Overload Base.literal_pow for AbstractQ

[dkarrasch]

Fixes #53989.

There's a decision to be made here. What do we want Q^-1 to return? Q'::AbstractQ or Q'*I::Matrix? On Julia v1.9 and older, Q^p returned the matrix representation for any p except p=-1, where it returned inv(Q) which was defined to be Q'. I have returned to this behavior after finding out.

[dkarrasch]

I changed things to the matrix approach: so only Q^(+/-1) are lazy, all other powers use power_by_squaring. @dlfivefifty Could you please take a quick look?

[mbauman]

Suggested change

	@inline Base.literal_pow(::typeof(^), Q::AbstractQ, ::Val{1}) = Q
	@inline Base.literal_pow(::typeof(^), Q::AbstractQ, ::Val{-1}) = inv(Q)

Is there a reason for these? The fallback should work just fine here, no?

[mbauman]

Ah, ok, I see. This is fallout from #46196, where we lost the AbstractArray ^ method.

[dkarrasch]

Hihi, technically, one could make Q^p::AbstractQPower <: AbstractQ and lazy, similar to Q^1 and Q^-1. 😄 If Q is not a matrix, why should Q^p be? For instance, we loose the size flexibility of the right-most copy of Q in Q*Q*...*Q.

[dlfivefifty]

Q^2 should be equal to Q*Q. Since the latter works so should the former.

[dkarrasch]

Ok, then I think this is ready to go, isn't it?

[dlfivefifty]

Looks fine to me! Thanks!