✨ Extend merge pass to non-rotation gates

[denialhaag]

The merge pass added in #1407 only acts on rotation gates. We could extend the pass to non-rotation gates so that single-qubit gates are fused into a single UOp or other rotation gates.

For more details, see #1407 (comment) and the replies thereof.

[J4MMlE]

I just dump the TODO i had for that here:

Gate  | Quaternion Q(w,x,y,z)                      | Global Phase 
------+--------------------------------------------+--------------
X     | Q(0, 1, 0, 0)                              |  π/2
Y     | Q(0, 0, 1, 0)                              |  π/2
Z     | Q(0, 0, 0, 1)                              |  π/2
S     | Q(cos(π/4), 0, 0,  sin(π/4))               |  π/4
Sdg   | Q(cos(π/4), 0, 0, -sin(π/4))               | -π/4
T     | Q(cos(π/8), 0, 0,  sin(π/8))               |  π/8
Tdg   | Q(cos(π/8), 0, 0, -sin(π/8))               | -π/8
H     | Q(0, 1/√2, 0, 1/√2)                        |  π/2
SX    | Q(cos(π/4),  sin(π/4), 0, 0)               |  π/4
SXdg  | Q(cos(π/4), -sin(π/4), 0, 0)               | -π/4

Special handling (no quaternion conversion):
Id: skip, don't break chain (contributes nothing)
Barrier: break chain
GPhase: skip rotation, accumulate phase only

[burgholzer]

I just dump the TODO i had for that here:

Gate  | Quaternion Q(w,x,y,z)                      | Global Phase 
------+--------------------------------------------+--------------
X     | Q(0, 1, 0, 0)                              |  π/2
Y     | Q(0, 0, 1, 0)                              |  π/2
Z     | Q(0, 0, 0, 1)                              |  π/2
S     | Q(cos(π/4), 0, 0,  sin(π/4))               |  π/4
Sdg   | Q(cos(π/4), 0, 0, -sin(π/4))               | -π/4
T     | Q(cos(π/8), 0, 0,  sin(π/8))               |  π/8
Tdg   | Q(cos(π/8), 0, 0, -sin(π/8))               | -π/8
H     | Q(0, 1/√2, 0, 1/√2)                        |  π/2
SX    | Q(cos(π/4),  sin(π/4), 0, 0)               |  π/4
SXdg  | Q(cos(π/4), -sin(π/4), 0, 0)               | -π/4

Special handling (no quaternion conversion):
Id: skip, don't break chain (contributes nothing)
Barrier: break chain
GPhase: skip rotation, accumulate phase only

Thanks!
Most of these gates carry a global phase contribution as well, right?

[J4MMlE]

Thanks! Most of these gates carry a global phase contribution as well, right?

Yes, the global phase contribution is given in the Global Phase column

[burgholzer]

Thanks! Most of these gates carry a global phase contribution as well, right?

Yes, the global phase contribution is given in the Global Phase column

Was that column always there? 😅 I must have overlooked that somehow.
Thanks!