Checkpoint: One Surface-Code Round

This checkpoint ties the module together: you will run one round of stabilizer measurements on a small chain, inject a single error, and read off a syndrome that pinpoints it — exactly the input a matching decoder consumes.

The setup

Take a one-dimensional chain of three data qubits, $0, 1, 2$ — the bit-flip "spine" of a small surface-code patch. Two neighbouring $Z$ -type checks watch it:

S_1 = Z_0 Z_1 \quad\text{(ancilla } 3\text{)}, \qquad S_2 = Z_1 Z_2 \quad\text{(ancilla } 4\text{)}.

The middle data qubit $1$ is shared by both checks. This is the key structural feature of the surface code: an error on a shared qubit lights up both adjacent checks, which is how the decoder localises it.

What a middle error does

Suppose a bit-flip $X_1$ strikes the middle data qubit, leaving the data in $|010\rangle$ . Then

S_1: \; q_0 \oplus q_1 = 0 \oplus 1 = 1, \qquad S_2: \; q_1 \oplus q_2 = 1 \oplus 0 = 1.

Both checks fire — syndrome $(1, 1)$ . Compare with an error on an end qubit, say $X_0$ , which would give syndrome $(1, 0)$ since only $S_1$ touches qubit $0$ . The pattern of which checks fire is therefore a fingerprint of where the error is:

| Error | $S_1$ | $S_2$ | |--------|-------|-------| | none | 0 | 0 | | $X_0$ | 1 | 0 | | $X_1$ | 1 | 1 | | $X_2$ | 0 | 1 |

A decoder reads the syndrome column and infers the most likely error. Here $(1,1)$ unambiguously points at the middle qubit.

Try it

Build the full round in one circuit on five qubits: data $0,1,2$ , ancilla $3$ for $S_1=Z_0Z_1$ , ancilla $4$ for $S_2=Z_1Z_2$ . Inject the error with c.x(1), extract each check with a pair of CNOTs into its ancilla, then measure both ancillas. The grader checks that the only outcome with non-zero probability is $|01011\rangle$ — data $010$ , syndrome $(1,1)$ .

Run your code to see the quantum state.

After running, the Probabilities tab should show a single bar at $01011$ : the middle qubit is flipped and both syndrome bits read $1$ , exactly localising the error.