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When the parameterized unitary is of the form $e^{-i\theta V}$, where $V$ is a Hermitian operator of the unitary, we can use parameter shift rule to calculate the gradient.

In this paper, it says: "We now move to analyze the case in which a parameter shift rule does not necessarily hold, but similar to the original analysis of standard barren plateaus the parameterized unitaries are sufficiently expressive. "

My question is, in which case the parameter shift rule does not hold? Can you give me some specific examples?

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Judging from the citations for the parameter-shift rule, it seems that the referenced paper only considers the rule for generators $V$ with two eigenvalues.

There is a number of works generalizing this, though:

Regarding the question "In which case the parameter shift rule does not hold?", the paper seems to talk about any cases in which $V$ has more than two unique eigenvalues. Beyond the paper's scope, the question is more interesting. There are particularly mean relationships between the generator eigenvalues that may make your life hard, but usually the automated recipe we give in the last reference above works out of the box, for example. The cost of the parameter-shift rule we developed depends crucially on the number of differences between eigenvalues of $V$, because those enter the cost function, which is a Fourier series, as frequencies.

I hope this helps. Let me know in case you are interested in particular types of gate generators $V$.

NB: If your gates are perturbed, as in $e^{i\theta V+iW}$ for example, there are additional generalizations to those, including the stochastic parameter-shift rule and the so-called "proper" shift rules, the latter including the so-called Nyquist shift rule.

*disclosure: I'm a coauthor of those.

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