Efficient Clifford representations via stabiliser theory can never capture global phase, since all of the data is based on externally observable behaviour (the stabilisers) and global phase is never observable. Nevertheless, since global phase tracking is situationally useful, we should aim for all areas of compilation to support it, even if it comes at a small computational cost (ideally toggleable if the cost is not negligible).

For example, in settings where we may wish to apply a controlled-Clifford unitary, the "global" phase in the Clifford unitary becomes local under control. We can't just handle this with stabiliser methods because the controlled-Clifford may longer be Clifford.

Solutions to this problem come from fixing a standardised statevector interpretation of each stabiliser process, from which we can track the relative phase whenever we apply gates. The "reduced affine with phases form" (reduced AP-form) described in [Chapter 5, “Picturing Quantum Software”, Kissinger & van de Wetering] would handle this nicely (relating closely to the CH-form of [“Simulation of quantum circuits by low-rank stabilizer decompositions”, Bravyi et al] and the quadratic form of ["The Clifford group, stabilizer states, and linear and quadratic operations over GF(2)", Dehaene & De Moor]).

Updating this data structure will be more expensive than a standard tableau as it takes more effort to keep in this standard form, so this is something we should provide alongside faster tableau methods for when it is needed rather than in place of.