{"paper":{"title":"Exact $T$-counts of Toffoli layers from an isotropy bound","arxiv_id":"2610.01024","primary_cat":"quant-ph","submitted_at":"2026-10-01T04:11:53+00:00","version_compiled":"v1"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v1","sentences":[{"i":0,"p":1.0,"id":"s0","own":false,"role":"background","text":"The $T$-count is the dominant cost of fault-tolerant Clifford+$T$ computation.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.05,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.35,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":1,"p":0.9,"id":"s1","own":true,"role":"result","text":"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.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.95,"main_claim_p":0.9,"quantitative":true,"quantitative_p":0.99,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.9,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.1,"interpretation":0.0}},{"i":2,"p":0.98,"id":"s2","own":false,"role":"background","text":"Campbell and Howard gave the matching construction.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.04,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.07,"role_probabilities":{"gap":0.01,"method":0.0,"result":0.01,"background":0.98,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":3,"p":0.83,"id":"s3","own":true,"role":"gap","text":"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 $m=2$).","hedge":{"p":0.74,"label":"asserted"},"own_p":0.56,"main_claim_p":0.2,"quantitative":true,"quantitative_p":0.96,"role_probabilities":{"gap":0.83,"method":0.0,"result":0.01,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.12,"interpretation":0.04}},{"i":4,"p":0.9,"id":"s4","own":true,"role":"result","text":"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.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.76,"main_claim_p":0.08,"quantitative":true,"quantitative_p":0.97,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.9,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.09}},{"i":5,"p":0.98,"id":"s5","own":true,"role":"method","text":"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.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.8,"main_claim_p":0.08,"quantitative":false,"quantitative_p":0.17,"role_probabilities":{"gap":0.0,"method":0.98,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.02,"interpretation":0.0}},{"i":6,"p":0.61,"id":"s6","own":true,"role":"result","text":"In general the constraint gives the isotropy floor $\\delta\\ge2(n-d^{\\ast})-r$ for every diagonal level-three gate, computed from the phase polynomial in polynomial time.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.86,"main_claim_p":0.1,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.0,"method":0.21,"result":0.61,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.18,"interpretation":0.0}},{"i":7,"p":0.85,"id":"s7","own":true,"role":"result","text":"The floor is never below stabilizer nullity $\\nu$, which equals $n-d^{\\ast}$ on this class, and a separate parity argument raises it to $2\\nu+1$ on non-Clifford pure-cubic gates.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.83,"main_claim_p":0.07,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.0,"method":0.02,"result":0.85,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.03,"interpretation":0.1}},{"i":8,"p":0.99,"id":"s8","own":true,"role":"result","text":"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.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.92,"main_claim_p":0.08,"quantitative":true,"quantitative_p":0.99,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.99,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.0}},{"i":9,"p":0.86,"id":"s9","own":true,"role":"limitation","text":"The floor also holds, under stated conditions, 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