Controlled-Z update in the binary phase representation ($b_j^{\prime }=b_j\oplus a_k,b_k^{\prime }=b_k\oplus a_j,s^{\prime }=s+2a_j{a}_k$)

Conjugation by a Controlled-Z gate leaves the $a$ bits unchanged and updates the displayed $b$ bits and phase, using the old bits. The phase occurs when the extra $Z$ from one line is moved past $X$ on the other line. These constant-size updates support Heisenberg propagation of a Pauli observable through a Clifford circuit.