Error correction in one lesson

The three-qubit repetition code, syndromes, and the road to a logical qubit.

You will learn to

Encode one logical bit in three physical qubits and detect a bit flip without measuring the data.
Explain what a syndrome is.
Place the surface code and the threshold in context.

You cannot copy a qubit, and you cannot look at it without collapsing it. Error correction gets around both: spread the information over several qubits (entangle, do not copy) and measure only parities, which reveal where an error happened but nothing about the encoded value.

∣ \overset{ˉ}{0} ⟩ = ∣000 ⟩, ∣ \overset{ˉ}{1} ⟩ = ∣111 ⟩, α ∣ \overset{ˉ}{0} ⟩ + β ∣ \overset{ˉ}{1} ⟩ = α ∣000 ⟩ + β ∣111 ⟩

The bit-flip repetition code. Parities Z₀Z₁ and Z₁Z₂, measured into two ancillas, form the syndrome: 00 = no flip, 10 = qubit 0, 11 = qubit 1, 01 = qubit 2.

Try it

Detect and correct a flip with feed-forward

q0 is the data, encoded into q0 q1 q2; an X on q1 simulates an error; q3 and q4 read the two parities; if-blocks apply the correction; the final measurement decodes. Simulate: the logical bit c[0] reads 1 every time despite the error. Move the X to q2 and watch the syndrome change.

OPENQASM 3.0;
include "stdgates.inc";
qubit[5] q;
bit[3] c;
x q[0];
cx q[0], q[1];
cx q[0], q[2];
barrier q;
x q[1];
barrier q;
cx q[0], q[3];
cx q[1], q[3];
cx q[1], q[4];
cx q[2], q[4];
c[1] = measure q[3];
c[2] = measure q[4];
if (c[1] && !c[2]) {
  x q[0];
}
if (c[1] && c[2]) {
  x q[1];
}
if (!c[1] && c[2]) {
  x q[2];
}
cx q[0], q[1];
cx q[0], q[2];
c[0] = measure q[0];

Open in the IDE Open in the Playground

The repetition code only fixes bit flips. Real codes such as the surface code protect against phase flips too, use hundreds of physical qubits per logical qubit, and work only if physical error rates are below a threshold near 1%. Today's best devices sit just below it, which is why the first logical qubits with lower error than their parts appeared only in 2023–2024.

Quiz 1 · 10 XP

Why measure Z₀Z₁ instead of Z₀ and Z₁ separately?

Quiz 2 · 10 XP

What does the threshold theorem say?

Key takeaway

Error correction spreads one logical qubit over many physical ones and reads only parities. Below the threshold, bigger codes mean fewer logical errors; that regime is where the field is arriving now.

Docs: Steane and Shor code benchmarks

Up next: Circuit cutting · Cutting and Quantum ML

OPENQASM 3.0; include "stdgates.inc"; qubit[5] q; bit[3] c; x q[0]; cx q[0], q[1]; cx q[0], q[2]; barrier q; x q[1]; barrier q; cx q[0], q[3]; cx q[1], q[3]; cx q[1], q[4]; cx q[2], q[4]; c[1] = measure q[3]; c[2] = measure q[4]; if (c[1] && !c[2]) { x q[0]; } if (c[1] && c[2]) { x q[1]; } if (!c[1] && c[2]) { x q[2]; } cx q[0], q[1]; cx q[0], q[2]; c[0] = measure q[0];