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This work proves a dimension-independent weighted stability theorem for a directed noncommutative analogue of Mantel's theorem, which connects distributed quantum computing with noncommutative extremal combinatorics by identifying local collision probabilities with the weighted multiplicative energy of matrix-space decompositions.
We show that one-way one-round quantum LOCAL algorithms cannot $4$-color directed cycles with high probability, even with unbounded local computation and quantum message length. This is the first lower bound in the high-probability quantum LOCAL setting that goes beyond the non-signaling and bounded-dependence models, exploiting the structure of distributed quantum algorithms. Our proof connects distributed quantum computing with noncommutative extremal combinatorics by identifying local collision probabilities with the weighted multiplicative energy of matrix-space decompositions. We obtain our lower bound by proving a dimension-independent weighted stability theorem for a directed noncommutative analogue of Mantel's theorem.
