[Feature Request] Sparse Representation of operations in the Pauli Formalism

[isolatedinformation]

Issue Description

Our pipeline pads single-qubit rotations with identities to span the whole width of the circuit. Switching to a sparse format (like in the gate-based pipeline) is likely to speed up this step and drastically reduce memory footprint. (as pointed out in #263 )

[gwwatkin]

I suggest looking at how gates are implemented as an example. Could even add the pauli rotation there, similarly to how the rz is implemented (inheriting from Gate), but with one more parameter for the axis (X/Z)

[alexnguyenn]

Should we save this for when we start migrating to C++ @gwwatkin?

[gwwatkin]

@alexnguyenn and @isolatedinformation, would be nice to get this done in python so we can squeeze it into the paper. It should be fairly quick to implement

[isolatedinformation]

lattice-surgery-compiler/src/lsqecc/gates/gates.py

Lines 53 to 64 in d59f447

	class RZ(Gate):
	phase: Fraction = Fraction(1, 1)
	def to_clifford_plus_t(self) -> Sequence[Gate]:
	if self.phase == Fraction(1, 1):
	return [Z(self.target_qubit)]
	elif self.phase == Fraction(1, 2):
	return [S(self.target_qubit)]
	elif self.phase == Fraction(1, 4):
	return [S(self.target_qubit)]
	else:
	return approximate.approximate_rz(self)

@gwwatkin shouldn't line 62 be the T gate and not the S gate?

[gwwatkin]

Yes you are right! Good catch

[isolatedinformation]

added the fix for this in #273

[gwwatkin]

@isolatedinformation I think we can can close this after this comment from 274 is adressed:

One last think to consider is the half angle conversion for Pauli Rotations. Right now all the angles are double those in the convention of the latest lattice surgery papers and our existing block based Pauli rotations. I think we should also switch here to half angles