Implementation of exponential extrapolator for the differentiable mitigate_with_zne functionality

Feature details

The newly added differentiable zero noise extrapolation (mitigate_with_zne) function only supports a polynomial extrapolation. mitiq equivalent of this function allows many other functions (e.g. exponential, linear, poly-exponential, etc. (see)). I believe it would be a great addition to add these options.

Implementation

No response

How important would you say this feature is?

2: Somewhat important. Needed this quarter.

Additional information

No response

[CatalinaAlbornoz]

Hi @cnktysz, this issue has now been labelled as participating in Hacktoberfest! This is a very large open-source event that is taking place during the month of October. If you register for the event and contribute at least 4PRs in any of the participating projects you get the chance to win a Hacktoberfest t-shirt or plant a tree.

Let us know if you have any questions about this! Remember that for your contribution to count it needs to be accepted by October 31st.

[Qottmann]

Hi @cnktysz good idea! As long as you can trace it back to a polynomial fit it should be relatively straight-forward. I.e. if you mean by exponential fit $y=c_0\exp(c_1 x)$ then you can write a dummy function

def exp_extrapolate(x, y):
    logy = qml.math.log(y)
    coef = _polyfit(x, logy, 1)
    return qml.math.exp(coef[1])

since $\log(y) = \log(c_1) + c_0 x$ and the extrapolated result is $y(0)=c_1$.

For extrapolation that cannot be reduced to polynomial fitting it gets more complicated. To preserve differentiability you have to stick to pennylane.math functions. Currently, the backbone in the ZNE is the custom _polyfit function (see pennylane/transforms/mitigate.py#L247), and you would have to write a more elaborate non-linear fitting function. This is totally possible, e.g. using backprop and doing a least-squares fit, but just a little more involved.

Alternatively, you can use any numpy/scipy fitting function if you can live without automatic differentiability!