Error mitigation in the code slot (ZNE, PEC) — not a code code_mitig

Error mitigation estimates an observable's noiseless expectation value from many noisy runs and classical post-processing; nothing is encoded, no syndrome is measured, no error corrected G[693]. Temme, Bravyi and Gambetta proposed extrapolation to zero noise by Richardson's method and cancellation by quasi-probability resampling in 2016 D[694]; Li and Benjamin independently extrapolated boosted errors to zero D[695]. Hardware ZNE followed in 2019 D[696]; then PEA, amplifying a learned noise model with injected Pauli errors D[308], TREX for readout C[693] and TEM, a tensor-network inverse of the global noise S[697]. Attributes: platform-agnostic, static, no transport or control modality; Pauli error class after twirling.

PEC writes each layer's inverse noise as a signed mixture of implementable operations of norm γ ≥ 1; the estimate is unbiased, but shots grow as γ² and γ exponentially with depth C[693]. For Pauli noise γ ≈ e^(2P), so shots multiply by ≈ e^(4P) S. At Heron r3's median CZ error of 0.15% D[698], 1,000 CZs in the light cone give P ≈ 1.5 and ≈ 400× the shots, 7,500 give ~10¹⁹× S; today's 10²–10³× budgets cap P near 1.2–1.7 S. ZNE samples at amplified noise — by default three factors, nominally ~3× C[693] — and extrapolates; it is biased by its fit and must resolve a signal attenuated by up to e^(−2P), so its cost grows comparably S. TEM claims the square root of PEC's overhead S[697], still exponential S. The limit is general: overhead grows exponentially with depth for any protocol under local depolarising noise D[699], nonlinear post-processing included D[700], and worst-case estimation needs superpolynomially many samples at shallow depth D[701]. Only lower physical error moves the exponent.

Under local depolarising noise, a circuit's noisy expectation value is the Laplace transform of its reactivity function — the contribution of Pauli paths sorted by weight (the number of qubits a Pauli operator acts on non-trivially); inverting it exactly is ill-conditioned, with coefficients growing exponentially in weight, and only smoothed features in weight are recoverable, at a sampling cost exponential in the ratio of weight to resolution P[891]. The Atlas's reading, not derived in the paper: noise-scaling mitigation attempts the same inversion, so its cost is set by the circuit's high-weight Pauli content, not by the error rate alone.

The 2023 claim of accuracy "beyond brute-force classical computation" D[308] was matched classically within weeks, and converged values exposed bias in its extrapolation D[704]; the 2026 claims rest on classical heuristics disagreeing, not on proven hardness D[698]. Dominant term: P. Faster shots — Nighthawk r2's 100,000 circuits per second, 25× Heron — shorten wall-clock time, not the exponent C[601].

Nothing is fabricated; the bill is QPU time and classical compute — 3.2 million shots, ~45 min of QPU time per point for Algorithmiq's rescaled estimate D[698]. Supply is the vendor runtime C[693] plus Qedma's QESEM, which learns device noise, adapts circuits and post-processes C[705], and Algorithmiq's TEM in IBM's Qiskit Functions Catalog C[706].

The requirement is a calibrated, stable noise model: PEA and PEC learn a sparse Pauli–Lindblad model per layer of entangling gates under random Pauli twirling D[308]D[702]; every circuit is compiled in many randomised instances S. TREX randomises measurements with X gates and classical bit flips, diagonalising the readout matrix for inversion C[693]. Drift is the failure mode: a model learned before a four-hour PEC run is wrong by whatever drifts during it S. There is no feed-forward, and the output is expectation values, not samples G[693].

Slot 7 of the Transmon lattice with tunable couplers architecture, beside code_surface, code_color, code_qldpc, code_magic and code_detect. It requires ct_rt, for calibrated noise models; it provides nothing downstream — no logical qubit, no syndrome for slot 8's decoders; it is replaced by code_surface, and the open conflict edge (2023-06) lists no remedy for its exponential overhead. code_detect discards flagged runs at a cost of 1/acceptance, also exponential in circuit size; mitigation keeps every run and reweights, removing bias only on average S. The gap ledger's "all three IBM July-2026 advantage claims" is two of three — QESEM on Heron r3 D[233] and PEC on ibm_boston D[698]; IBM–UChicago's post-selects on spacetime codes, a code_detect result C[45]. The ledger's ion placement was not taken, though Qedma repeated cycles on Quantinuum H2 and Helios D[233]. Register: IBM Eagle r1–r3 (primary, retired); Heron r3 and Nighthawk r2 (alternate).

PEC is unbiased, with error bars, if its noise model is right, so verification becomes validating the model C[45]. Of the register's three cells, Heron r3's and Nighthawk r2's are verified ✅ — the first by the July-2026 ibm_boston preprints D[233], [698], the second by a section of the r2 blog C[601] — and Eagle's is 🔎, though its source supports it D[308]. Heron r1 and Nighthawk r1 hold no code: no primary source shows a named mitigation method run on r1 itself, and the press article on r3 that its cell cites names none P[707]. Zuchongzhi 3.0 and Tianyan-287 run random-circuit sampling scored on raw fidelity; their abstracts mention neither mitigation nor correction D[35]D[708], so as of 2026-09-26 their code slot is empty, not mitigated.

IBM's 100× PEC-overhead reduction by samplomatic (2025-11-12) is a blog claim. Not published: its baseline and circuit class, and the shot count, QPU time and observable of Nighthawk r2's 7,500-gate PEA run. The e^(4P) shot scaling and the P values are derived from the PEC construction and published error rates, not quoted from a source.

Who.
| Organisation | Role | Country | What exactly they do with this technology | Evidence |
|---|---|---|---|---|
| IBM | developer | US | TREX, ZNE, PEA, PEC in Qiskit Runtime; 2023 utility experiment; PEA at 7,500 gates | C[693]D[308]C[601] |
| Qedma | software vendor | IL | QESEM: noise learning, PEC- and ZNE-based estimators | D[233] |
| Algorithmiq | software vendor | IT (formerly FI) | TEM in IBM's catalogue; PEC in the Loschmidt-echo claim | C[706]D[698] |
| Flatiron Institute | research | US | Contributor to the open advantage tracker | C[486] |

Money.

• 2025-07-03 · Qedma · Series A led by Glilot Capital Partners, IBM participating · USD 26 M · announced C[705]
• 2026-05-11 · Algorithmiq · round led by United Ventures, with CDP and Inventure · EUR 18 M (EUR 36 M in total) · announced C[709]

Market & supply chain. Sold as runtime options and third-party functions on the hardware vendor's cloud, with IBM holding equity in Qedma: a vertically tied market. It pays into G2 only; G3–G4 need a code.

IP & standards. No standard defines a figure of merit for mitigated estimates or a noise-model disclosure format; no dated patent count from a named database, as of 2026-09-26.

Roadmaps & track record. IBM promised Nighthawk revisions at 5,000, 7,500, 10,000 and 15,000 gates R[486]; the 7,500-gate step, its 2026 milestone, shipped on 2026-08-31 with PEA C[601].

Strategic reading. Mitigation makes the noise model the product and keeps value inside the hardware vendor's stack. Only noise learning carries over to fault tolerance; when a code runs, the technology empties.

Confirm if an unbiased PEC estimate on ≥5,000 gates appears with error bars by 2027-12-31, or a July-2026 claim survives to 2027-07-31 without converged classical reproduction; demote those claims to utility if converged tensor-network or Pauli-propagation values match them, as in 2023.
Open questions. (1) How stable is a learned Pauli–Lindblad model over a multi-hour run at 10³–10⁴ gates? (2) Does TEM's quadratic saving survive model error on hardware? (3) At what size does a detection code plus mitigation beat mitigation alone in shots? (4) Can a mitigated expectation-value task be proven classically hard? (5) What shot multiplier did Nighthawk r2's 7,500-gate PEA run spend?