High-rate concatenated codes with transversal gates code_highrate

Block codes whose rate k/n is a large fraction of unity and whose logical Cliffords run transversally — depth-one layers of two-local physical gates — instead of d rounds of lattice surgery. Four members define the node. The iceberg code [[k+2,k,2]] detects any single-qubit error (Quantinuum: Self, Benedetti, Amaro; Nature Physics 20, 219, 2024-01-05) D[673]. Two-level concatenation gives [[(k₁+2)(k₂+2), k₁k₂, 4]]; [[80,48,4]] (k₁=6, k₂=8) is the largest cycle executed anywhere D[105]. The tesseract colour code [[16,6,4]] is a doubly-even self-dual 4D CSS code from the [16,5,8] Reed–Muller code (subsystem variant [[16,4,2,4]]), its logical Clifford group depth-one D[674]. Goto's many-hypercube codes concatenate [[4,2,2]] blocks to ~30% rate S[675].

Attributes: no carrier of its own (affinity 0.0), it inherits the platform's; no characteristic time and no entangling determinism here; no readout of its own; mobility by physical transport of the carriers, which supplies the required all-to-all; no control modality, placed nowhere; error structure as the code sees it, depolarising Pauli noise; no manufacturing.

[[80,48,4]] costs 1.67 physical qubits per logical at distance 4 D[105], the tesseract 2.67 D[674]; a teraquop surface code (sized for 10¹² reliable operations) costs 650 under 0.1% noise S[655], IBM's [[144,12,12]] 24 at distance 12 D[248]. The comparison is not like-for-like, and that is the limitation: distance 4 corrects one error, suppression goes roughly as p², and there is no Λ-style exponential handle. Distance 2 corrects nothing — it detects, and detection means discarding.

Hence acceptance: the [[80,48,4]] cycle accepted 0.62(2) of shots after post-selecting uncorrectable errors D[105], and the 64-logical XY-model simulation accepted 3.2% at algorithmic depth D[97]. Acceptance falls exponentially in circuit volume, so a claim here that omits it is unreadable.

Transversality is not universality (Eastin–Knill): magic states come from outside the block — [[6,2,2]] at 7×10⁻⁵ on ions D[107], 5-to-1 colour-code distillation on atoms D[147]. Transversal CNOT correlates errors across blocks, so decoding must be block-wide; algorithmic fault tolerance is the compensating theory, a constant number of syndrome rounds per logical gate rather than d S[152]. Ion leakage at 1.1×10⁻⁵ per Clifford D[97] and atom loss are not Pauli errors, so loss-aware decoding is a precondition.

Best demonstrated as of 3 Sep 2026: 48 error-corrected logical qubits in one [[80,48,4]] block on a 98-qubit machine, logical gate infidelity (1.0–1.2)×10⁻⁴ against ~8×10⁻⁴ bare physical two-qubit error D[97], [105]. Typical at scale is thinner — that platform's 94 "logical" qubits are distance-2 iceberg, detection only D[105].

The dominant budget term is the residual weight-2 error at distance 4 plus the discarded fraction: at 62% acceptance per cycle, a hundred-cycle circuit is out of reach.

No fabrication of its own; the dependency is the connectivity hardware. On ions that is the QCCD trap — Helios runs 1,228 electrodes and at least seven laser wavelengths, and Sol (2027) moves to a Honeywell-fabricated 2D grid trap D[97]R[118]. On atoms it is AOD/SLM tweezer optics and the laser chain D[4]. Single points of failure: ion-trap chip fabrication (Honeywell in-house; IonQ bought SkyWater's foundry, ~$1.8 B, closed 2026-07-31 fact), acousto-optic deflectors, high-power lasers. Export control attaches to the machines, not the mathematics: US EAR quantum-computer controls and EU dual-use equivalents cover traps and lasers; the codes are published and ship as software.

The burden is decoding, not wiring. Syndrome extraction is shallow, but each transversal gate correlates errors across the block, so the decoder handles block-wide syndromes and, on atoms, loss flags, inside the reaction time. Not yet binding: ion cycles run 1–5 ms and atom cycles ~1–4.5 ms D[97]D[4] against decoders at tens of microseconds. Pinnacle instead assumes a 1 µs cycle and 10 µs reaction time S[29] — superconducting or photonic numbers, where real-time correlated decoding becomes the wall. At 10³ physical qubits one [[80,48,4]] block plus routing fits; at 10⁴ the constraint is all-to-all between blocks; at 10⁶ the family does not close without long-range couplers or bounded-degree qLDPC.

Four architectures: primary on trapped-ion QCCD (Quantinuum) and alkali neutral atoms (Harvard/MIT, QuEra, Pasqal, Infleqtion, Google); alternate on ions in a linear Paul trap with laser gates (AQT, Quantum Art, Innsbruck) and ions with electronic (laser-free, microwave-driven) gates and chip control (IonQ/Oxford Ionics, eleQtron). It requires transport-supplied connectivity — AOD tweezer transport in a zoned architecture, ion shuttling through QCCD junctions and grid traps, or a single chain's motional bus — and provides for correlated, loss-aware decoding. It replaces the surface code at an explicit price: the surface code needs only nearest-neighbour coupling and has a mature ~1% threshold with exponential distance scaling, while this family needs all-to-all and buys width instead of depth, and it conflicts with nearest-neighbour connectivity outright.

Across families: this node explains the largest divergence in the 2026 fault-tolerance table — 48 and 96 logical qubits on ions and atoms against 1–2 on superconducting — a connectivity fact, not a qubit-quality fact. It adds no clock of its own; it sets the QEC cycles per logical layer, ≈1 with transversal gates plus correlated decoding versus ≈d for lattice surgery S[152]. Derived clock (= sum of the syndrome round: gate layers + transport + readout + reset): 9.66 ms on the ion QCCD architecture, transport-set, against the ~55 ms Helios full-width layer, and 1.31 ms on the alkali atom architecture against a measured ~1–4.5 ms round. Empty slots nearby: no high-rate transversal code on a nearest-neighbour platform, no member above distance 4, no magic-state factory inside a high-rate block.

Headline numbers come from repeated logical preparation and measurement in both bases, post-selected on detected-but-uncorrectable events. Two cautions: the Helios ≤4×10⁻⁵ cycle figure is a confidence upper bound from zero logical errors in 5,000 shots per basis, not a measured rate D[105]; and no number is interpretable without its acceptance D[105]D[97]. Leakage and loss sit outside the Pauli model — IonQ's break-even, within error bars in one code the paper does not identify (3.95 ± 0.68 s against 3.84 ± 0.48 s physical, while its [[18,4,3]] memory reaches 2.48 ± 0.40 s), needed leakage post-selection D[109]. Nothing here reports Λ-style distance scaling. Replication is good: the tesseract has run on two unrelated platforms D[667]D[4].

Conflicts. The tesseract is [[16,6,4]], not [[16,4,4]] — the Error Correction Zoo entry and the arXiv abstract agree D[674]D[667]. "48 corrected" and "94 detected" on Helios are different codes at different distances, routinely conflated D[105]. Iceberg's sub-100,000-qubit RSA-2048 claim assumes p=10⁻³ with 1 µs cycles S[29], not comparable to Gidney's <1 M S[7].

Not published: Iceberg Quantum's headquarters — trade press gives a Berlin base with a US presence, the founding institution is the University of Sydney, and the company site names no location P[677]C[676]; whether the Oxford Ionics–Iceberg Quantum partnership continues under IonQ — as of 2026-09-30 Iceberg Quantum lists PsiQuantum and Diraq as partners and names neither Oxford Ionics nor IonQ C[676]; the acceptance of the Harvard [[16,6,4]] 96-logical-qubit runs; and a dated patent-family count for this code family. Quantinuum's Q2-2026 claim of "near five-nines logical fidelity with a novel code family" has no public paper as of 2026-09-03. Goto's many-hypercube codes have no hardware demonstration; their 30% rate (64 logical in 216 physical qubits) is simulation only S[675].

Money.

• 2025-03-24 · Iceberg Quantum · pre-seed · $2 M · Blackbird lead, LocalGlobe · closed P[678]
• 2025-09-04 · Quantinuum · equity · $600 M at $10 B pre-money · NVentures, Quanta, QED, JPMorgan · closed fact
• 2025-09-17 · IonQ · M&A, Oxford Ionics · $1.075 B · closed fact
• 2025-11-06 · DARPA · QBI Stage B, up to $15 M each, eleven performers · announced fact
• 2026-02-13 · Iceberg Quantum · seed · $6 M · LocalGlobe lead, Blackbird, DCVC · cumulative ~$8 M · closed P[677]
• 2026-06-03 · Quantinuum · IPO · $1.68 B gross, Nasdaq QNT; Q2-2026 revenue $8.0 M · closed fact
• 2026-06-16 · Atom Computing · $100 M Series C + $100 M DoC CHIPS LOI · >$300 M · closed + LOI fact
• 2026-07-31 · IonQ · M&A, SkyWater Technology · ~$1.8 B · closed fact

Market & supply chain. Nobody sells a code. Value accrues to machine vendors that can quote logical-qubit counts (Quantinuum, QuEra, Atom/Microsoft), to one architecture licensor (Iceberg Quantum C[676]), and to the decoder layer (Riverlane, NVIDIA's NVQLink transport fact). Concentration risk is extreme: the experimental record rests on four groups — Quantinuum with Microsoft, Harvard/MIT with QuEra, Infleqtion, Innsbruck — and two hardware families, and one group authored both the code and the machine. Unit economics are the rate — 1.67 physical per logical at d=4 D[105] against 650 for a teraquop surface code S[655]. Goals paid for: G3 primarily, G2 (bare iceberg as a cheap detection layer D[673]), G7 where a machine is sold in logical qubits, G4 only if concatenation beyond d=4 survives leakage.

IP & standards. The iceberg code is published (Nature Physics 2024) D[673] and the tesseract catalogued in the open Error Correction Zoo P[674]; Microsoft and Quantinuum publish jointly rather than cross-licensing visibly D[667]. Pinnacle is a preprint with no disclosed patent position S[29]. No dated patent count exists for this code family — the PatSnap figures cited are device-class Pfact. Tooling is open; no standards body governs code choice.

Roadmaps & track record. Quantinuum (promised 2024-09 · Sol 2027 ~100 logical, Apollo 2029 · Helios on schedule 2025-11, [[80,48,4]] delivered 2026-02) Rfact — credible; code results ran ahead of the hardware. QuEra/Harvard (promised 2024-01 · 100 logical in 2026 · slipped to Libra 2028) Rfact — the 96 logical qubits were physics, not product. IonQ (promised 2025-06 · 25–50 physical per logical by 2030) Rfact — needs high-rate codes; its 2020 roadmap missed 2026 by ~40×. Iceberg Quantum (promised 2026-02 · partners at utility scale in 3–5 years) P[677] — seed-stage, no hardware.

Strategic reading. Winners: all-to-all platforms and anyone selling logical-qubit counts. Losers: 2D-local platforms, which must buy long-range couplers to compete on rate. The substitution threat is near — bounded-degree qLDPC gets most of the rate and real distance scaling without all-to-all D[248]; if IBM's gross-code module works in 2026–27, this family retreats to cheap distance-2/4 detection. Bargaining power sits with hardware vendors; a code-architecture supplier has none unless its IP proves essential.

Confirm within 12–24 months if: a distance-6 or -8 member of the concatenated iceberg family runs with acceptance above 0.5 (Sol, 2027); a high-rate block is sustained over ≥50 QEC cycles without post-selection; or a partner names Pinnacle in a published roadmap by end-2027. Demote if 2027 machines still quote logical counts only at single-digit-percent acceptance at depth, or if IBM's gross-code module matches the rate at distance 12 on degree-≤7 connectivity. Best case by 2029: Apollo- and Libra-class machines run hundreds of logical qubits at constant syndrome rounds per logical gate S[29]S[7]. Worst case: the family stays a detection layer and bounded-degree qLDPC takes the fault-tolerant road.

Open questions: (1) does concatenation to d≥6 keep the rate advantage once leakage and loss are modelled honestly? (2) can a magic-state factory live inside a high-rate block? (3) what is the true acceptance scaling with circuit volume? (4) can block-wide loss-aware decoders keep up at 1 µs cycles? (5) is any nearest-neighbour route cheaper than long-range couplers? Watch Sol's first logical results, Quantinuum's unpublished "near five-nines" code-family claim, IBM Kookaburra, and the first partner adoption of Pinnacle.