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

Arul Rhik Mazumder

arXiv:2610.01024v3quant-phcs.CC

TL;DR

T gates consume substantial fault-tolerant resources, but compiled circuits provide only upper bounds on the count. This paper combines parity-matrix self-orthogonality with a rank bound to prove the exact cost of disjoint CCZ layers in the Hadamard-free model. It also gives a computable lower bound for diagonal third-level gates and reports certificates on optimized phase blocks.

  • Problem

    Compiled circuits give only compiler-specific upper bounds, while a cheap, tight lower bound is needed to assess how much T-count optimization remains.

    Source: §1 · p. 2

  • Method

    The isotropy floor combines parity-matrix self-orthogonality with a flattening-based rank bound, strengthened by one for non-Clifford gates.

    Source: §1.2 · p. 3

  • Results

    6m + 1 T gates are necessary and sufficient for a layer of m disjoint CCZ gates over {CNOT, T} circuits.

    Source: §1 · p. 2, §2.1.3 · p. 6

  • Takeaways & Limitations

    193 of 311 PyZX phase blocks are certified T-optimal at the best of five seeds, compared with 113 by nullity.

    Source: §2.3 · p. 8

    The bound is for Hadamard-free resynthesis, does not constrain measurement-and-feedforward compilers, and its extension to internal Hadamards remains open.

    Source: §1 · p. 2, §4 · p. 10

Abstract

from arXiv · show

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.

1 Introduction

T gates are a dominant fault-tolerant cost, and compiler outputs alone do not show how much optimization remains. The paper proves the 6m + 1 count optimal for disjoint CCZ layers in the Hadamard-free model and develops lower bounds for broader diagonal gates.

1 Introduction

  • 6m + 1 T gates are necessary and sufficient for m disjoint CCZ gates in the Hadamard-free model, matching the known construction.Clean ancillas do not lower the count in this model.Source: §1 · p. 2

1.1 Existing lower bounds

  • Exact diagonal T-count is a decoding problem with no known polynomial-time algorithm, while stabilizer nullity gives only 3 for CCZ, whose T-count is 7.Earlier lower bounds for Toffoli layers grew like 3m for nullity or about 3.71m for the amortized stabilizer Rényi entropy bound [4].Source: §1.1 · p. 2, §1 · p. 2

1 Introduction

  • Measurement and feedforward change the scope: temporary AND gates implement CCZ with four T gates, so this bound does not constrain those circuits.The bound concerns Hadamard-free resynthesis of phase blocks.Source: §1 · p. 2, §1.1 · p. 3

1.2 The isotropy argument

  • The isotropy argument combines parity-column self-orthogonality with a flattening-derived rank bound, then adds one for non-Clifford gates.For pure-cubic phases, the parity vectors are even-weight and pairwise even-overlapping; the rank bound gives t ≥ 2(n − d) [2].Source: §1.2 · p. 3

1.3 Contributions

  • For diagonal third-level gates, the isotropy floor is 2(n − d*) − r, is never below stabilizer nullity, and equals 2ν + 1 on non-Clifford pure-cubic gates.The relevant quantities take O(n^3) word operations to compute.Source: §1.3 · p. 3
  • The floor is exact for certain shared-qubit layers, and for m Toffolis fanned out from one control it recovers Campbell and Howard’s 4m + 3 count [2].For the shared-qubit layers, the count is 6m + 1 − 2η when the sharing pattern meets the stated forest condition.Source: §1.3 · pp. 3–4
  • 193 of 311 PyZX phase blocks are certified T-optimal at the best of five seeds, compared with 113 by nullity; among 72 too-large-to-decode blocks, the counts are 37 and 15.The certificates are relative to the optimizer’s block partition; across two optimizers, the floor is within two of the true T-count on all 515 blocks small enough to decode.Source: §1.3 · p. 4

2 Technical overview

The section develops a lower bound for diagonal-gate T-counts from parity-matrix self-orthogonality and rank, then extends the argument from CCZ to general diagonal level-three gates. It establishes the exact count 6m + 1 for disjoint CCZ layers and applies the floor to other gate families and optimizer outputs.

2.1 Warm-up: the T-count of CCZ

  • The warm-up expresses CCZ’s phase as a sum of seven parity terms, yielding a seven-T circuit; finding the minimum over all such expressions is a decoding problem.The parity sets whose T gates multiply to a diagonal Clifford form a punctured Reed–Muller code coset.Source: §2.1 · pp. 4–5

2.1.2 The rank, and the extra one

  • For CCZ, self-orthogonality and a rank bound first give six T gates, while a parity argument rules out six and proves the exact count is seven.The contradiction uses the trivial radical and the argument extends without decoding to any number of qubits.Source: §2.1.2 · p. 6

2.1.3 Many gates at once

  • 6m + 1 T gates is both the lower bound and achieved count for m disjoint CCZ gates, proving the layer’s T-count exactly.The cubic moment form is block-diagonal with trivial radical, and Campbell and Howard’s construction meets the floor.Source: §2.1.3 · p. 6
  • A layer of disjoint CCZ gates has a block-diagonal cubic moment form with trivial radical, giving the floor 2n + 1 = 6m + 1.Campbell and Howard’s construction attains this bound.Source: §2.1.3 · p. 6

2.2 The general floor

  • For every diagonal level-three gate, the isotropy floor is 2(n − d∗) − r, is never below stabilizer nullity, and is computable in O(n^3) word operations.Here d∗ is the radical dimension of the full moment tensor and r is the rank of the quadratic moments.Source: §2.2 · p. 7
  • For non-Clifford pure-cubic gates the floor reaches 2ν + 1, while quadratic moments weaken the bound by r and the improvement over nullity is ν − r.When r = ν, the floor gives no improvement over nullity.Source: §2.2 · p. 7

2.2.1 Where the floor is exact

  • The floor is exact for certain shared-wire layers: it recovers 4m + 3 for a fan-out, but can be loose when block sharing forms a cycle.Three pairwise-sharing blocks have floor 13 and T-count 15; an upper bound W(s) can also be constructed in O(n^3) time.Source: §2.2.1 · p. 7

2.3 Applications

  • 193 of 311 optimizer-output phase blocks are certified T-optimal after repeated parities are merged, versus 113 by nullity, at the best of five seeds.Among 72 blocks too large for exhaustive decoding, the floor certifies 37 and nullity certifies 15; certificates are relative to the optimizer’s Hadamard placement and blocks.Source: §2.3 · p. 8

3 Related work

The paper relates its bound to prior T-count lower bounds, tensor-decomposition methods, and self-orthogonality techniques used in code design. It distinguishes the isotropy floor by its efficient computation and by the gate families and circuit models its guarantees cover.

3 Related work

  • The factor-of-two improvement over nullity applies to pure-cubic gates without feedforward, while stabilizer nullity also bounds circuits with adaptive feedforward.The paper reports equality at 2ν + 1 for non-Clifford pure-cubic gates.Source: §3 · p. 8
  • Stabilizer nullity and amortized stabilizer Rényi entropy give weaker known lower bounds for disjoint CCZ layers, while the new bound is exact for every layer size.The amortized entropy bound is exactly about 3.71m; the nullity bound is 3m.Source: §3 · p. 8
  • Heyfron and Campbell [5] formulate T-count optimization as symmetric-rank decomposition of the signature tensor, which is the paper’s full moment tensor; the floor bounds that rank.The paper combines a flattening-rank bound with isotropy, and Theorem 25’s −r term uses the count of alternating forms associated with Lempel’s result.Source: §3 · p. 9
  • Prior magic monotones can bound T-count but generally take exponential time to compute, whereas stabilizer nullity is efficiently computable but gives only 3 for CCZ.The diagonal gates considered have eighth-root entries, so the denominator-exponent certificate discussed by Kam et al. [28] gives no bound for them.Source: §3 · p. 9
  • Related self-orthogonality conditions in code design constrain code parameters, whereas this work uses pairwise column conditions to lower-bound the T-count of a given gate.Triorthogonality requires even triple overlaps; generalized triorthogonality and synthillation allow odd overlaps that encode CCZ, but are used to construct codes and circuits.Source: §3 · p. 9

4 Discussion and open problems

The discussion identifies where the isotropy floor is exact and where its scope or tightness remains open. The appendices organize proofs of the floor, exact-count results, field multiplication, and benchmarks.

4 Discussion and open problems

  • The floor is not monotone under stabilizer operations with feedforward, so its stronger-than-nullity bounds depend on the Hadamard-free model.The supplementary notes extend the result to some circuits with internal Hadamards and higher hierarchy levels.Source: §4 · p. 9
  • The general extension to measurement-free circuits with internal Hadamards remains open, beyond stated cases with at most two Hadamards and additional conditions.The supplementary notes establish the bound when the correction is Clifford, at maximal output rank, and for certain temporary-AND compilations.Source: §4 · p. 10
  • The floor is exact for every pure-cubic gate on at most five qubits and for several structured families, but fails on some six-qubit cubic classes and intersecting blocks.The authors leave open which non-Clifford pure-cubic gates attain δ = 2ν + 1.Source: §4 · p. 10
  • For non-pure-cubic controlled diagonal Cliffords with odd μ, the floor is one below Campbell and Howard’s exact count; whether a bound recovers their parity step remains open.For field multiplication, slice ranks sharpen the floor and also bound Toffoli counts in Hadamard-free circuits, even with measurement and feedforward.Source: §4 · p. 10

Organization of the appendices

  • The appendices collect notation and preliminaries, prove the floor and ceiling, establish exact Toffoli-layer counts, compare controlled-unitary bounds, and treat field multiplication and benchmarks.The discussion also identifies open problems on internal Hadamards, exactness, and the one-unit gap for controlled unitaries.Source: Organization of the appendices · p. 10, §4 · p. 10

A Preliminaries

The preliminaries establish the algebraic and coding-theoretic language used to analyze diagonal gates. They define the Hadamard-free T-count through phase-polynomial representations and state its key invariance and ancilla properties.

A Preliminaries

  • The appendix fixes notation and recalls bilinear forms, Reed–Muller codes, the Amy–Mosca correspondence, cubic moments, radicals, and stabilizer nullity.CCZ, CS, and T are used as running examples.Source: A Preliminaries · p. 10

A.1 Bilinear forms and isotropic subspaces

  • A totally isotropic subspace lies inside its orthogonal complement, so its dimension is at most half the ambient dimension.The restricted dot-product rank is the Gram-matrix rank, and its radical has dimension dim C − rank G.Source: §A.1 · pp. 10–11

A.2 Linear and Reed–Muller codes

  • RM(n −4, n)*, the punctured code used below, has minimum distance 15 for every n ≥4.Reed–Muller codes consist of evaluations of bounded-degree Boolean polynomials; their duals and cosets support the later decoding formulation.Source: §A.2 · pp. 11–12

A.3 Phase polynomials and the T-count

  • A Hadamard-free circuit assigns T rotations to input parities, and its T-count is the number of odd coefficients in a phase-polynomial representation.The minimum over representations defines δ; it is invariant under the stated Clifford transformations, subadditive, and lower-bounds circuits with clean ancillas and no internal Hadamards.Source: §A.3 · pp. 12–13

A.4 The Amy–Mosca correspondence and the moment reformulation

The moment reformulation encodes diagonal level-three gates through low-order moments of their odd-parity residues. These moments also define the radicals and nullity quantities used in the isotropy-floor argument.

A.4 The Amy–Mosca correspondence and the moment reformulation

  • Odd-parity residues encode a gate through moments of orders one to three: matching those moments is equivalent to belonging to the same punctured Reed–Muller coset.The Amy–Mosca correspondence makes the moments invariant across residues of a gate.Source: §A.4 · pp. 13–14
  • The cubic moment form records triple moments, while the quadratic moment matrix records singleton and pair moments; their radicals capture directions invisible to those forms.For pure-cubic gates the lower moments vanish, so the two radicals coincide; CCZ has zero cubic radical, while adding an idle qubit adds a radical direction.Source: §A.4 · pp. 14–15

A.5 Stabilizer nullity and the shift group

  • The unitary stabilizer nullity is defined from the shift group and equals n minus the dimension of that group; the group is identified with the full-moment radical.For CCZ, the shift group is trivial and nullity is 3; the examples also give nullity 1 for T and 2 for CS.Source: §A.5 · pp. 15–16

B The isotropy floor

  • The isotropy-floor proof proceeds from pure-cubic gates to all diagonal level-three gates, identifies the relevant radical with nullity, and derives rigidity and a computable ceiling.The proof fixes odd parities in a representation and uses their incidence matrix and columns to establish its key identities.Source: B The isotropy floor · p. 16

B.1 The pure-cubic case

For pure-cubic gates, self-orthogonality and a rank bound yield a lower bound on T-count; a parity argument adds one for every non-Clifford gate.

The cubic lower bound

  • Self-orthogonality of the parity columns gives t ≥ 2 rank Y, while the cubic moments imply rank Y ≥ n − d.Together these give t ≥ 2(n − d).Source: §B.1 · p. 16
  • For a pure-cubic gate with cubic-tensor radical dimension d, δ(U) ≥ 2(n − d), and δ(U) ≥ 2n + 1 when d = 0.For every non-Clifford pure-cubic gate, the bound is δ(U) ≥ 2(n − d) + 1.Source: §B.1 · p. 17

The extra one

  • When d = 0, the all-ones vector forces a contradiction if t = 2n, so the lower bound increases by one.The argument uses the all-ones vector in the column code and the fact that its associated index set must lie in the zero radical.Source: §B.1 · p. 17
  • For d > 0, radical directions split off as idle qubits, and the non-degenerate bound applies to the remaining n − d qubits.The resulting bound is δ(U) ≥ 2(n − d) + 1 for every non-Clifford pure-cubic gate.Source: §B.1 · p. 17

B.2 The general case

For general diagonal level-three gates, the lower bound accounts for both the full moment-tensor radical and the rank of the quadratic moments. The pure-cubic bound follows when the lower moments vanish.

General lower bound

  • The column code’s restricted dot-product radical has dimension ρ − r, yielding t ≥ 2ρ − r; the rank bound ρ ≥ n − d* gives the stated floor.Source: §B.2 · p. 18
  • For a diagonal level-three gate, δ(U) ≥ 2(n − d*) − r, where d* is the full moment-tensor radical dimension and r is the quadratic-moment matrix rank.The bound uses the full-tensor radical, not merely the cubic radical.Source: §B.2 · p. 18
  • For pure-cubic gates, r = 0 and d* = d, so the general bound reduces to the first part of the cubic lower bound.Source: §B.2 · p. 18

Why the full-tensor radical matters

  • For the example with d = 0, r = 2, and d* = 1, the theorem gives a lower bound of 2, while substituting d for d* would incorrectly give 4 above δ = 3.The example shows why the full-tensor radical is needed off the pure-cubic locus.Source: §B.2 · p. 18

B.3 Stabilizer nullity and the radical

The shift group equals the radical of the full moment tensor, making stabilizer nullity directly comparable to the isotropy floor. The section also gives a rigidity threshold for exact T-counts.

B.3 Stabilizer nullity and the radical

  • The shift group A equals the full-tensor radical R*, so ν = n − d*.Membership in A is checked by testing the quadratic moments modulo 2 and the cubic-moment contractions modulo 4.Source: §B.3 · p. 19
  • On non-Clifford pure-cubic gates, the floor is 2ν + 1; off that locus, the +1 need not hold, and even δ ≥ 2ν can fail.CS and Example 26 each have ν = 2 and δ = 3.Source: §B.3 · p. 20
  • The floor 2(n − d*) − r is never below stabilizer nullity ν = n − d*, with equality iff r = n − d*.The floor is computable from the moments in O(n^3) time by Gaussian elimination.Source: §B.3 · pp. 19–20

B.4 Rigidity

  • A residue of Hamming weight ℓ ≤ 7 is optimal, so δ = ℓ; above weight seven, a weight-15 codeword can reduce the residue weight.The threshold seven is the unique-decoding radius.Source: §B.4 · p. 20

B.5 An upper bound, also in O(n3)

A canonical residue gives an efficiently computed upper bound that brackets the isotropy floor, and the bounds coincide for residues of weight at most seven. The section also states exact counts for specified Toffoli-layer structures.

B.5 An upper bound, also in O(n3)

  • Solving the moments downward gives a canonical residue and an upper bound W(s) on δ, computable in O(n^3) time without decoding.Together with the lower floor, it brackets every gate; when the bounds coincide, they determine δ.Source: §B.5 · p. 21
  • For residues with W(s) ≤ 7, the canonical-residue ceiling is exact: δ(U) = W(s).The construction sets z_y = 0 for parities of weight at least four.Source: §B.5 · p. 21
  • On pure-cubic gates, the ceiling is w(F), the number of subsets of sizes one through three contained in an odd number of monomials; on pure quadratics it is the edge count plus odd-degree vertices.Source: §B.5 · p. 21

C Exact T-counts of Toffoli layers

C.1 Toffoli layers

  • For blocks sharing single qubits routed along a forest of maximum degree three, the floor is exact and the count is 6m + 1 − 2η.Here η counts excess incidences; a Toffoli layer corresponds to CCZ blocks up to Hadamards on the targets.Source: §C.1 · p. 22

C.1.1 Disjoint blocks

For every m, the Hadamard-free T-count of m disjoint CCZ gates is exactly 6m + 1, including with clean ancillas. This proves the optimality of Campbell and Howard’s construction for general m.

C.1.1 Disjoint blocks

  • For m ≥ 3, optimality was previously open; the cited lower bounds grew only as 3m or about 3.71m.The cited discussion reports the 7-gate single-CCZ value and the 13-gate two-CCZ value as known.Source: §C.1.1 · p. 22
  • 6m + 1 T gates are necessary and sufficient for every layer of m disjoint CCZ gates, even with clean ancillas.The lower bound and Campbell and Howard’s construction match; the CCZ layer is the diagonal core of a parallel Toffoli layer.Source: §C.1.1 · pp. 22–23
  • A trivial radical gives d = 0, so the flattening bound supplies the lower bound 6m + 1.The construction splices seven-parity CCZ circuits to obtain the matching upper bound.Source: §C.1.1 · pp. 22–23
  • Splicing two disjoint seven-T CCZ blocks deletes one parity and uses the lifted parities to restore its phase, costing 13 rather than 14.This is the m = 2 instance of the general 6m + 1 construction.Source: Figure 2 · p. 23

C.1.2 Overlapping blocks

For linear-overlap CCZ layers, the isotropy floor decreases with excess qubit incidences. The blocks’ T-count is bounded below by 6m + 1 − 2η.

C.1.2 Overlapping blocks

  • Each excess incidence lowers the floor by two, while sharing a full pair of qubits makes the layer Clifford-equivalent to a single CCZ with δ = 7.For layers sharing a full pair, the floor is attained for every m.Source: §C.1.2 · p. 24
  • 6m + 1 − 2η is a lower bound when every pair of qubits occurs in at most one block, where η = 3m − n.Under this linear-overlap condition, the cubic tensor is non-degenerate and the equivalent bound is 2n + 1.Source: §C.1.2 · p. 24
  • The floor is a lower bound for layers with linear overlap; the stated result does not claim that every such layer attains it.The section gives attainment conditions separately through a forest-based construction.Source: §C.1.2 · p. 24

C.1.3 Splicing along a forest

Splicing parity sets attains the sharing floor for specified forest-routable overlap patterns, and it gives exact counts for fan-out and chain layers. The section also compares the floor with controlled-unitary counts and applies it to field-multiplication oracles.

C.1.3 Splicing along a forest

  • A splice deletes one parity and lifts the new block’s parities; when the blocks share a qubit, a coincident pair cancels and reduces the added cost.The two cases add 6 or 4 parities, respectively.Source: §C.1.3 · p. 25
  • For linear-overlap layers routed by a maximum-degree-three spanning forest, splicing attains δ = 6m + 1 − 2η.Each of the η shared-qubit edges uses a case-(ii) splice; the remaining m − 1 − η splices use case (i).Source: §C.1.3 · p. 26
  • The chain layer of carry triples has T-count 4m + 3, while the construction also yields 6m + 1 for disjoint blocks.The chain’s shared carries form a path; a disjoint fifth block starts a new component.Source: §C.1.3 · p. 27, Figure 4 · p. 28
  • The forest construction has a scope boundary: each overlap-graph component must satisfy η_C = m_C − 1, and the routing must also meet the degree condition.A component with η_C ≥ m_C has no such forest; the six-qubit triangle has floor 13 but exact count 15.Source: §C.1.3 · pp. 27–28
  • The proposed dichotomy conjectures that a linear-overlap layer attains the floor exactly when every overlap-graph component has η_C = m_C − 1.The m = 9 example shows that this condition alone does not make Proposition 41’s routing argument work.Source: §C.1.3 · p. 28

C.2 Controlled unitaries

  • For controlled unitaries, the floor is exact on pure-cubic gates and gates with even μ, and is one below the exact count otherwise.The exact counts cited from Campbell and Howard are 2μ for even μ and 2μ + 1 for odd μ.Source: §C.2 · pp. 28–29

D Applications

  • The appendix applies the floor to GF(2^k) multiplication oracles and certifies optimizer outputs block by block.This passage states the application scope rather than its quantitative findings.Source: D Applications · p. 29

D.1 Field multiplication

The isotropy floor applies directly to the pure-cubic phase oracle for multiplication in GF(2^k). It gives a linear lower bound, and exact decoding establishes the merged-residue count for GF(4).

D.1 Field multiplication

  • 6k + 1 is a lower bound on the T-count of the GF(2^k) multiplication phase oracle on 3k qubits.Its trilinear tensor is non-degenerate, so the pure-cubic floor applies.Source: §D.1 · pp. 29–30
  • At k = 2, the merged phase-polynomial residue has weight 17 and is its own coset leader, so δ(U_2) = 17.The term-by-term expansion is optimal once parities are merged; the unmerged schoolbook expansion has weight 35.Source: §D.1 · p. 30
  • The GF(4) multiplier’s best published T-count is 17, matching the certified value.The cited circuits use FastTODD optimization or AlphaTensor-Quantum.Source: §D.1 · p. 30
  • The T-count grows as Θ(k), although the known constants in the linear bound are too large to compete with explicit circuits.The cited construction-based lower bound has the right asymptotic order but not competitive constants.Source: §D.1 · p. 30
  • Sharper floors from individual slice ranks and brackets against published multipliers are deferred to §4.Source: §D.1 · p. 30

D.2 Certifying the optimizers

The optimizer study applies the floor to PyZX phase-polynomial blocks and compares its certificates with nullity and exact decoding. The resulting guarantees are tied to the chosen Hadamard positions and block partition, with separate examples outside that scope.

D.2 Certifying the optimizers

  • Across five seeds, the floor certifies 186–193 blocks among 309–311, versus nullity’s 111–114; seed 4 has the most certificates and lowest total T-count.Seed 4 has total T-count 1878, compared with 1880 to 1891 for the other seeds.Source: §D.2 · p. 31
  • Certificates apply only to resynthesis on the same qubits with the same block partition and Hadamard positions; on csum mux 9, AlphaTensor-Quantum reports 71 T gates versus block floors totaling 73.On mod5 4, the whole-circuit floor is 7 and matches AlphaTensor-Quantum’s reported optimum, while the phase-block optimizer splits the circuit into two blocks totaling 8 T gates.Source: §D.2 · p. 31
  • For all twelve GF(2^k) benchmark multipliers, k = 4 to 256, the output-register structure permits the paper’s multiplier result to apply.Each Toffoli targets an output register never used as a control, and the modular-reduction CNOT network separates two Hadamard-conjugated blocks.Source: §D.2 · p. 31
  • Among blocks with 7 to 28 active qubits, the floor certifies 37 of 72, versus 15 for nullity; for phase teleportation, it certifies 33 of 61 beyond decoding.For phase teleportation, the floor certifies 252 of 337 blocks versus 141 for nullity, and 248 of 318 on the 24 circuits both optimizers finish.Source: §D.2 · p. 32
  • The 237-T-gate total gap splits into 81 gates of floor slack on 239 decoded blocks, 8 gates of known excess, and 148 gates across 72 undecodable blocks.Thirty-five of those 72 undecodable blocks have a nonzero gap; Table 5 reports floor-to-exact-count distances for blocks with at most six active qubits.Source: §D.2 · p. 32, Table 5 · p. 32

AI Disclosure

  • The author reports conceiving the isotropy floor and proving the main results, while generative AI assisted with secondary results, verification code, benchmark experiments, and drafting.The author says all assistance was reviewed and verified and takes full responsibility for the work.Source: AI Disclosure · pp. 32–33
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