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Wednesday, October 28, 2026

The Error-Correction Threshold: When More Qubits Start to Help

Adding qubits to an error-correcting code only helps if each part is already good enough. “Below threshold” means a machine has crossed that line. It does not mean the work is done.

A machine that seems possible

Here is a plan that sounds easy on paper. Every part of a quantum computer makes mistakes, so store information redundantly, keep checking it, and repair the slips. Need more protection? Use a bigger code.

The catch is that every added part is another place for something to go wrong. The error-correctionTechniques that combine many shaky physical qubits into fewer reliable ones, so a long calculation stays correct. threshold is the point where adding parts stops hurting and starts helping.

The simplest race

Take the smallest code: one bit stored as three copies and read by majority vote, the toy from our decoderThe classical algorithm that reads the pattern of error-correction alarms and bets on which errors most likely caused them. Note. Each copy flips with probability p. The vote fails only when two or more copies flip, which happens with probability 3p² − 2p³.

At p = 10 percent, one copy fails 10 percent of the time; the vote fails only 2.8 percent. Five copies, voting the same way, fail about 0.9 percent of the time. Redundancy wins, and more redundancy wins more. At p = 60 percent the picture reverses: three copies fail about 65 percent of the time, worse than one alone. For this toy the crossover sits at exactly 50 percent. Below it, bigger is better. Above it, bigger is worse.

That crossover is the threshold. The threshold theorem says real fault-tolerantThe milestone where a quantum computer can run long calculations correctly despite ongoing errors. It is the field’s holy grail. schemes behave the same way: if every component errs less often than some threshold rate, making the code larger can push the logical error rateHow often the protected information itself fails, as opposed to any single physical qubit. as low as you like, in principle.3

Two hand-drawn plots titled Below Threshold. Panel A, the crossover: logical error rate against physical error rate, both on log scales, with three rising curves for codes of distance 3, 5 and 7. All three meet at one red point marked threshold. To the left of it, the distance 7 curve is lowest, labeled bigger code helps; to the right, the order flips and the distance 7 curve is highest, labeled bigger code hurts. Panel B, how far below: logical error per round against code distance 3, 5 and 7, with a shallow line labeled divided by 2.1 per step and a steep line labeled divided by 12 per step.
Figure 1: The threshold is a crossover. A: below it, each bigger code makes fewer logical errors; above it, each bigger code makes more. B: below threshold, the gain per step of code distance depends on how far below the hardware sits. The two lines come from two experiments with different codes and platforms, so they show steepness, not a ranking.

Why the real number is small

Quantum codes are far less forgiving. The checks are built from the same noisy operations they police, and errors come in more than one kind. With perfect checks, the leading surface codeAn error-correcting code that spreads one protected qubit over a square grid of physical qubits. tolerates about 11 percent error per qubit.1 Count faulty gates and measurements too, and its threshold drops to about 0.57 percent per operation, often rounded to “about 1 percent.”2

So there is no single threshold. Each belongs to a particular code, noise model and decoder, and rests on assumptions, such as independent errors, that real hardware only approximates.

How far below matters

Just below the threshold, a bigger code barely helps; well below it, it helps a lot. Experiments summarize this with one factor: how many times the logical error rate falls each time the code distanceThe fewest errors that can fool the code; a larger code has a larger distance. grows by two. Roughly, that factor is the threshold divided by the physical error rate.2

In one superconducting experiment, the error per cycle fell by a factor of about 2.1 per step, from distance 3 to 7.4 A 2026 trapped-ion preprint, not yet peer reviewed, reports about 12 per step for a different code, the color code, at distances 3, 5 and 7.5 Its uncertainty spans roughly 10 to 15, and other decoders give less.5 With different codes and platforms, these numbers measure headroom, not rank.

Headroom compounds. At about 2 per step, the superconducting team projected that one error per million cycles needs distance 27, 1,457 physical qubits per logical qubitA reliable ‘qubit’ built by bundling many error-prone physical qubits together with error correction. These are the units that actually matter for useful computing..4

A line, not a finish line

Below threshold is necessary, not sufficient. In that experiment, rare correlated bursts of errors, about once an hour, set a floor near one error in ten billion for a simpler code.4 Such correlated errors break the independence the theorem leans on. And these results protect stored information and, in the preprint, a limited set of operations, not full computations. The preprint itself notes that preparing magic statesSpecial resource states that a fault-tolerant computer consumes to perform the operations it cannot do directly. below threshold, and decoding in real time, remain to be shown.5

Why it matters

The threshold turns error correction from hope into arithmetic: below it, every step up in code size buys compounding protection. When you read “below threshold,” ask three questions. Which threshold: what code, noise model and decoder? How far below: what factor per step? And below threshold for what: memory, or computation?

Sources

  1. E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, “Topological quantum memory,” Journal of Mathematical Physics 43, 4452 (2002). arXiv:quant-ph/0110143. (Threshold as a phase transition; about 11% per qubit with perfect syndrome measurements.) ↩
  2. A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, “Surface codes: Towards practical large-scale quantum computation,” Physical Review A 86, 032324 (2012). arXiv:1208.0928. (Crossover with distance; 0.57% per step in a circuit-level model, versus “about 1%” in earlier estimates; scaling exponent (d+1)/2.) ↩
  3. D. Aharonov and M. Ben-Or, “Fault-tolerant quantum computation with constant error rate,” arXiv:quant-ph/9906129; E. Knill, R. Laflamme, and W. H. Zurek, “Resilient quantum computation: error models and thresholds,” arXiv:quant-ph/9702058. (The threshold theorem.) ↩
  4. Google Quantum AI and Collaborators, “Quantum error correction below the surface code threshold,” Nature 638, 920 (2025). arXiv:2408.13687. (Factor 2.14 per step, distance 3 to 7; distance 27 and 1,457 qubits projected for 10^-6; correlated bursts about hourly, floor near 10^-10 in repetition codes.) ↩
  5. A. Lavasani, A. P. Reed, J. M. Dreiling, et al. (Quantinuum), “Quantum error correction well below the color code threshold,” preprint, arXiv:2610.24077 (October 16, 2026). (Color codes, distance 3 to 7; factor 12 (+3/−2) per step with one decoder, 7 to 11 with others; magic states and real-time decoding still to be shown.) ↩