factor in Catalyst with the literature as above, by either renaming the scale_factors argument in the API, or assigning a new meaning to it.

Footnotes

1. https://arxiv.org/abs/2005.10921 ↩

2. https://mitiq.readthedocs.io/en/v0.38.0/guide/zne-3-options.html#what-additional-options-are-available-when-using-zne ↩ ↩²

[cosenal]

Implementation in non-catalyst pennylane, for reference: https://github.com/PennyLaneAI/pennylane/blob/1d220cf6d69cbf60c99c5318b5137b69f7d6d30d/pennylane/transforms/mitigate.py#L214-L217

[rmoyard]

I will tackle that in the upcoming days 👍

[cosenal]

Nice, thanks @rmoyard. Please keep me in the review loop of the design and PRs.
Is the plan to do the same as Pennylane, that is, convert the scale factor in a folding factor and keeping the folding as is? Is it going to be in the frontend or in the dialect?

[rmoyard]

Yes I think that is the best, I will either add some restrictions because we don't support the fraction scale or just just ignore the fraction scale.

num_global_folds, _ = _divmod(scale_factor - 1, 2)

[cosenal]

num_global_foldsnum_folds, since this will have to be the same way for both global and local folding.