The problem, in plain words
A qubit is fragile. Stray fields, heat, control errors and, in superconducting chips, the occasional cosmic ray all push its state away from where it should be. The obvious classical fix, keeping several copies and taking a majority vote, is closed off by the no-cloning theorem: an unknown quantum state cannot be copied. Measuring a qubit to check it destroys the superposition you were trying to protect.
Peter Shor's 1995 nine-qubit code showed a way round this. Spread one logical qubit across nine physical qubits, then measure only the relationships between them, the so-called syndromes, rather than the data itself. The syndrome tells you which error occurred without telling you what the state is. Andrew Steane's seven-qubit code followed the same year. Both are small and neither is practical at scale, but they settled the question of principle.
The threshold theorem and why it reframes everything
In the late 1990s several groups, including Aharonov and Ben-Or, Kitaev, and Knill, Laflamme and Zurek, proved that if the physical error rate per operation is below a certain threshold, the logical error rate can be driven arbitrarily low by making the code bigger, at a cost that grows only polynomially in the size of the computation. The threshold value depends on the code and on the assumed noise; for the surface code under the usual circuit-level noise model it is commonly quoted at around one percent per gate.
The practical consequence is that the interesting figure of merit is not how many qubits a machine has but how far below threshold its operations sit. A processor with a thousand qubits above threshold cannot be made reliable by adding more. A processor with a hundred qubits comfortably below it can, in principle, be scaled. This is why the reporting on this site treats a demonstration of scaling with code size as more significant than a new record qubit count.
The surface code
The surface code descends from Kitaev's toric code of 1997. Qubits sit on a two-dimensional lattice; half of them hold data and half measure parity checks on their four neighbours. Everything is nearest-neighbour, which suits superconducting chips, and the threshold is high compared with most alternatives. The price is overhead. A code of distance d, which can correct up to (d−1)/2 errors, needs roughly 2d² physical qubits for one logical qubit. At the distances people expect to need for long computations, that is hundreds to a thousand or more physical qubits per logical qubit.
Google Quantum AI published a paper in Nature in 2023 reporting that a distance-5 surface code on their Sycamore-class processor produced a lower logical error rate than a distance-3 code. The margin was small, and the paper was candid about it, but the direction was right for the first time in a superconducting device. A follow-up paper in Nature in 2024, on the processor they call Willow, reported distance-3, -5 and -7 codes with the logical error falling by a factor a little above two each time the distance rose by two. That is what "below threshold" means operationally, and it is the result on this sheet we would weight most heavily.
Codes that need fewer qubits
The surface code's overhead has pushed interest towards quantum low-density parity-check codes, which pack several logical qubits into one block. IBM described a family of these, the bivariate bicycle codes, in Nature in 2024; the example they call the gross code encodes twelve logical qubits in 144 data qubits with distance twelve, plus a matching set of check qubits. That is a large saving on the surface code at similar distance.
The trade-off is connectivity. These codes require each check qubit to talk to qubits that are not its physical neighbours. On a flat superconducting chip that means additional long-range couplers, which are hard to fabricate and add noise. Neutral-atom and trapped-ion machines, which can physically move qubits or address any pair, are more naturally suited to such codes, and that is a large part of why the platform question on sheet 02 is not settled.
Logical qubits on atoms and ions
The Harvard, MIT and QuEra collaboration reported in Nature in late 2023 a reconfigurable neutral-atom processor running circuits on up to 48 logical qubits, using colour codes and other small codes with atoms shuttled by optical tweezers. Quantinuum and Microsoft reported in 2024 logical qubits on the trapped-ion H2 machine whose error rates were below those of the underlying physical qubits. Both results are real progress; both also rely, to differing degrees, on error detection with post-selection, discarding runs where an error was flagged rather than correcting it. Post-selection is legitimate, but it does not scale in the same way, because the fraction of surviving runs collapses as circuits lengthen.
What is still open
Three things, chiefly. First, decoding. The syndrome measurements must be turned into corrections in real time, faster than errors accumulate, and the decoders that perform best are computationally heavy. Second, the gates that make a computer universal. The surface code handles Clifford operations cheaply, but non-Clifford gates such as the T gate require magic state distillation, whose overhead has not been demonstrated at scale by anyone. Third, correlated errors. A single high-energy particle striking a superconducting chip can flip many qubits at once, which no local code can fix; mitigations exist but the problem is documented and unsolved.
When reading a new error-correction result, we would ask: correction or detection? What distance, and did the error fall as distance rose? Were results post-selected, and what fraction survived? Was the decoder running in real time? A paper that answers all four is rare, and rarity is exactly the point.