Exact $T$-counts of Toffoli layers from an isotropy bound

Abstract

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 matching construction. To our knowledge this is the first proof that it is optimal for general $m$ (for $m=1$ the value $7$ is classical, and Campbell and Howard state the value $13$ at $m=2$). On every controlled unitary the same floor comes within one of their exact count, and it recovers their $4m+3$ for a fan-out of $m$ Toffolis from one control. The proof rests on an isotropy constraint: for a pure-cubic phase, the vectors recording which $T$ gates touch each qubit span a totally isotropic subspace. In general the constraint gives the isotropy floor $δ\ge2(n-d^{\ast})-r$ for every diagonal level-three gate, computed from the phase polynomial in polynomial time. The floor is never below stabilizer nullity $ν$, which equals $n-d^{\ast}$ on this class, and a separate parity argument raises it to $2ν+1$ on non-Clifford pure-cubic gates. On the output of the TODD optimizer for the $24$ benchmark circuits it completes, the floor certifies $193$ of its $311$ merged phase-polynomial blocks optimal for their Hadamard layering ($186$ to $193$ across five optimizer seeds), against $113$ for nullity. The floor also holds, under stated conditions, for circuits whose only internal Hadamards form unitarily uncomputed temporary AND blocks, while under adaptive feedforward only $t\geν$ is proved.