QEC lab · learn 01/15 · the surface code

Catch the errors without looking at the data.

A surface code spreads one logical qubit over a grid of physical ones and measures only parities of neighbours. Bit flips light up the parity checks around them; a decoder pairs the lit checks along the shortest paths and flips those qubits back. If the pairing guesses wrong in a way that crosses the whole grid, the logical qubit is lost. Every shot below was sampled and decoded by PyMatching in the tested Python (qll/circuits/surface_code_capacity.py); the page re-checks each one.

the lattice · data qubits (dots), Z checks (amber, they catch bit flips), X checks (violet, idle in this model)

1 · errorsbit flips hit random qubits

2 · syndromechecks with odd parity light up

3 · matchingpair the lit checks, shortest total path

4 · correctionflip the qubits on those paths

controls

code distance

bit-flip probability

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physical qubits

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errors this shot

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lit checks

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logical errors, these 16 shots

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logical error rate (20 000 shots)

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page's own check

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Corrects any ⌊(d−1)/2⌋ errors: 1 for d = 3, 2 for d = 5, 3 for d = 7. More errors may still be caught, or may form a string the decoder closes the wrong way, which is a logical error.

logical error rate vs physical bit-flip rate · below the threshold, bigger codes win

Code capacity: independent bit flips, perfect parity measurements, minimum-weight perfect matching. The curves cross near 10 %, the threshold (about 10.3 % for very large codes; small planar codes cross a little lower). With noisy measurements the threshold drops to about 1 % (learn 01/15, qll/circuits/decoders.py), which is why the hardware race is to get physical errors well below it: Google's 2024 processors did, and each step from d = 3 to 5 to 7 halved the logical error.