Phase-aware tableau updates for Clifford generators

For $H$, [A1] gives the exact coordinate swap and phase $2x_a{z}_a$. For $S$, $SX{S}^{†}=iXZ$ and $SZ{S}^{†}=Z$ give the stated shear and phase increment. The second identity and unitarity give $S^{†}ZS=S^{†}(SZ{S}^{†})S=Z$. Then $X=S^{†}(SX{S}^{†})S=i(S^{†}XS)Z$, so it follows that $S^{†}XS=-iXZ$; hence the inverse shear decrements the phase by $x_a$. All other tensor factors are unchanged.

For CNOT, conjugation sends $X_c\mapsto X_c{X}_t$, $X_t\mapsto X_t$, $Z_c\mapsto Z_c$, and $Z_t\mapsto Z_c{Z}_t$ by [F3]. The image of the ordered $X^x$ part is a product of $X$ factors and the image of $Z^z$ is a product of $Z$ factors; the order $X^x{Z}^z$ is retained, so no $X$ crosses a $Z$ and $k$ is unchanged. This gives the stated toggles.

For $H$, swapping $x_a,z_a$ in both labels leaves their contribution $x_a{z}_a^{\mathrm{\prime }}+z_a{x}_a^{\mathrm{\prime }}$ unchanged. For $S$, replacing $z_a,z_a^{\mathrm{\prime }}$ by $z_a+x_a,z_a^{\mathrm{\prime }}+x_a^{\mathrm{\prime }}$ adds $x_a{x}_a^{\mathrm{\prime }}+x_a{x}_a^{\mathrm{\prime }}=0$. For CNOT, the control and target contributions change by, respectively, $x_c{z}_t^{\mathrm{\prime }}+z_t{x}_c^{\mathrm{\prime }}$ and $x_c{z}_t^{\mathrm{\prime }}+z_t{x}_c^{\mathrm{\prime }}$, whose sum is zero in ${\mathbb{F}}_2$. Thus each update preserves the displayed form [F4] and all row pairings.

Each label update is invertible (the H swap is its own inverse and the S and CNOT shears are invertible over ${\mathbb{F}}_2$); conjugation also preserves Hermiticity and fixes $-I$. Consequently row independence, commutation, exclusion of $-I$, and every extended pairing remain valid.