Information Pathways for Learning Structural Predicates on Quantum Circuit DAGs

Structural prediction on directed acyclic graphs (DAGs) is an information-access problem before it is an architecture-selection problem. Critical-path membership decomposes exactly into forward longest-path depth $d_v$, reverse longest-path depth $r_v$, and a graph-level maximum $L^{\star }$. Because $d_v+r_v+1\le L^{\star }$ always holds with equality only at critical nodes, the predicate is a one-sided threshold, which splits the target: within-circuit ranking requires descendant-side path information and provably cannot depend on $L^{\star }$, while converting that order into a decision requires only $L^{\star }$. Quantum circuit DAGs make all three quantities deterministically computable, so the split is directly testable. We show that any linear estimate of $r_v$ from quantities a strong baseline already receives is rank-equivalent to forward depth alone, so exact reverse depth supplies a direction of information rather than added precision on a quantity already present. Holding a node-wise MLP fixed up to a $385$-parameter projection, broadcasting $L^{\star }$ raises critical-path F1 from $0.573$ to $0.826$ and adding exact $r_v$ to $0.974$, with $0.518$, $0.581$, $0.846$ under a same-generator upper-tail depth shift; macro AUPRC reproduces the ordering ($0.832$, $0.898$, $0.997$), so it is not an artifact of a fixed operating point. The identical increment on a message-passing encoder improves the operating point without improving node ranking in distribution while improving both under shift, exactly as the decomposition predicts. Architecture experiments then locate the same quantities in distinct pathways: a DAG-native encoder obtains reverse depth by topological propagation, reaching depth-shift macro AUPRC $0.975$ without an explicit broadcast, the evaluated latent-bottleneck Perceiver acquires essentially no rank-relevant descendant information and a $16\times$ latent sweep does not repair this, and a graph transformer is near ceiling with supplied structure but substantially degraded without it. Structural-prediction capability is therefore governed not by architecture labels alone, but by whether the representation obtains reliable access to the information the target demands.