Add "RandomUnitary" gate

[mtfishman]

Add a "RandomUnitary" gate like in PastaQ: https://github.com/GTorlai/PastaQ.jl/blob/v0.0.6/src/circuits/gates.jl#L301-L313.

We would have to think about how it might work for QN ITensors (maybe the user could also specify the flux with a keyword argument, which would default to QN(0)).

[emstoudenmire]

Would be great to have. Would the flux not always be zero since it is a unitary?

[mtfishman]

Probably it isn't common for gates, but what's wrong with having unitaries with nonzero flux?

julia> i = Index(QN(0) => 2)
(dim=2|id=88) <Out>
 1: QN(0) => 2

julia> j = Index(QN(1) => 2)
(dim=2|id=237) <Out>
 1: QN(1) => 2

julia> U = ITensor(qr(randn(2, 2)).Q * I, i, dag(j))
ITensor ord=2
(dim=2|id=88) <Out>
 1: QN(0) => 2
(dim=2|id=237) <In>
 1: QN(1) => 2
NDTensors.BlockSparse{Float64,Array{Float64,1},2}

julia> flux(U)
QN(-1)

[emstoudenmire]

Agreed it's technically possible. It's just that in every example I can think of which is tied to a specific problem or use-case I know about, a QN-conserving unitary has flux zero (isometric tensors of a canonical MPS, time-evolution operator, etc.). Also philosophically a unitary is a change of basis, and just viewing a tensor in a different basis shouldn't change its flux which is a scalar quantity.

But if we can think of a case where it's in some sense natural for the flux to be non-zero that could be interesting to know about.

Mostly I would just say if this is about making a random gate, then zero flux is probably what we want. Otherwise we'd have to get into a separate question of also choosing the flux randomly, and then what distribution should that be drawn from? Maybe there's an answer but I'm not sure.

[mtfishman]

Sure, that's fair. Definitely not pushing the need for a nonzero flux unitary gate. The proposal was to use zero flux by default, and allow someone to specify a nonzero one if the need arises.