Noise and error correction
Lesson 18 of 23 in the course “Quantum”. Free, with animations and a quiz.
Why there’s no way around error correction
Today’s best physical qubits make roughly one error per thousand two-qubit operations. Sounds fine, but Shor’s algorithm needs billions of operations in a row, and with that error rate the calculation falls apart after a couple of thousand steps. You need an error rate around 1 in a billion, or even lower. Making hardware a million times better isn’t realistic, so errors are fixed with a trick: many noisy physical qubits are combined into one reliable qubit, called a logical qubit.
The idea: majority vote
Ordinary communication has an old trick: send the bit three times, 000 instead of 0. If one flips and 010 arrives, the majority gives you the right answer. The quantum world has two problems: a qubit can’t be copied (remember no-cloning?), and measuring it destroys the superposition. The way out is clever: the state is “spread” across several qubits with entanglement, and you never ask “what’s inside?”, only “are the neighbours equal?” The answers are called the syndrome. It shows where the error is without touching the information itself.
The surface code: a chessboard
The most studied protection scheme is the surface code. Qubits sit in a grid: some hold data, others sit between them and repeatedly check groups of four neighbours. The grid size is called the distance d: the bigger it is, the more errors the code survives. A code with d = 7 fixes any 3 errors at once. On Willow such a logical qubit takes 101 physical ones: 49 data qubits, 48 check qubits and 4 more helpers. The big plus: each qubit only needs its nearest neighbours, which is perfect for a flat chip.
The threshold: when bigger means better
Here’s the catch: check qubits make errors too. If the hardware is too noisy, a bigger code just adds errors. But below a certain level, the threshold (about 1% for the surface code), everything flips: each step up in code size cuts errors. In December 2024 Google showed this on Willow: going from d = 3 to 5 to 7, the logical error dropped about 2.14× each time, down to 0.143% per cycle. The logical qubit lived 291 microseconds, 2.4× longer than the best physical one. It’s the first convincing “below threshold” on superconductors.
IBM: a thriftier code
The surface code’s downside is its appetite: one reliable qubit eats hundreds of physical ones. IBM bet on a different family, qLDPC codes. Their star is the gross code: 288 physical qubits hold 12 logical ones, while the surface code would need nearly 3,000 for the same job. The paper made the cover of Nature in March 2024. The price of the savings: each qubit needs links not just to neighbours but also “long-range” wires across the chip. That’s exactly what IBM is testing on its experimental Loon chip (2025), and it promises the Starling machine with 200 logical qubits by 2029.
Magic states
The last piece of the puzzle. Codes protect “simple” gates well, but those alone aren’t enough: circuits made only of such gates are easy to simulate on an ordinary computer, so there’d be no advantage. You need at least one “non-simple” gate, usually called T. It’s made with magic states: you prepare lots of noisy ones, then “distil” them, keeping the cleanest, like water run through several filters. Such “factories” may take up a noticeable share of a future machine, which is one reason Shor estimates need so many qubits.
Check yourself
- Why can’t you protect a qubit by just copying it three times?
Correct answer: A quantum state can’t be copied, and measuring it directly destroys it. No-cloning and measurement collapse are the two reasons you need a syndrome: “are the neighbours equal?” checks instead of peeking. - What does “operating below threshold” mean?
Correct answer: Making the code bigger lowers the logical qubit’s errors. Below threshold, growing the code works for you: on Willow each step d = 3 → 5 → 7 cut the error about 2.14×. - Why does IBM use qLDPC codes instead of the surface code?
Correct answer: They store logical qubits about 10× more cheaply. Gross code: 288 physical for 12 logical versus nearly 3,000 for the surface code. The cost is long-range links on the chip.
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