Exponential of Pauli object requires conversion to Matrix

[dieris]

expm(X) does not convert a Pauli (X) to a matrix.

Need to type expm(complex(X))

[marcusps]

I was under the impression that maybe there would be a way to make this work through type promotion, but that does not appear to be the case -- is that right, @blakejohnson ?

It seems like right now we don't even support addition between a Pauli object and a matrix (without explicit conversion) -- although multiplication between a Pauli object and a matrix is supported.

The tricky things is that we want 1im*Pauli(j) to be a Pauli object, and in order to do that, we forbid multiplication by anything that is not 1,-1,1im,-1im. So you would not be able to write im*(0.5*X+0.2*Y) for example, much less the exponential of that.

Perhaps we can define a macro that will define Pauli matrices of type Matrix{Complex128}, something like

@paulimatrices(I,X,Y,Z,2)

that would define the matrices

II
IX
IY
IZ
...

you would still be able to access the Pauli objects using Pauli.X, Pauli.Y, etc. Maybe we can make some sort of prefix optional too

@paulimatrices(I,X,Y,Z,2,prefix=m)

that would define the matrices

mII
mIX
mIY
mIZ
...

[blakejohnson]

It would be easy enough to add a method to expm so that the conversion step would be unnecessary, but I suspect that wouldn't be enough, because then we'd want things like expm(im*pi/2*X) to work and that runs into the multiplication issue you mention.

Maybe a better solution is to have a special internal method for Number * Pauli that returns a Pauli and have the usual operator return a Matrix?

[blakejohnson]

Closed by #5.