Fault-Tolerant Quantum Computing

Established Fault tolerance is the discipline of computing reliably on hardware that fails constantly, and it rests on a proven theorem. A logical qubit is encoded across many physical qubits; stabiliser measurements extract error syndromes every cycle without disturbing the encoded data; a classical decoder infers what went wrong; corrections are applied or tracked in software. The threshold theorem (Shor, and Aharonov and Ben-Or, in the mid-1990s) guarantees that if the physical error rate sits below a code-dependent threshold, the logical error rate can be driven down exponentially by enlarging the code, at polynomial cost in qubit count.

Established In December 2024 the theorem became a measurement. Google's 105-qubit Willow processor ran a distance-7 surface code on 101 of its qubits and recorded a logical error rate of 0.143 ± 0.003 per cent per cycle, with each two-step increase in code distance suppressing error by a factor Λ = 2.14 ± 0.02 (Quantum error correction below the surface code threshold, Nature). The encoded memory outlived the best physical qubit on the same chip by a factor of 2.4 ± 0.3. Adding qubits made the machine better rather than worse — the inversion the field had chased since 1995.

Frontier Five numbers now define the state of the art, and this brief tracks them instead of qubit counts. They are: the physical error rate against the code threshold (roughly 1 per cent for the surface code, 0.7 per cent for the leading qLDPC codes); the suppression factor Λ per distance step; decoder latency against the 1.1-µs syndrome cycle; the overhead ratio of physical to logical qubits (101 to 1 in the Willow memory, roughly 1,000 to 1 in resource estimates for useful machines); and magic-state infidelity, which prices the non-Clifford gates every useful algorithm consumes in bulk.

Established The distance from here to useful is large, and for the first time it is measurable. Below-threshold memory exists on one superconducting chip. Universal fault-tolerant gate sets exist on trapped ions at error rates near 10⁻⁴. Resource estimates for factoring or industrial chemistry assume logical error rates of 10⁻⁹ to 10⁻¹², sustained across roughly a million physical qubits for days. Vendor roadmaps promising such machines by 2029 appear throughout this brief as claims with dates attached, never as findings.

Established Below threshold, logical error falls exponentially with code distance; above it, error correction makes a computer worse. For the surface code the logical error per cycle scales approximately as (p/p_thr)^(d+1)/2, where p is the physical error rate, p_thr the threshold near 1 per cent, and d the code distance. Every measurement in this section is a test of that scaling law, and since 2024 the law has been winning.

Established Google's Willow result is the cleanest below-threshold demonstration on record. Its 2023 predecessor experiment (Suppressing quantum errors by scaling a surface code logical qubit, Nature) found distance-5 only marginally better than distance-3. Willow, with better qubits, crossed decisively: distance-3 logical error near 0.3 per cent per cycle, distance-5 at 0.35 per cent with the real-time decoder (0.269 per cent with an offline neural-network decoder), distance-7 at 0.143 ± 0.003 per cent. The suppression factor was Λ = 2.14 ± 0.02 with the best offline decoder and 2.0 ± 0.1 with the real-time one — the decoder is now a measurable part of the machine's error rate.

Established Beyond break-even: the logical qubit beat every physical qubit it was made of. The distance-7 memory retained information for 291 ± 6 µs against 119 ± 13 µs for the best constituent physical qubit, a ratio of 2.4 ± 0.3. This is the quantum-memory analogue of a repeater finally gaining rather than losing signal, and it held while the code ran continuously rather than in post-selected snippets.

Established Real-time decoding worked, with measured latency. Willow's distance-5 code was decoded live at a mean latency of 63 ± 17 µs while syndrome rounds arrived every 1.1 µs, and below-threshold performance held across runs of up to one million cycles — about a second of continuous correction. Latency and throughput, not decoding accuracy alone, are now first-class engineering metrics: the most accurate decoders in the study (neural networks) ran only offline.

Frontier Correlated errors put a measured floor under everything. Willow's high-distance repetition codes, which should have reached vanishing error, instead hit a floor near one error in 10¹⁰ cycles, set by high-energy events arriving roughly once per hour, each depressing a neighbourhood of about 30 qubits for around 400 µs. The 2022 baseline was far worse: McEwen and colleagues (Nature Physics) measured cosmic-ray and radioactivity bursts striking a 26-qubit processor roughly every ten seconds and collapsing coherence chip-wide. Gap engineering in the qubits bought orders of magnitude of both rate and locality, but the residual floor sits about four orders of magnitude above the 10⁻¹⁴-class error budgets that factoring estimates assume — correlated errors are the strongest known argument that scaling is not just more of the same.

Frontier Colour codes now run below threshold too, with worse suppression but cheaper geometry (vendor figures). In June 2025 Google reported distance-3 to distance-5 colour codes on Willow with Λ = 1.56, against 2.31 for surface codes on the same platform, plus injection of a magic state into a colour-code logical qubit at 99 per cent fidelity. The trade is deliberate: colour codes need fewer physical qubits per logical qubit and execute some logical Cliffords in tens of nanoseconds that cost the surface code far more.

Frontier Magic states — the priced ingredient of universal computation — got two orders cheaper. In December 2025 Google reported magic-state cultivation on a superconducting processor at about 1 × 10⁻⁴ infidelity, a roughly 40-fold improvement over injection protocols, with cultivated states code-switched into a distance-5 surface code (arXiv:2512.13908). The cost is post-selection: only about 8 per cent of attempts survive filtering. Applications need magic-state infidelities of 10⁻⁶ to 10⁻⁸, so cultivation must improve a further two to four orders while raising, not lowering, throughput.

Frontier Trapped ions hold the record for logical gate quality; superconducting circuits hold the record for speed. Microsoft and Quantinuum reported 12 logical qubits with better-than-physical error rates in 2024. In June 2025 Quantinuum claimed the first fully fault-tolerant universal gate set with repeatable error correction: magic-state infidelity of 7 × 10⁻⁵ and a fault-tolerant two-qubit non-Clifford gate at 2 × 10⁻⁴, roughly ten times below the underlying physical error, using code switching on just 28 qubits (all vendor figures). In March 2026, work on the Helios trapped-ion machine reported GHZ entanglement across 94 error-protected logical qubits at about 95 per cent fidelity, 48 logical qubits under concatenated codes, and logical gate errors around one in ten thousand operations (arXiv:2602.22211, reported by The Quantum Insider). The caveat is clock speed: ion operations run at millisecond scales, roughly a thousand times slower than superconducting cycles, and no demonstrated architecture yet closes that gap.

Frontier Neutral atoms demonstrated the largest logical circuits — under a weaker standard. Bluvstein and colleagues (Nature, 2024) ran up to 48 logical qubits encoded in 280 atoms, executed 228 logical two-qubit gates and 48 logical CCZ gates, and showed a transversal CNOT improving as surface-code distance grew from 3 to 7. Much of the gain, however, came from error detection with post-selection rather than repeated correction — the 48-qubit circuits retained a cross-entropy benchmark score of only about 0.1 — so these results measure a different, easier quantity than Willow's continuously corrected memory.

Established qLDPC codes promise a tenfold overhead cut, so far on paper. Bravyi and colleagues (Nature, 2024) constructed bivariate bicycle codes storing 12 logical qubits in 288 physical qubits for nearly a million syndrome cycles in simulation, with a 0.7 per cent threshold; matching that performance with surface codes would take nearly 3,000 physical qubits. The price is connectivity: a degree-6 coupling graph that no planar superconducting chip yet provides. IBM's Loon processor (2025) is the stated hardware test vehicle, with the company's Relay-BP decoder claiming a five-to-tenfold efficiency gain over rivals (vendor figures).

Frontier Bosonic cat qubits attack the overhead from the hardware side. AWS's Ocelot chip (Nature, February 2025) pairs five cat data qubits, whose engineered noise bias pushed bit-flip lifetimes toward one second against 20-µs phase-flip times, with a repetition code for the remaining error channel. Measured logical error was 1.65 per cent per cycle at distance 5 versus 1.72 per cent at distance 3 — suppression barely resolved — while the company projects up to 90 per cent overhead reduction at scale (vendor). The bias changes which code you need, not whether you need one.

Established The ledger of what has not happened disciplines the rest of this brief. No logical qubit has sustained an error rate at or below 10⁻⁶ per cycle. No complete algorithm has run fault-tolerantly end to end and beaten its unencoded equivalent. No qLDPC memory has run below threshold on hardware. Below-threshold operation with growing distance has been shown for one code family at scale on one 105-qubit device, and every logical-qubit count above about fifty involves post-selection, error detection rather than correction, or very short circuits.

The open questions are quantitative, which is itself the news — each one names a number somebody will measure this decade.

• Frontier Does Λ survive scaling? At Λ = 2.14, reaching 10⁻⁶ per cycle from distance-7's 1.43 × 10⁻³ requires roughly ten more distance steps — codes near distance 25 to 31 on thousands of qubits — with no new error mechanism appearing across three orders of magnitude. The 10⁻¹⁰ repetition-code floor already shows one mechanism waiting.
• Frontier Can decoders keep up? Syndrome volume grows with the square of distance per cycle; Willow's 63-µs real-time latency must not grow into a backlog at distance 25. Riverlane's Local Clustering Decoder claims one round decoded in under a microsecond on FPGAs and is deployed with Rigetti, OQC, Infleqtion and Oak Ridge (vendor); independent at-scale benchmarks do not yet exist.
• Frontier Can qLDPC codes compute, not just remember? Bivariate bicycle codes store cheaply, but addressable fault-tolerant logic on them — IBM's proposed logical processing units — must not surrender the tenfold overhead win to gate machinery.
• Frontier Magic-state economics. Cultivation delivers 10⁻⁴ infidelity at 8 per cent acceptance; algorithms need 10⁻⁶ to 10⁻⁸ at production rates, and magic-state throughput dominates the floor plans of most resource estimates.
• Frontier Clock speed is destiny. Gidney's one-week factoring runtime assumes 1-µs cycles. At millisecond trapped-ion speeds the identical computation takes decades unless parallelism or hybrid architectures compensate; nobody has published a convincing closure of this gap.
• Speculative Do biased-noise and bosonic platforms change the constants? The claimed up-to-90-per-cent overhead reduction from cat qubits rests on extrapolation from a five-qubit device whose distance-5 code barely outperformed distance-3.
• Frontier What exactly causes the bursts? Partitioning the once-per-hour high-energy events between muons, gamma rays and ambient radioactivity determines whether shielding, underground siting or on-chip mitigation is the right spend; a 2025 Nature Communications experiment synchronously tagging cosmic rays and qubit errors began this program.

Every bottleneck below is a measured gap between a demonstrated number and a required one, not a metaphor.

• Established Decoder throughput. Real-time decoding exists at distance 5 with 63-µs latency; useful machines need distance 25 or more decoded continuously for days, with syndrome data arriving at terabytes per hour per logical qubit block. The accurate decoders are slow (neural networks, offline) and the fast ones are less accurate — Willow paid a Λ penalty of about 0.14 for decoding live.
• Established Wiring and cryogenic input/output. Present dilution refrigerators route a few control lines per qubit; a million-qubit machine at that ratio implies order-of-magnitude-a-million cables into the coldest stage, which no fridge supports. Cryo-CMOS multiplexing at 4 K exists in prototype and cuts this by orders of magnitude, at a heat and noise budget still being measured.
• Established Fabrication spread. Josephson-junction parameter variance and two-level-system defects set both the physical error rate and the yield of acceptable qubits per wafer; Willow's below-threshold margin came substantially from materials and gap engineering, and nobody has published defect statistics at hundred-thousand-qubit volume.
• Frontier Correlated events. The once-per-hour, 30-qubit bursts flooring repetition codes at 10⁻¹⁰ are unhandled by standard decoders, which assume independent errors; mitigations (gap engineering, phonon traps, shielding, burst-aware decoding) each address part of the spectrum and none is demonstrated at algorithm length.
• Established Slow platforms must parallelise. Trapped-ion and neutral-atom systems buy their gate quality with millisecond transport and imaging steps; atom loss and reloading interrupt continuous correction, and mid-circuit measurement remains the rate limiter on both platforms.
• Frontier Magic-state factories. In factoring-scale floor plans, distillation or cultivation occupies a large fraction of the machine; cultivation's 8 per cent acceptance means twelve attempts per delivered state, and the pipeline from cultivated state to consumed T gate has never run at rate.
• Established No qLDPC hardware. The tenfold overhead cut requires degree-6 connectivity with long-range couplers whose crosstalk and loss budgets are unproven; IBM's c-couplers on Loon are the first physical test (vendor).

Progress here draws on fields that publish on their own schedules.

• Established Materials science of superconducting circuits — two-level-system defect physics, junction uniformity, quasiparticle and phonon dynamics — sets the physical error rate that everything downstream multiplies.
• Established Cryogenic classical electronics: control ASICs at 4 K, low-heat amplifiers and multiplexers are prerequisites for wiring beyond a few thousand channels.
• Established Classical computing co-design: FPGA and ASIC decoder implementations, and the real-time software stack binding them to control hardware, are as load-bearing as any quantum component — the decoder is part of the error rate.
• Frontier Coding theory continues to move the goalposts favourably: bivariate bicycle codes, yoked surface codes for idle storage and magic-state cultivation each cut published resource estimates by integer factors within the last three years.
• Established Atomic physics infrastructure — laser stability, optical tweezers, vacuum lifetime, photonic interconnects — governs the ion and atom platforms' path to continuous operation.
• Frontier Error-model science: leakage, crosstalk, drift and burst characterisation feed decoder priors; adaptive decoders that learn the noise (as Riverlane's does (vendor)) depend on this measurement discipline.

Established This field is unusually rich in scheduled, falsifiable tests; the next five years of hardware generations are effectively pre-registered experiments. Vendors have published dated checkpoints — IBM's Loon (2025), Kookaburra (2026), Cockatoo (2027) and Starling (2029), Quantinuum's Apollo (2029) — that will either land or slip in public.

Frontier The decisive demonstration is endurance at scale: a single logical qubit at distance 15 or more, decoded in real time, holding a logical error rate near one in a million cycles for hours, through the radiation bursts that currently floor repetition-code performance at one error in ten billion cycles. Willow proved the scaling law over two distance steps and one second of correction; a useful machine needs it over ten steps and days of runtime, through the correlated-error weather. Nobody has scheduled this run as such, but the hardware generations Google and IBM have announced for 2026–2028 are the machines that could attempt it, which puts a verdict inside this decade if the roadmaps hold.

Frontier Second: a qLDPC memory on hardware beating surface-code overhead. Twelve logical qubits held below threshold in 288 physical qubits would convert the Bravyi result from simulation to engineering and cut every superconducting resource estimate roughly tenfold; IBM's Kookaburra (2026) is the stated vehicle (vendor).

Frontier Third: a universal logical gate set below 10⁻⁵ with magic states supplied at rate. Quantinuum's 7 × 10⁻⁵ magic states and 2 × 10⁻⁴ non-Clifford gates, and Google's 10⁻⁴ cultivation, must improve one to three orders and run as a pipeline, not a hero shot.

Frontier Fourth: one complete algorithm, run fault-tolerantly end to end, beating its own unencoded execution. The March 2026 Helios simulation of quantum magnetism cut effective error by about 30 per cent against unencoded circuits — the closest anyone has come; a clean win on a problem with a checkable answer would end the era in which fault tolerance is demonstrated only in fragments.

Established Fifth, the cheap one: operate an error-corrected processor underground. An identical chip run in a surface lab and a shielded or underground site would partition the burst-error budget between cosmic rays and local radioactivity in weeks; the synchronous-detection experiments of 2025 started, but no below-threshold processor has done the A/B test. It remains unscheduled.

Established The best-audited target machine: factoring RSA-2048 in under a week with fewer than a million noisy qubits. Gidney's 2025 estimate (arXiv:2505.15917) assumes 0.1 per cent physical gate error, 1-µs surface-code cycles and 10-µs control reaction, and reaches under one million qubits via approximate residue arithmetic, yoked surface codes for idle storage and magic-state cultivation. The 2019 Gidney–Ekerå baseline needed 20 million qubits for 8 hours: a twentyfold qubit reduction in six years from algorithms and codes alone, with no hardware improvement assumed.

Established The chemistry benchmark is the same order of machine. Simulating the FeMoco nitrogenase cofactor — the standard proxy for industrially useful quantum chemistry — takes about four million physical qubits for under four days of runtime at the same 0.1 per cent error and 1-µs cycle assumptions (Lee and colleagues, PRX Quantum 2021, via tensor hypercontraction). Reiher and colleagues' 2017 PNAS estimate opened this literature with far larger gate counts; the direction of travel has been steadily downward.

Established Those estimates imply roughly a thousand physical qubits per logical qubit. At 0.1 per cent physical error, the assumed logical error budgets require surface-code distances near 25 to 31, and a distance-27 patch is about 1,460 physical qubits — which is why a hundred-logical-qubit machine is a hundred-thousand-qubit machine, and why the qLDPC tenfold cut matters so much.

Frontier No single chip gets there; modularity is now the announced architecture everywhere. IBM's roadmap runs through long-range c-couplers (Loon, 2025), qLDPC memory with a logical processing unit (Kookaburra, 2026) and module-to-module entanglement via universal adapters (Cockatoo, 2027) toward Starling's 200 logical qubits and 100 million gates in 2029 (vendor); ion and atom platforms lean on photonic interconnects with their own loss budgets.

Frontier The classical half of the machine is a supercomputer with a deadline. Continuous decoding at distance 25-plus across hundreds of logical qubits, with 10-µs feedback into control electronics, demands decoder ASICs, deterministic networking and a calibration system that tracks drift across a million channels — engineering that resembles a particle-physics trigger farm more than a data centre.

Established Energy is a non-problem at this scale. A dilution-refrigerator plant plus controls for a million-qubit machine draws megawatts — large for a lab, invisible next to AI-training clusters; the interesting thermodynamic questions live at the interface with classical computing efficiency, covered in the adjacent brief on computing energy systems.

Adjacent work on this map bears on fault tolerance mostly through shared infrastructure and shared physics.

• Established Cryogenics and the superconducting industrial base. Dilution refrigerators, superconducting cabling and junction fabrication are the same supply chain examined in superconducting infrastructure; quantum computing is currently that supply chain's most demanding customer.
• Frontier Materials. Transmon qubits use aluminium and niobium at millikelvin, not high-temperature superconductors, but the condensed-matter toolbox — defect spectroscopy, interface control — overlaps the programme described in high-temperature superconductors.
• Frontier Computing thermodynamics. Quantum computation is reversible computing's extreme case, and the energy accounting of error correction connects to ultra-efficient computing energy systems.
• Speculative Fundamental physics. Error-correcting codes have become a working vocabulary in holographic approaches to quantum gravity, and error-corrected processors double as exquisitely sensitive particle detectors — the cosmic-ray burst literature is jointly a nuisance study and an instrument paper.

Established The institutional response that matters most is already codified: post-quantum cryptography. NIST finalised its first post-quantum standards (FIPS 203, 204 and 205) in August 2024; migration is insurance whose premium is rational under exactly the trajectory Gidney's falling estimates describe, and it converts the worst civilizational downside of this technology into a plumbing project.

Frontier There is no standard for the field's central claim. "Logical qubit" is unstandardised: counts of 12, 48, 94 and 200 in circulation mix error detection with correction, post-selected with continuous operation, and memory with computation. A benchmark regime with the authority of Top500 — specifying code distance, decoding mode, duration and acceptance rate — does not exist, and until it does, cross-vendor comparisons are marketing.

Established Vendor roadmaps now carry dated, checkable milestones, which is institutional progress in itself. IBM has named chips and years through Starling (2029, 200 logical qubits, 100 million gates, with a stated quantum-advantage claim for the end of 2026); Quantinuum has named Apollo (2029) (vendor claims, both). Public, falsifiable checkpoints let outsiders score the field annually — a discipline fusion and AI roadmaps largely lack.

Frontier Concentration and controls. Below-threshold capability currently lives in three or four firms and a handful of state programmes; export controls on dilution refrigerators and control electronics are tightening, and the verification problem — how a customer or rival audits a logical-error claim without the vendor's own decoder — is unsolved.

Established Harvest-now-decrypt-later is the live harm, and it operates today. Ciphertext recorded now is readable by whoever first fields a cryptographically relevant machine; the falling resource estimates (20 million qubits to under one million in six years) make long-lived secrets — state, medical, genomic — the correct planning horizon, independent of anyone's Q-day forecast.

Frontier First access will be concentrated and probably quiet. A state or cloud vendor reaching cryptanalytic capability has incentives not to announce it; governance proposals (disclosure norms, key-rotation mandates) exist mostly as white papers, and nothing binds a first mover.

Frontier Hype has measurable costs in both directions. Overclaiming misprices public research portfolios and invites a funding winter when checkpoints slip; underclaiming delays PQC migration. The honest communication problem — a technology that is simultaneously overhyped in application and underhyped in engineering progress — is unusually acute here.

Speculative Dual-use chemistry is the standard simulation dilemma, no worse. A machine that models nitrogenase also models energetic materials and agents; the marginal risk over classical simulation and experiment is real but modest, and controls will look like existing dual-use research governance rather than anything quantum-specific.

Frontier The defensible upside is chemistry and materials as calculable science. FeMoco-class simulations at four million qubits and four days would let catalysis, battery electrochemistry and enzyme mechanisms be interrogated numerically before synthesis — compressing, not replacing, the experimental loop that currently gates fertiliser, energy-storage and pharmaceutical progress.

Established The cryptographic transition is disruptive but precedented. Civilization has re-plumbed its security infrastructure before (DES to AES, SHA-1 retirement); with standards finalised in 2024 and a decade-scale warning clock running in public, an orderly migration is the expected path, and panic is priced only for laggards holding long-lived secrets.

Handwave "Trillion-dollar quantum economy by 2035" projections are assertion, not analysis. They extrapolate from consulting surveys, assume applications (optimisation, machine learning) for which no verified quantum advantage exists, and should be read as marketing for capital formation.

Frontier The deepest long-run effect may be epistemic. A fault-tolerant simulator is a new class of scientific instrument — the first tool that computes quantum many-body dynamics beyond any classical check — and fields from correlated-electron materials to lattice field theory would inherit a verification culture problem worth solving early.

These horizons track logical-error engineering — distances, decoders, gate sets — not qubit-count announcements.

• 10 yr: Frontier The pre-registered decade: IBM's Loon/Kookaburra/Cockatoo/Starling sequence and Quantinuum's Apollo either land by 2029–2031 or slip in public (vendor timelines). On measured trend lines — Λ near 2 sustained, qLDPC moving to hardware, cultivation improving — hundreds of logical qubits at 10⁻⁶ and a first verified logical-beats-physical algorithm are credible by the early 2030s; first commercially interesting chemistry runs late in the window if nothing floors.
• 25 yr: Speculative A cryptographically relevant machine (under a million physical qubits, sub-week runs, per Gidney's assumptions) is plausible within this window if Λ-scaling and modular integration both hold; failure to get there by mid-window would be strong evidence that correlated errors or decoder economics impose a wall the theorem does not.
• 50 yr: Speculative Fault-tolerant computation as cloud utility, with overheads cut a further order of magnitude by qLDPC-class codes or biased-noise hardware, and quantum simulation as routine as density functional theory is today — conditional on the 25-year milestone, not on new physics.
• 100 / 250+ yr: Handwave Projections at this range — room-temperature logical qubits, quantum computing woven through general infrastructure, simulation of arbitrary matter — are unconstrained by anything measured and are recorded here as speculation in circulation, not forecast.

• Depends on nothing on this map blocks it: fault tolerance is upstream of its neighbours, gated by its own internal engineering. Its nearest live couplings are the cryogenic industrial base surveyed in superconducting infrastructure and the energy accounting of ultra-efficient computing energy systems — enabling suppliers, not prerequisites.
• Requires (not on this map) six things nobody has yet produced: exponential suppression sustained from today's distance-7 codes to distance 25 and beyond without hitting a correlated-error floor; decoders that keep pace with a one-microsecond syndrome cycle at those distances for days; magic states below 10⁻⁶ infidelity manufactured at production rate rather than 8 per cent acceptance; Josephson-junction fabrication with acceptable yield at million-qubit volume; cryogenic control wiring and multiplexing beyond a hundred thousand channels per machine; and an institutional benchmark that makes "logical qubit" claims comparable across vendors.
• Enables verified quantum simulation for catalysis, batteries and enzyme chemistry at the FeMoco scale; cryptanalysis of RSA-class systems (and therefore the completed post-quantum migration); and error-corrected quantum sensing and simulation instruments science does not yet have names for.
• Adjacent to quantum gravity, where error-correcting codes supply working mathematics for holography, and to high-temperature superconductors, which shares its condensed-matter measurement culture but not its materials.

Handwave “Willow did in five minutes what would take a supercomputer ten septillion years — quantum computers have arrived.” The 10²⁵-year figure attaches to random circuit sampling, a benchmark with no known application, and classical-simulation costs for such benchmarks have repeatedly collapsed under better algorithms. The substantive Willow result is the below-threshold error correction; the septillion is marketing (vendor framing).

Handwave “Quantum computers try all answers at once, so they will solve optimisation and NP-hard problems.” Complexity theory's standard position is that quantum computers are not expected to solve NP-hard problems efficiently; the known exponential advantages are structured (factoring, quantum simulation), and Grover-type speedups are quadratic — usually erased by error-correction overhead at realistic clock rates.

Frontier “Q-day by 2030.” The gap is four orders of magnitude: 105 physical qubits below threshold today against roughly a million required under Gidney's assumptions. What the claim gets right is direction and speed — the required-resource estimate fell twentyfold between 2019 and 2025, and vendors have staked 2029 for first fault-tolerant machines — so prudent cryptographic planning treats the 2030s as live without treating 2030 as a deadline.

Handwave “Error correction is now solved; the rest is manufacturing.” What exists is below-threshold memory on one chip and small universal gate sets elsewhere. Unsolved: sustaining suppression over ten more distance steps, decoding at scale in real time, magic-state supply two to four orders better, correlated-error floors, and qLDPC logic. Each is research, not procurement.

Frontier “Company X has N logical qubits, so it leads.” Logical-qubit counts are not comparable across platforms: 48 atom-array logical qubits ran with post-selection and a 0.1 cross-entropy score; 94 ion logical qubits held a GHZ state at 95 per cent fidelity; one superconducting logical qubit ran continuously corrected below threshold. Distance, decoding mode, duration and acceptance rate are the comparison; the count alone is noise.

Handwave “Cat qubits (or topological qubits) eliminate the need for error correction.” Ocelot's noise bias still required a repetition code and still showed 1.65 per cent per-cycle logical error; Microsoft's 2025 topological-qubit announcements remain contested measurements with no error-corrected demonstration behind them. Hardware can cheapen the code; nothing measured eliminates it.

Handwave “Quantum computers will supercharge AI training.” No quantum algorithm with verified advantage for training modern machine-learning models exists; data-loading costs alone erase most proposed speedups, and every serious resource estimate targets chemistry or cryptanalysis, not AI.

Established What the record actually supports believing: the threshold theorem works on real hardware; suppression factors near 2 per distance step have been measured; resource estimates for useful work have fallen, not risen, under scrutiny; and the field now fails or succeeds against published numbers on published dates. That is more than most frontier technologies can say, and it is enough — without a single additional promise.