Encoding Circuits

An encoding circuit is the unitary $U_{\text{enc}}$ that maps an unprotected qubit onto a logical codeword. Every code in this module begins with one. Get the encoder right and the rest of the pipeline — syndrome extraction, decoding, logical gates — has something well-defined to act on.

What an encoder must do

A code is fixed by its two logical basis states $|0_L\rangle$ and $|1_L\rangle$. The encoder is any unitary satisfying

where the data qubit carries the input and the remaining qubits start in $|0\rangle$. Because $U_{\text{enc}}$ is linear, it then automatically encodes any superposition:

This is the whole reason a single fixed circuit protects an unknown state.

The bit-flip encoder

The three-qubit bit-flip code uses

Its encoder copies the data qubit onto two partners with CNOTs:

Check it on basis inputs: $|000\rangle \mapsto |000\rangle$ and $|100\rangle \mapsto |111\rangle$. By linearity $\alpha|000\rangle + \beta|100\rangle \mapsto \alpha|000\rangle + \beta|111\rangle$.

Encoding a non-trivial state

To make the exercise checkable we encode the logical $|+_L\rangle$. First prepare $|+\rangle$ on the data qubit with a Hadamard, then run the bit-flip encoder:

The two nonzero amplitudes are each $1/\sqrt{2} \approx 0.707$. Note this is a genuinely entangled three-qubit state — measuring any one qubit instantly fixes the other two.

Order matters

Apply the Hadamard before the CNOTs. Doing it after would put the redundancy qubits into superposition and produce a different (and unprotected) state. An encoder is a specific sequence, not a bag of gates.

Try it

Build the bit-flip encoder for $|+_L\rangle$ on three qubits. The grader checks the full statevector: amplitudes of $|000\rangle$ and $|111\rangle$ must each be $1/\sqrt{2}$ and all six other amplitudes must be zero.

After running, the statevector panel should show exactly two bars, at indices 0 (000) and 7 (111), each at height $\approx 0.707$.

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