Exact T-counts of CCZ layers from an isotropy bound
Abstract
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.
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BibTeX
@misc{mazumder2026exact,
title = {{Exact $T$-counts of CCZ layers from an isotropy bound}},
author = {Arul Rhik Mazumder},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01024},
eprint = {2610.01024},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01024},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exact T-counts of CCZ layers from an isotropy bound
AU - Arul Rhik Mazumder
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.01024
UR - https://arxiv.org/abs/2610.01024
AB - 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.
ER -