Other combinations for decomposition to rotation matrices

[purva-thakre]

In the single qubit gate decomposition module, if the gate is supposed to decomposed to product of rotation matrices then two possibilities are ZYZ and ZXZ.

Map the angles from ZYZ to other combinations - YXY, YZY, XZX, XYX.

[purva-thakre]

Possible combinations (screenshot from page 6).

If instead of using _angles_for_ZYZ, want to calculate the angles for each combination individually, Appendix A of linked doc has the calculations. The only issue here is that the rotation matrices used are SO(3) but the rotation matrices of interest are SU(2). It might be easier to define a mapping function between the two groups as they are isomorphic i.e. there should exist an equivalence between their respective adjoint representations.

[BoxiLi]

Is there a way to define only a few of them explicitly and then use unitary transformation to get the rest of them? Although unitary transformation may introduce some minor numerical error, having 12 distinct implementations seems very hard to maintain and debug.

[purva-thakre]

Yeah, I have been working over the calculations. I am trying to figure out which ones need to be specified explicitly or if all can be calculated via zyz.

12 distinct implementations seems very hard to maintain and debug

Thought the same after I made my previous comment. If I use these for ABC decomposition as well then there will be 24 private functions in total !! It's better to do the calculations and create a dictionary of the angles for each combination.

[Buffoni]

Hey @purva-thakre @BoxiLi , I'm interested in contributing to this package and doing a project for GSoC '22
Since this issue doesn't seem to be resolved yet could I jump in and help?

We can probably create a method _angles_for_rotation that defaults to output the angles of a zyz rotation and make the other ones available as alternative decomposition if requested by the user. It shouldn't be too difficult to do the calculations from that paper.

[purva-thakre]

Hi @Buffoni ! I have worked on a working form of this. Haven't had the chance to push it yet because first I need to fix a phase error in the existing ZYZ method.

[Buffoni]

Ok, I'll see how I can contribute to other issues or open a new one, thanks!