Backward Pauli updates for Hadamard and phase gates ($H:(a,b,s)\mapsto (b,a,s+2ab),S:(a,b,s)\mapsto (a,b\oplus a,s-a)$)

For the binary phase representation of a Pauli string, these local rules implement $P\mapsto U^{†}PU$ for the Hadamard gate and the phase gate $S=\mathrm{diag}(1,i)$. In particular $S^{†}XS=-iXZ$, so the phase update differs in sign from forward conjugation. Use the old input bits, and reduce $s$ modulo four.