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

Arul Rhik Mazumder

REVIEW

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

Arul Rhik Mazumder

Quantum Physics · Submitted 1 Oct 2026

Pith's verdict: The 6m+1 lower bound for disjoint CCZ layers is new, cleanly proved, and honestly scoped to the Hadamard-free model; the transfer to internal-Hadamard circuits stays open and the paper says so.

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arXiv ID2610.01024 (version 3)

Pith ID533ASI32, Pith's permanent record of this paper and its review

CitationArul Rhik Mazumder. "Exact $T$-counts of CCZ layers from an isotropy bound." arXiv:2610.01024 (2026). https://pith.science/paper/533ASI32

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Editor's note

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The headline result holds up: a layer of m disjoint CCZ gates costs exactly 6m+1 T gates in every Hadamard-free Clifford+T circuit with clean ancillas. For m≥3 this is new; m=1 is classical and m=2 was stated by Campbell and Howard without proof. The matching upper bound comes from their published construction, so the lower bound is the contribution.

The proof is clean and worth knowing. The isotropy argument (the columns recording which T gates touch each qubit span a totally isotropic subspace), the radical rank bound giving rank ≥ n−d, and the parity +1 step together yield δ ≥ 2(n−d)+1 on non-degenerate pure-cubic gates, computable in O(n³) by Gaussian elimination. Identifying stabilizer nullity with n−d* on this class is tidy bookkeeping, and the floor being never below nullity, with a genuine factor-of-two on pure-cubic gates, is the real content. The paper is unusually honest: it states its scope precisely and lists its open problems rather than blurring them.

The one load-bearing limitation is the Hadamard-free restriction. The exact 6m+1 and the 2ν+1 floor hold in the model where the gate is its phase polynomial. Transfer to circuits with internal Hadamards requires Nker ≥ δ(κ) of Proposition 81, proved only at maximal output rank, for Clifford corrections, for at most two internal Hadamards, and for AND-compiled circuits. The strata with three or more internal Hadamards and sub-maximal output rank stay open (Open Problems 6 and 7), and the paper says so. It also notes that under adaptive feedforward only t ≥ ν holds and CCZ costs four T gates, so the factor of two genuinely disappears there. That is an accurate account of where the method stops, not a defect.

Two smaller things. The benchmark certification is relative to the optimizer's Hadamard layering, not an absolute lower bound on the full circuit; the paper flags this, but the abstract's "certifies 193 of 311 blocks" is easy to read as stronger than it is. And the floor is not tight in general — at n=6 it misses two of five pure-cubic orbits, with a gap of four. Both are disclosed in the text.

My verdict matches the reader's: accept. The scoped theorem is rigorous, checks out, and the open problems are honestly identified. The cited construction covers the upper bound, so no circularity worry. This deserves a serious referee; a careful one will find the argument sound and the scope statements accurate.

Referee report

Significance

If it holds, the main result is a genuine advance: it gives the first exact T-count for general m of the parallel Toffoli layer, a universal building block in reversible arithmetic and phase oracles, and only the 6m+1 upper bound was previously published (Campbell–Howard). The derivation is parameter-free—d, d*, and r are computed invariants of the phase polynomial via Gaussian elimination, with no fitted values—and the lower bound doubles stabilizer nullity on the pure-cubic locus at the same O(n^3) cost. The paper is unusually scrupulous about its own limits: exactness of the floor is verified exhaustively for n ≤ 6 (with the 39.4% success rate at n=6 reported honestly), the failure of the sharing floor on cyclic merge patterns is documented (Remark 48), the internal-Hadamard transfer is proved only in the stated regimes (Corollaries 82, 85) and the remaining strata are labelled open, and the benchmark certification is repeatedly described as relative to the optimizer's Hadamard layering. The computational claims are backed by named scripts and a reproducibility index in Appendix L, and the novelty (the lower bound) is cleanly separated from the cited upper bound.

Major objections (1)

[§2.4.3, Proposition 81, Open Problems 6–7] The exact values 6m+1 (Theorem 40) and 4m+3 (Theorem 43) are proved in the Hadamard-free model, and their validity for circuits with internal Hadamards is established only under the hypotheses of Corollaries 82 and 85 (maximal output rank, Clifford corrections, h ≤ 2, AND-compiled temporary-AND blocks). I recommend that Theorem 40 and Theorem 43 restate the model hypothesis ('Hadamard-free Clifford+T circuits with clean ancillas') directly in their statements and point to Open Problem 6 for the internal-Hadamard case, so the scope is visible at the point of use. The paper already states this scope correctly in the abstract and in §2.4.3, so this is a prominence/usability request rather than a defect in the scoped results.

Minor objections (5)

[§2.1.2 vs Appendix H] The symbol d is used for the radical dimension in §2.1.2 and again (as d_F) for the fiber dimension in Appendix H; please add one line at the first use of d_F noting the clash is deliberate.
[§2.3.4 / Table 2] The certification counts (193/311 and 252/337 blocks) are block-local and layering-relative, which the paper states; I suggest adding one sentence in §2.3.4 reminding readers that a certified block is optimal only within the fixed Hadamard layering and block partition, with the csum mux 9 example (floor 73 vs AlphaTensor-Quantum's 71) as the canonical caveat.
[Table 3 and §B.4] The column 'low(f)' is explained only in the table caption; please define it once in §B.4 near Corollary 58 so readers do not need to reverse-engineer it from the caption.
[Appendix L] Please make explicit in one sentence that the 6m+1 upper bound for all m is the cited Campbell–Howard construction ([13, Cor. 1]) and that the paper's own Proposition 47 construction supplies an alternative proof verified computationally for m ≤ 9; this removes any ambiguity about the provenance of the matching upper bound.
[§1, first paragraph] The opening paragraph emphasizes the general Toffoli-layer cost; a one-clause pointer to §2.4.3 (internal Hadamards) in that paragraph would help readers who skim the introduction before reaching the formal scoping in §2.

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The paper

The paper claims

A layer of m disjoint CCZ gates needs exactly 6m+1 T gates in Hadamard-free Clifford+T circuits, and a computable isotropy bound on the phase polynomial proves it.

The reading

The paper proves the first optimal T-count for arbitrary layers of parallel Toffoli gates: a layer of m disjoint CCZ gates, the diagonal core of a parallel Toffoli layer, requires exactly 6m+1 T gates in any Hadamard-free Clifford+T circuit with clean ancillas. The lower bound comes from a new invariant, the cubic moment form of the phase polynomial, whose radical dimension forces the synthesis matrix's columns to span a totally isotropic subspace of dimension at least n-d, giving delta >= 2(n-d) and, after a parity step, delta >= 2n+1 on non-degenerate pure-cubic gates. A disjoint CCZ layer has d=0, so the floor is 2n+1 = 6m+1; a matching construction attains it. The method applies broadly: it certifies 193 of 311 optimized benchmark blocks optimal for their Hadamard layering, versus 113 for stabilizer nullity, and it provides the only known polynomial-time bound that closes the factor-of-two gap to nullity on this class.

What carries the argument

The cubic moment form is a symmetric trilinear map T_abc in F_2 on F_2^n recording, for each triple of qubits, the parity of the number of odd phase-parities containing all three; its radical R is the kernel of the linear map w -> T(w), with d = dim R. For a pure-cubic gate the column vectors of the synthesis matrix satisfy even weights and even pairwise overlaps, hence span a totally isotropic subspace, so t >= 2 rank Y; the projection from F_2^n to F_2^t has kernel inside R, giving rank Y >= n-d, and the all-ones vector argument adds one. Both d and the ceiling are computed by Gaussian elimination in O($n^{3}$) time, so the bound needs no search.

Extended reading notes

Core claim

The central claim is that the T-count of a diagonal level-three gate is pinned from below by an isotropy constraint: for a pure-cubic phase, the vectors recording which T gates touch each qubit span a totally isotropic subspace of F_2^t, so t >= 2 rank Y; the cubic moment form's radical R bounds the rank from below by n-d, and a parity argument adds one when d=0, yielding the isotropy floor delta(U) >= 2(n-d)+1. Applying this to a layer of m disjoint CCZ gates on n=3m qubits, whose radical is trivial, gives delta = 6m+1, matching Campbell and Howard's construction. The floor subsumes stabilizer nullity, which has the closed form nu = n-d* here, and the paper proves exact counts for overlapping layers, fan-outs (4m+3), factorizable cubics, and field-multiplication oracles, along with an O($n^{3}$) ceiling that brackets every diagonal level-three gate.

Load-bearing premise

The exact 6m+1 value and the isotropy floor are proved in the Hadamard-free model, where the gate is fully described by its phase polynomial; moving to circuits with internal Hadamards requires the inequality N_ker >= delta(kappa), proved only at maximal output rank, for Clifford corrections, for at most two internal Hadamards, and for AND-compiled circuits, leaving the strata with three or more internal Hadamards and sub-maximal output rank open.

Editorial extensions

If this is right

A parallel Toffoli layer of m gates costs exactly 6m+1 T gates, so any compiler output meeting that count is certified optimal for its Hadamard layering.
Stabilizer nullity, previously the standard efficient bound, is off by a factor of two on every non-degenerate pure-cubic gate; the isotropy floor closes that gap at the same O(n^3) cost.
The floor certifies 193 of 311 merged optimizer blocks (186 to 193 across seeds) optimal for their Hadamard layering on the benchmark circuits TODD completes, against 113 for nullity.
A non-degenerate cubic phase oracle costs at least 2n+1 T gates per call, the GF(2^k) multiplication oracle at least 6k+1, and the fan-out of m Toffolis from one control exactly 4m+3.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the open strata (three or more internal Hadamards with sub-maximal output rank) were settled by the conjectured inequality N_ker >= delta(kappa), the 6m+1 value would transfer to general Clifford+T circuits, sharply tightening the floor for practical reversible arithmetic.
Because the floor is computed in polynomial time from the phase polynomial alone, it is available before any circuit exists, which suggests using it inside compilers as a per-block optimality certificate rather than only as an a-posteriori check.
The linear-in-k bounds on field-multiplication oracles imply that subquadratic multipliers, though faster asymptotically, are eventually a growing factor from optimal T-count; testing whether that gap is real or a floor artifact would settle the GF(8) question at 20 versus 23.
The floor's exactness is specific to level three, since past level four no radical-based bound can be tight; any tight higher-level bound would need a classification of alternating trilinear forms that does not exist today.

Original 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 $\nu$ and reaches $2\nu+1$ 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 ($\nu=3$) 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.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-02: 1 request, 7,948 input tokens.

What the abstract itself states, as probabilities: claims first 0.87 · states limitation 0.93 · human participants 0.01 · controlled comparison 0.31 · randomized assignment 0.01 · states quantitative result 0.99 · compares with prior results 0.98 · mentions code or data availability 0.03.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

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 $\delta\ge2(n-d^{\ast})-r$ for every diagonal level-three gate, computed from the phase polynomial in polynomial time. The floor is never below stabilizer nullity $\nu$, 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. 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 $t\ge\nu$ is proved.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The $T$-count is the dominant cost of fault-tolerant Clifford+$T$ computation.	background 1.00	0.05	0.03	asserted 1.00
s1	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.	result 0.90	0.95	0.90	asserted 1.00
s2	Campbell and Howard gave the matching construction.	background 0.98	0.04	0.04	asserted 1.00
s3	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 $m=2$).	gap 0.83	0.56	0.20	asserted 0.74
s4	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.	result 0.90	0.76	0.08	asserted 1.00
s5	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.	method 0.98	0.80	0.08	asserted 1.00
s6	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.	result 0.61	0.86	0.10	asserted 1.00
s7	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.	result 0.85	0.83	0.07	asserted 1.00
s8	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.	result 0.99	0.92	0.08	asserted 1.00
s9	The floor also holds, under stated conditions, for circuits whose only internal Hadamards form unitarily uncomputed temporary AND blocks, while under adaptive feedforward only $t\ge\nu$ is proved.	limitation 0.86	0.80	0.05	asserted 0.98

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** The isotropy floor, end to end, with the

**Figure 2.** The 6m + 1 construction, drawn at m = 2: bracketing the second block onto a qubit of the first removes one gate, so the pair uses thirteen Ts where two independent blocks would use fourteen. The circuit restores its wires and implements CCZabcCCZdef (checked by simulation, Appendix L). Implemented separately, the m gates cost 7m

**Figure 3.** How sharing lowers the T-count. Two CCZ blocks both act on wire 0, and their circuits run one after the other. Each block places one of its Ts on wire 0 itself. Wire 0 is only ever a CNOT control, so it carries x0 throughout and the two boxed Ts act on the same parity. They combine into S, a Clifford gate, and the pair is removed. linear 3-uniform hypergraph: pairwise intersections have size at most one). Then the union’s cubic tensor is non-degenerate, so δ Y i CCZBi ≥ 2n + 1 = 2(3m − s) + 1, where s = 3m − n is the number of merges, the qubit-incidences in excess of one per q

**Figure 4.** Sharing qubits interpolates the cost of a

**Figure 5.** The standard ancilla-based CCZ circuit, built from a temporary AND on one clean ancilla, with the quantities of Remark 88. Here h = 4. Each Toffoli is a CCZ between two Hadamards on u, and u must start and end in |0⟩, so none of the four Hadamards can be pushed to the ends, and they introduce the branch bits y1, . . . , y4. The ancilla ends holding y4, so the fiber is {y4 = 0} with dF = 3, and only y4 reaches the output, so ρG = 1, below the maximum 2. The first CCZ acts on a, b and y1 and contributes the seven parities of {a, b, y1}, and the second contributes the same seven with…

**Figure 6.** Dilation. Left, the outcome of measuring

Figures reproduced from arXiv:2610.01024 by the authors.

Discussion

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