The Quantum Error Curve

Qubit counts are the wrong headline. The curve that matters is logical error against code size. It has just started bending the right way, and most of its descent still lies ahead.

Research through September 2026

Figure 1

Bigger codes finally make fewer errors

Below threshold, every increase in code distance divides the logical error by Λ. The hardware cost rises quadratically; the reliability improves exponentially. That bargain is the whole case for fault tolerance.

Target:

At Λ 2.14

Reaching 10⁻⁹ needs distance ≈ 45, or about 4,049 physical qubits for one bare logical patch, before routing and magic-state factories. Selected measurement: Distance seven, 101 qubits: 0.143% per cycle, Λ = 2.14 ± 0.02, and a logical lifetime 2.4 times the best physical qubit in the patch.

Figure 1: Measured logical error per error-correction cycle for surface-code memories (points), and extrapolations from the distance-7 result (dashed) at the measured suppression factor and at hypothetical higher ones. Choose a Λ and a target to see the distance and bare-patch size required. The extrapolations assume Λ stays constant far beyond the measured range, which correlated errors may prevent, and they apply a memory curve to targets that ultimately require logical gates. They are editorial calculations, not forecasts.

Λ 1.04 → 2.14From a larger code barely beating a smaller one in 2023, to each step of two in distance more than halving the error in 2024.

~10⁶-10⁹×The further fall needed, from 1.43×10⁻³ per cycle to the 10⁻⁹-10⁻¹² that hundred-million- to trillion-operation algorithms require.

Λ is the leverAt the measured Λ, a 10⁻⁹ patch needs distance 45 and ~4,000 qubits. At Λ = 4 it needs distance 29 and ~1,700.

What the record shows

The threshold transition is now experimentally real. On superconducting hardware, a distance-7 surface code reached 0.143% error per cycle with Λ = 2.14, and a logical lifetime 2.4 times its best physical qubit. Ions and atoms have demonstrated complementary pieces.
A logical memory is not a computer. The best result supports roughly 700 cycles per logical failure; useful algorithms need 10⁸ to 10¹² reliable operations.
Software moved the target faster than hardware. The estimated cost of breaking RSA-2048 fell from about 20 million physical qubits to under a million in six years, under the same physical assumptions.
The bottleneck has migrated from physics to systems: correlated errors, leakage, decoding, wiring, magic states, and whether all of it holds as the machine grows.

This report separates measured results, resource estimates, editorial calculations, and company targets. A physical qubit, a logical qubit, an algorithmic qubit, and a detection-only encoded qubit are different objects and are not placed on one axis.

Figure 2: The architecture trade space

No architecture wins every axis

Fidelity, speed, connectivity, and manufacturability trade against one another. Each platform moves difficulty to a different layer of the stack.

The strongest direct evidence that larger codes suppress error, on a platform fast enough to run billions of cycles in hours

Physical error: 99.9%-class two-qubit gates on selected devices
Speed: Tens of nanoseconds per gate; ~1 µs per QEC cycle
Connectivity: Local, planar; natural fit for the surface code
Logical evidence: Distance-7 memory below threshold: 0.143% per cycle, Λ = 2.14

Binding constraint: Millikelvin refrigeration, wiring, leakage, and rare correlated events such as radiation bursts that hit many qubits at once.

Physical qubits per logical qubit cannot be compared across platforms by one constant. A fast platform can afford a large code; a slow one needs a compact code; an experiment that discards failed runs is measuring something else again.

Part I: First principles

A qubit is a resource. A reliable operation is the product.

A thousand noisy qubits can be worth less than a hundred clean ones. At an error of 10⁻³, a thousand operations already make failure likely, and a billion make it certain.

Useful computation≈reliable logical operations delivered at a fixed success probability÷physical qubits × time × control resources

01

Build the physical qubit

A controllable two-level system: a superconducting circuit, an ion, an atom, a photon, or a spin. Its errors are relaxation, dephasing, leakage out of the two levels, and loss.

Measure: T₁ and T₂ against gate time, leakage, loss
Failure boundary: Materials defects, two-level systems in dielectrics, and noise that is correlated rather than random

What the record shows at this step

Reported evidence: 99.9%-class two-qubit operations have been shown on several platforms, and one small device held above 99.9% for 24 days without recalibration.
Where it is moving: Better materials, noise bias engineered into the qubit, and intrinsic protection.
Principal risk: Materials defects, two-level systems in dielectrics, and noise that is correlated rather than random

Evidence statements describe published experiments and resource estimates. “Where it is moving” is an editorial reading of direction, not an announced capability.

02

Operate it at scale

Fault tolerance needs record-level performance on every qubit, simultaneously, for hours: not the best pair on the chip, but the worst tail across thousands.

Measure: Median and tail two-qubit error under simultaneous operation, calibration uptime
Failure boundary: Crosstalk and calibration parameters that grow faster than the number of qubits

What the record shows at this step

Reported evidence: Published records usually come from isolated one- and two-qubit devices; errors measured across whole arrays running simultaneously are typically higher.
Where it is moving: Automated continuous calibration and fabrication uniform enough that bad qubits are rare.
Principal risk: Crosstalk and calibration parameters that grow faster than the number of qubits

Evidence statements describe published experiments and resource estimates. “Where it is moving” is an editorial reading of direction, not an announced capability.

03

Measure the syndromes

Error correction repeatedly measures parity checks that reveal where errors occurred without revealing the encoded state. Every cycle needs good measurement and reset, not only good gates.

Measure: Readout error, reset error, cycle time
Failure boundary: Leakage that silently corrupts every check a qubit touches

What the record shows at this step

Reported evidence: Superconducting cycles run in about a microsecond; ion and atom cycles take far longer, which sets how deep a computation can be in a day.
Where it is moving: Leakage-removal circuits and codes tailored to each platform's dominant errors.
Principal risk: Leakage that silently corrupts every check a qubit touches

Evidence statements describe published experiments and resource estimates. “Where it is moving” is an editorial reading of direction, not an announced capability.

04

Decode in real time

A classical computer must infer the most likely errors from the syndrome stream as fast as it arrives, or a backlog builds and the machine stalls.

Measure: Decoder latency and throughput against cycle time
Failure boundary: A real-time supercomputing problem at millions of qubits

What the record shows at this step

Reported evidence: Willow ran a real-time decoder at distance five with about 63 µs average latency, over runs up to a million cycles.
Where it is moving: Parallel, windowed decoders on dedicated hardware.
Principal risk: A real-time supercomputing problem at millions of qubits

Evidence statements describe published experiments and resource estimates. “Where it is moving” is an editorial reading of direction, not an announced capability.

05

Compute with logical qubits

A memory that survives is not a computer. Logical gates between patches, and non-Clifford operations fed by magic states, must also be fault tolerant.

Measure: Logical gate error, magic-state output error and rate
Failure boundary: Magic-state factories that consume much of the machine

What the record shows at this step

Reported evidence: Neutral atoms executed 48 logical CCZ gates using error detection; newer cultivation protocols cut projected factory cost substantially.
Where it is moving: Lattice surgery, transversal gates where codes allow them, and cheaper magic states.
Principal risk: Magic-state factories that consume much of the machine

Evidence statements describe published experiments and resource estimates. “Where it is moving” is an editorial reading of direction, not an announced capability.

06

Compile the algorithm

An algorithm becomes a count of logical qubits, operations, and non-Clifford gates. Better arithmetic and representations shrink all three.

Measure: Logical qubits, Toffoli or T count, runtime
Failure boundary: Classical algorithms improve and move the crossover

What the record shows at this step

Reported evidence: RSA-2048 estimates fell from ~20 million physical qubits to under a million under the same hardware assumptions.
Where it is moving: Problem-specific algorithms with large, provable advantages.
Principal risk: Classical algorithms improve and move the crossover

Evidence statements describe published experiments and resource estimates. “Where it is moving” is an editorial reading of direction, not an announced capability.

Physical errors do not need to reach zero. They need to sit far enough below threshold that each added layer of redundancy buys an exponential gain in reliability.

Figure 3 · The wall of depth

A computation is only as long as its error rate allows

With independent faults at probability p, a circuit of G operations finishes cleanly with probability of roughly e^−pG. The curve falls off a cliff at about 1/p operations, wherever that is. Error correction exists to move the cliff to the right.

Error per operation:

At p = 1.4×10⁻³, a 90% chance of finishing allows about 73 operations. That is enough to demonstrate error correction and nowhere near enough to run a useful algorithm.

Figure 3: An editorial toy model with independent, identical faults. Real algorithms have non-identical operations, structured faults, retries, and verification, and a logical memory error per cycle is not the same quantity as a logical gate error. The position of the cliff, not its exact shape, is the point.

How the bottleneck climbed the stack

Each era answered one question and exposed the next. None of the answers was a larger qubit count.

Coherence was the question

From the mid-1990s, the problem was whether a quantum state could be prepared, controlled, and read before it thermalised. Transmons and trapped atoms answered it.

Then gate fidelity

Until roughly 2014, two-qubit errors of several percent sat above the surface-code threshold. Pulse shaping, couplers, and materials brought them below 1%.

Then the crossover

Qubits at 99% still produced codes where bigger was worse. Crosstalk, leakage, and drift kept Λ below one until 2023, and clearly above it only in 2024.

Now systems scaling

The question is whether Λ holds while distance, logical count, gate set, cycle count, and machine size grow together, with decoders keeping pace.

What the logical experiments proved, and what they did not

Λ = 2.14Google’s distance-7 memory on 101 qubits: the central scaling signature of the threshold theorem, measured. A memory, not a universal logical computer.

48 logicalHarvard, MIT, and QuEra encoded qubits running 228 logical two-qubit gates and 48 CCZ gates, largely with error detection and discarded runs rather than continuous correction.

Below physicalQuantinuum and Microsoft logical errors under the corresponding physical errors, with repeated correction cycles, on small codes with some post-selection.

Ratios such as “48 logical from 280 physical” are not efficiency metrics unless the logical error, the allowed post-selection, the gate set, the number of correction rounds, and the circuit depth are all fixed. Without those, the comparison measures vocabulary, not capability.

The discovery chain

The theory arrived in the 1990s. The experiments needed nearly thirty years to reach the regime the theory described.

1995
Quantum error correction
Shor showed that a qubit could be protected by encoding it across nine, despite the no-cloning theorem. Steane and others generalised the idea.
1996-1999
The threshold theorem
Below a constant physical error rate, arbitrarily long computation is possible with manageable overhead. The field's entire engineering bet rests on this result.
1997-2012
The surface code
Kitaev's topological codes became a practical architecture with local checks and a threshold near 1%, the baseline for most resource estimates.
2014
Gates at threshold
Superconducting two-qubit gates reached 99.4%, the regime where surface-code fault tolerance first looked plausible.
2019
The 20-million-qubit estimate
Gidney and Ekerå costed RSA-2048 at about 20 million noisy qubits for eight hours, making the scale of a useful machine concrete.
2023
Bigger finally better
A distance-five surface code beat distance three, barely. Neutral atoms ran circuits on 48 encoded logical qubits.
2024
Below threshold
Willow's distance-seven memory reached 0.143% per cycle with Λ = 2.14, outliving its best physical qubit by 2.4 times.
2025-2026
The target moves
RSA-2048 estimates fell below a million qubits. Roadmaps began quoting logical qubits and operation counts rather than physical qubits.

Who is building what

The major programmes have begun describing goals in logical qubits and operation counts rather than physical qubits. Search the record, or filter by platform.

Google Quantum AIWillowTransmon surface codes with real-time decoding

Reported evidence: A 101-qubit distance-7 memory at 0.143% error per cycle, Λ = 2.14, and a logical lifetime 2.4 times its best physical qubit; real-time decoding at about 63 µs latency at distance 5.
Announced next step: Long-lived logical qubits, then logical gates between them.
Unresolved risk: Rare correlated events set an error floor in long repetition-code runs; memory is not computation.

IBM QuantumStarling roadmapHigh-rate qLDPC (bivariate bicycle) codes with long-range couplers

Reported evidence: Published codes that store 12 logical qubits in 288 physical qubits at thresholds comparable to the surface code, in simulation.
Announced next step: 200 logical qubits running 100 million gates in 2029 (announced target).
Unresolved risk: Non-local connectivity on a chip is unproven at scale; the code advantage must survive real circuit noise.

AWSOcelotBosonic cat qubits with strongly biased noise

Reported evidence: A 2025 chip combining cat qubits with a repetition code for the remaining error type.
Announced next step: Lower overhead by suppressing one error type in hardware.
Unresolved risk: Bias must be preserved through every gate for the saving to hold.

Quantinuum and MicrosoftH-series trapped ionsAll-to-all connectivity with compact colour and concatenated codes

Reported evidence: Logical error rates below physical, with repeated error-correction cycles, on small codes in 2024.
Announced next step: Larger logical registers with continuous correction.
Unresolved risk: Gate speed: deep circuits take weeks rather than hours, and optics grow with the machine.

Harvard, MIT, and QuEraReconfigurable atom arraysRydberg gates on atoms moved by optical tweezers

Reported evidence: Up to 48 encoded logical qubits, 228 logical two-qubit gates and 48 logical CCZ gates in 2023, largely using error detection and post-selection.
Announced next step: Hundreds of logical qubits and around a million operations (announced, ‘Megaquop’-class).
Unresolved risk: Post-selection hides the failure rate; continuous correction with atom replacement is only beginning.

Atom ComputingAC1000Large neutral-atom arrays with mid-circuit reloading

Reported evidence: Over 1,200 physical atoms advertised; a 2026 preprint reported repeated toric-code correction with atom replacement for up to 90 cycles.
Announced next step: Logical qubits with Microsoft’s qubit-virtualisation stack.
Unresolved risk: Vendor-quoted fidelities; repeated-correction results are a preprint.

PsiQuantumFusion-based photonicsMeasurement-based computing from small entangled photon states

Reported evidence: Silicon-photonic components made in a commercial foundry; the architecture tolerates substantial loss per fusion in theory.
Announced next step: Utility-scale facilities announced in Brisbane and Chicago.
Unresolved risk: No repeated logical demonstration yet comparable with matter-qubit results; loss compounds through every component.

Diraq, imec, and IntelSilicon spin qubitsQuantum dots made on 300 mm CMOS lines

Reported evidence: Two-qubit fidelities above 99% in foundry-fabricated devices.
Announced next step: Dense arrays with cryogenic control electronics.
Unresolved risk: Device variability and wiring millions of dots within a millikelvin heat budget.

Microsoft Station QMajorana parity readoutTopologically protected qubits from superconductor-semiconductor wires

Reported evidence: A 2025 single-shot parity measurement with ~1% assignment error and millisecond dwell times.
Announced next step: A measurement-only topological qubit.
Unresolved risk: The paper states the readout does not by itself distinguish topological modes from trivial states.

Algorithm and resource-estimation researchGidney, Ekerå, Beverland and othersCompiling algorithms down to physical qubits and seconds

Reported evidence: RSA-2048 fell from ~20 million physical qubits (2019) to under a million (2025) under the same physical assumptions.
Announced next step: Lower-cost chemistry and materials algorithms.
Unresolved risk: Improving classical algorithms can move the crossover after an estimate is published.

NISTPost-quantum cryptographyStandardised replacement public-key algorithms

Reported evidence: First post-quantum standards finalised in 2024.
Announced next step: Migration before a cryptographically relevant machine exists.
Unresolved risk: Migration of long-lived systems takes a decade regardless of the quantum timeline.

DARPAQuantum Benchmarking InitiativeIndependent verification of utility-scale claims

Reported evidence: A government programme evaluating whether proposed architectures can reach useful fault tolerance.
Announced next step: Separating credible utility-scale roadmaps from announcements.
Unresolved risk: Evaluation is not demonstration; the machines still have to be built.

Roadmap targets, such as IBM’s 200 logical qubits and 100 million gates in 2029, are company announcements, not results. Vendor-quoted fidelities are reproduced as stated.

Part II: What useful computation needs

Width, length, and the redundancy that carries them

Application cost is roughly logical qubits × logical operations × error-correction overhead. The third term is where Λ, codes, and factories live, and where the largest recent savings came from.

Figure 4 · Interactive model

How many physical qubits does an answer need?

Logical qubits set the width. Error-correction rounds set the length. The suppression factor Λ decides how much redundancy each logical qubit needs to survive the whole run.

Logical qubits6,000Error-correction rounds3.0×10¹⁰ · ~8.3 h at 1 µsSuppression factor Λ10.00Share for factories and routing50%

RSA-2048, 2019 assumptions26mphysical qubits

Distance 33 surface code, 2,177 qubits per logical patch

Logical error budget: 5.8×10⁻¹⁶ per qubit-round
Data and syndrome qubits: 13m
Factories and routing: 13m
Physical per logical: 4,354 : 1

The RSA scenario uses the assumptions of Gidney and Ekerå (2019): 10⁻³ physical error against a roughly 1% threshold, which is a Λ of about 10, and a 1 µs cycle. This model lands in the same tens-of-millions range as their ~20 million qubits. Drop Λ to the measured 2.14 and the same computation needs several times more.

Calculation & assumptions

Logical error per qubit-round starts from Willow’s measured 1.43×10⁻³ at d = 7 and falls by a factor of Λ for every increase of two in distance: p_L ≈ 1.43×10⁻³ · Λ^−(d−7)/2. The budget gives the whole run a 90% chance of success: p_L ≤ 0.105 ÷ (logical qubits × rounds). Each logical qubit is one rotated surface-code patch of 2d² − 1 physical qubits; factories and routing are added as a share of the total.

What this model cannot capture is the lever that mattered most recently. Gidney’s 2025 estimate cut RSA-2048 to under a million qubits under the same physical assumptions by redesigning the arithmetic and replacing much of magic-state distillation with cultivation, trading qubits for a longer runtime of under a week.

Figure 4: Illustrative structure, not a resource estimate. Workload widths and lengths are rounded from the published estimates cited below; real machines need logical gates, not only memories, and their error budgets are set per operation type.

20m → <1mPhysical qubits estimated for RSA-2048, 2019 to 2025. Approximate arithmetic, denser storage, and magic-state cultivation, not better hardware, made the difference; runtime rose from hours to under a week.

10⁵-10⁶+Physical qubits for practically significant chemistry and materials problems in full-stack studies, even when the science involves only hundreds of logical degrees of freedom.

No generic numberOptimisation has no credible machine size, because quantum advantage for NP-hard problems in general is not established. The absence is itself a finding.

What pushes the curve down next

01

Better physical operations

Because suppression compounds through the code, a modest fall in physical error removes thousands of qubits per logical qubit at useful targets.

Condition: improvements in the tail across whole arrays, not only record pairs

02

Higher-rate codes

qLDPC codes store many logical qubits in a shared block, with simulated overhead up to ten times lower than the surface code.

Condition: non-local connectivity that can be built and survives real noise

03

Cheaper magic states

Cultivation and native non-Clifford operations shrink the factories that dominate many estimates.

Condition: output fidelity and rate demonstrated in hardware, not only in simulation

04

Better algorithms

Chemistry estimates fell by orders of magnitude through qubitisation and better Hamiltonian representations; the RSA estimate fell twentyfold.

Condition: advantages large enough to survive improving classical methods

Part III: Where progress is stuck

The remaining constraints appear only after the easy errors are gone

Suppressing ordinary errors exposes rarer ones. Most of what now limits the curve is invisible in a one- or two-qubit benchmark.

Correlated errors

Surface-code scaling assumes faults are local and independent. Radiation strikes and common-mode disturbances hit many qubits at once; long repetition-code runs already show a floor from rare events.

Leakage

A qubit that leaves its two computational levels is not a simple bit flip. It persists and contaminates its neighbours' checks, and removing it costs extra circuit depth.

Real-time decoding

At microsecond cycles across millions of qubits, the syndrome stream is enormous. Decoders must keep up indefinitely and answer before latency-sensitive logical operations.

Wiring and heat

A dilution refrigerator has microwatts of cooling at its coldest stage. Tens of thousands of control lines, or cryogenic electronics to replace them, strain that budget.

Magic-state throughput

Non-Clifford gates need purified ancilla states. Factories producing them can dominate the machine's area and set its speed.

The moving classical baseline

Better tensor networks, Monte Carlo methods, and GPUs keep reclaiming claimed advantages. A polynomial quantum speedup with large constants is especially exposed.

An optimistic view, with conditions

Quantum computing has become an engineering problem, mostly

Quantum computers do not need perfect qubits. They need an operating regime in which every added layer of redundancy makes the logical machine more reliable. That regime now exists in hardware. Lower physical error, better codes, faster decoders, cheaper magic states, and better algorithms multiply rather than add, and several of them are improving at once.

Now to 2029

The scaling test

Λ held or rising at distance 9 and 11 without a new floor; a logical two-qubit gate with lower error than the physical gate; decoders that keep pace for more than 10⁵ cycles.

Early 2030s

The megaquop machine

Tens to hundreds of logical qubits running about a million reliable operations, with continuous magic-state production: the first scientifically useful fault-tolerant regime.

Longer horizon

Beating the best classical answer

10⁸ to 10⁹ reliable operations on a chemistry or materials problem, at a total cost below the best classical alternative. That is the only finish line that pays for the machine.

View the annual scorecard

Metric	Record through September 2026	What a good curve looks like
Suppression factor Λ	2.14 ± 0.02 at d = 3-7 (superconducting)	≥ 3-4 sustained at d = 9 and 11
Logical memory error	1.43×10⁻³ per cycle at d = 7	Falling by orders of magnitude, not percentages
Logical gate error	Early; mostly detection-based	Below the physical two-qubit error, then 10⁻⁵ and beyond
Simultaneous two-qubit error	~1.4-5×10⁻³ across leading arrays	≤ 5×10⁻⁴ across more than 500 qubits
Reliable logical operations	~10²-10⁴ in demonstrations	10⁶ (megaquop), then 10⁸-10⁹
Real-time decoding	~63 µs latency at d = 5	Sustained with margin at d ≥ 9
Correlated burst rate	Visible as a floor in long repetition-code runs	Falling as systems grow

Baselines are reported results. Targets are editorial intermediate goals derived from published resource estimates, not company guidance.

There is more than one finish line

Below thresholdA larger code gives a lower logical error. Demonstrated for memory.
Beyond breakevenA logical gate outperforms the physical gate it is built from.
Universal and continuousClifford and non-Clifford logical gates run with ongoing correction and no discarded runs.
MegaquopAbout a million reliable logical operations in one computation.
GigaquopAbout a billion: the scale of the chemistry and cryptanalysis estimates.
UsefulAn answer cheaper, faster, or otherwise unobtainable against the best classical method.

Sources, method, and boundaries

Measured points in Figure 1 are reported results. Extrapolations in Figures 1 and 4 and the reliability curves in Figure 3 are editorial calculations with simplified independent-fault models, stated so their assumptions are visible; they are not resource estimates. Resource estimates quoted in the text are from the cited literature and depend on their stated physical error, cycle time, connectivity, and code. Company roadmaps are targets, not evidence.

Physical qubit: An individual two-level quantum system used to carry information.
Logical qubit: Information encoded across many physical qubits so that errors can be detected and corrected without measuring it.
Code distance: Roughly, the smallest number of physical faults that can cause an undetected logical error. A surface-code patch uses about 2d² physical qubits.
Suppression factor Λ: The ratio of logical error at distance d to that at d + 2. Above one, larger codes help.

Nature: Quantum error correction below the surface code threshold (2024)Nature: Suppressing quantum errors by scaling a surface code logical qubit (2023)Nature: Logical quantum processor based on reconfigurable atom arrays (2024)arXiv: Logical qubits with better-than-physical error rates (2024)Nature: Superconducting circuits at the threshold for fault tolerance (2014)Nature: State preservation by repetitive error detection (2015)Physical Review A: Surface codes, towards practical computation (2012)arXiv: Fault-tolerant computation with constant error rate Quantum: Factoring RSA integers with 20 million noisy qubits (2021)arXiv: Factoring RSA integers with less than a million noisy qubits (2025)arXiv: Assessing requirements to scale to practical quantum advantage (2022)PNAS: Elucidating reaction mechanisms on quantum computers (2017)Nature: High-threshold, low-overhead quantum memory (2024)Nature: Interferometric single-shot parity measurement (2025)IBM: Roadmap to large-scale fault-tolerant computing arXiv: Validating quantum computers using randomized circuits

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