Classification and Exact Local Masking in Finite-Field Clifford Dual-Unitary Circuits

We classify two-qudit finite-field Clifford dual-unitary gates and apply the classification to exact local masking and operator transport in homogeneous brickwork circuits. Under ordered one-qudit Clifford equivalence, the dual-unitary locus contains $q-2$ perfect-tensor cores, one rank-one core, and one SWAP core. The perfect cores are indexed by $$\delta=\det B=\det C,$$ and are related to the ordered cross-ratio $\lambda$ of the associated $[4,2,3]_q$ MDS configuration through $$\lambda=\frac{\delta}{\delta-1}.$$ Homogeneous repetition separates these cores into five transport phases whose algebraic codimensions in $\operatorname{Sp}(4,q)$ are $0,1,2,3,4$. The one-site Weyl edge channels determine exact local-masking distances. Perfect-tensor circuits attain $$d_1(t)=4t, \qquad d_2(t)=4t-2,$$ whereas delayed erasers satisfy $$d_1(t)=4t-2, \qquad d_2(t)=4t-4$$ for $t\geq 2$. Consequently, sufficiently short quantum messages are completely hidden from every one- or two-qudit output subsystem, even when the input is entangled with a reference, while remaining exactly recoverable from the full output. For $q=3$, we construct an explicit perfect-tensor Clifford gate from two inverse SUM gates. Exhaustive Weyl-support searches for $t=1,2,3$ reproduce the predicted masking distances, and a one-period Choi-channel calculation for a four-qutrit periodic circuit gives numerical residual leakage below $2\times 10^{-16}$. Under the coherent perturbation considered here, local leakage is linear in the perturbation strength, whereas the infidelity of recovery using the ideal inverse is quadratic near the perfect point.