If we keep track of the scalar we have the following (conjugate) action of the Clifford generators on matrices (a,b):= X^a Z^b:
H : (a,b) |-> (-1)^{ab} (b,a)
P: (a,b) |-> (-i)^a (a,a+b)
CZ: (a,b) x (c,d) |-> (-1)^{ac} (a,b+c)x (c,a+d).
In particular, we could keep track of the phases by adding two additional bits p, q: (-1)^p i^q X^a Z^b. If we keep track of the phases, we find that P^2 : (a,b) |-> (-1)^a (a,b) which precisely encodes Z .
However, it is unclear to me what states correspond to and how measurement works. In the stabilizer formalism, to a matrix A=(-1)^p i^q X^a Z^b you would assign the space (+1-eigenspace of A). E.g. (0,1) corresponds to |0> and -(0,1) to |1>. However, e.g. (0,0) corresponds to a 2d space. Also, this leads to a completely different interpretation of measurement. But then again, we probably shouldn't be able to perfectly map our little boxes onto Clifford stuff. Our box model definitely has a local hidden variable which Clifford QM doesn't have. In particular, in the stabilizer formalism (and the Knill-Gottesmann theorem) entangled states are not treated locally, you have to store the combined phase information of both qubits.
If we keep track of the scalar we have the following (conjugate) action of the Clifford generators on matrices (a,b):= X^a Z^b: H : (a,b) |-> (-1)^{ab} (b,a) P: (a,b) |-> (-i)^a (a,a+b) CZ: (a,b) x (c,d) |-> (-1)^{ac} (a,b+c)x (c,a+d). In particular, we could keep track of the phases by adding two additional bits p, q: (-1)^p i^q X^a Z^b. If we keep track of the phases, we find that P^2 : (a,b) |-> (-1)^a (a,b) which precisely encodes Z . However, it is unclear to me what states correspond to and how measurement works. In the stabilizer formalism, to a matrix A=(-1)^p i^q X^a Z^b you would assign the space (+1-eigenspace of A). E.g. (0,1) corresponds to |0> and -(0,1) to |1>. However, e.g. (0,0) corresponds to a 2d space. Also, this leads to a completely different interpretation of measurement. But then again, we probably shouldn't be able to perfectly map our little boxes onto Clifford stuff. Our box model definitely has a local hidden variable which Clifford QM doesn't have. In particular, in the stabilizer formalism (and the Knill-Gottesmann theorem) entangled states are not treated locally, you have to store the combined phase information of both qubits.