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Exact $T$-counts of Toffoli layers from an isotropy bound
The $T$-count is the dominant cost of fault-tolerant Clifford+$T$ computation. We prove that a layer of $m$ disjoint CCZ gates, the diagonal core of a parallel Toffoli layer, needs exactly $6m+1$ $T$ gates in every Hadamard-free Clifford+$T$ circuit with clean ancillas. Campbell and Howard gave the
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