Surface Code: Syndrome Extraction & Threshold

Code distance

Noise

Physical error rate p 0.08

Auto-run rounds

This round

Flipped qubits

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Syndrome defects

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Result

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Trials run

Logical fail rate

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Physical qubits

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Threshold scan

Runs many quick trials at d=3,5,7 across a range of p and plots the logical failure rate. Where the curves cross approximates the code's threshold p_th.

d=3 d=5 d=7

How it works

Data qubits sit on a d×d grid. Each 2×2 block of neighbours is one stabilizer — a parity check that reports whether an odd number of its four qubits have flipped. Independently, each data qubit flips with probability p (the bit-flip channel).

The pattern of triggered stabilizers is the syndrome — it pinpoints the boundary of the error without ever revealing the qubits' actual quantum state. A decoder pairs up triggered stabilizers (here, by nearest-neighbour matching, a simplified stand-in for minimum-weight perfect matching) and applies a correction chain between each pair.

Correction succeeds if it exactly undoes the error. It fails silently when error + correction together form a chain that runs from one side of the lattice to the other without ever triggering a stabilizer — an undetectable logical error. Larger d gives the decoder more room to distinguish nearby errors, so the logical failure rate falls faster below the threshold p_th and rises faster above it — the crossing point in the scan.

Flipped qubit Intact qubit Triggered stabilizer Correction chain

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