arXiv:2610.01024 · v3
Exact $T$-counts of CCZ layers from an isotropy bound
摘要
The $T$-count is a dominant cost of fault-tolerant Clifford$+T$ computation. We prove that every Hadamard-free Clifford$+T$ circuit with clean ancillas for a layer of $m$ disjoint CCZ gates, the diagonal core of a parallel Toffoli layer, uses at least $6m+1$ $T$ gates. Campbell and Howard gave a circuit with $6m+1$, and to our knowledge this is the first proof that it is optimal for general $m$. The proof combines Campbell and Howard's self-orthogonality of the columns of the parity matrix with a rank (flattening) bound, sharpened by one. More generally, the argument gives a lower bound on the $T$-count of every diagonal gate in the third level of the Clifford hierarchy, the isotropy floor, computable by Gaussian elimination. It is never below the stabilizer nullity $ν$ and reaches $2ν+1$ on non-Clifford gates whose phase is purely cubic. No bound that also holds for circuits with measurement and feedforward can do this, since such circuits implement CCZ ($ν=3$) with four $T$ gates and the layer with $4m$, where the floor gives seven and $6m+1$. The floor also recovers Campbell and Howard's count $4m+3$ for a fan-out of Toffolis. On the output of PyZX's TODD-based optimizer, relative to its Hadamard placement, it certifies $193$ of $311$ phase-polynomial blocks $T$-optimal once repeated parities are merged (at the best of five optimizer seeds), against $113$ for nullity, including $37$ of the $72$ blocks too large for exhaustive search.
作者备注：35 pages, 5 figures, 5 tables; supplementary notes (35 pages) as an ancillary file. v3: condensed to the core results, with the remaining material moved to the supplementary notes; field-multiplication slice-rank bounds deferred to a separate paper