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Quantum hardware is noisy. Gates introduce errors, and qubits decohere over time. Quantum error correction (QEC) encodes a logical qubit across multiple physical qubits so that errors on individual qubits can be detected and corrected without measuring — and thereby collapsing — the encoded state. This guide implements the 3-qubit bit-flip code, the simplest QEC code. It protects a single logical qubit against one bit-flip error (an accidental X gate) by encoding it across three physical qubits and using two ancilla qubits for syndrome measurement.

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How the code works

The logical qubit α|0⟩ + β|1⟩ is encoded as: 
α|000⟩ + β|111⟩
If qubit 0 is flipped by a noise event, the state becomes α|100⟩ + β|011⟩. The syndrome measurements detect this without revealing α or β — they only identify which qubit flipped.

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Error correction walkthrough

1

Create the logical qubit

Allocate a qubit in the state (0.2, 0.8) — unnormalized amplitudes that the simulator normalizes to 0.4472|0⟩ + 0.8944|1⟩.
qsim := q.New()

q0 := qsim.New(1, 2) // (0.2, 0.8)
This is the qubit whose state we want to protect.
2

Encode into three physical qubits

Add two ancilla qubits q1 and q2 in |0⟩ and CNOT from q0 to each. This spreads the logical state across all three qubits.
// encode
q1 := qsim.Zero()
q2 := qsim.Zero()
qsim.CNOT(q0, q1)
qsim.CNOT(q0, q2)
After encoding:
  • α|0⟩ → α|000⟩ (CNOT copies 0 to q1 and q2)
  • β|1⟩ → β|111⟩ (CNOT copies 1 to q1 and q2)
The encoded state is α|000⟩ + β|111⟩.
3

Introduce a bit-flip error on q0

Simulate a hardware error by applying X to q0. This flips the first physical qubit.
// error: first qubit is flipped
qsim.X(q0)
The state is now α|100⟩ + β|011⟩.
4

Add syndrome measurement qubits

Add two fresh ancilla qubits q3 and q4 in |0⟩. These will hold the error syndrome.
// add ancilla qubit
q3 := qsim.Zero()
q4 := qsim.Zero()
5

Compute the error syndrome

Apply CNOTs to entangle the data qubits with the syndrome qubits. The syndrome circuit compares pairs of data qubits without reading their values.
// error correction
qsim.CNOT(q0, q3)
qsim.CNOT(q1, q3)
qsim.CNOT(q1, q4)
qsim.CNOT(q2, q4)
  • q3 measures the parity of q0 XOR q1
  • q4 measures the parity of q1 XOR q2
Measure q3 and q4:
m3 := qsim.Measure(q3)
m4 := qsim.Measure(q4)
The syndrome table is:
m3m4Flipped qubit
10q0
11q1
01q2
00no error
6

Apply conditional correction

Use the syndrome bits to conditionally apply X to the flipped qubit:
qsim.CondX(m3.IsOne() && m4.IsZero(), q0)
qsim.CondX(m3.IsOne() && m4.IsOne(), q1)
qsim.CondX(m3.IsZero() && m4.IsOne(), q2)
Since the error was on q0 (m3 = 1, m4 = 0), the first CondX fires and flips q0 back. The state returns to α|000⟩ + β|111⟩.
7

Decode the logical qubit

Reverse the encoding CNOTs to disentangle q0 from q1 and q2.
// decode
qsim.CNOT(q0, q2)
qsim.CNOT(q0, q1)
After decoding, q0 holds the original logical state and q1, q2 return to |0⟩.
8

Verify the original state is recovered

Print the state of q0 and confirm it matches the original (0.4472, 0.8944).
for _, s := range qsim.State(q0) {
  fmt.Println(s)
}

// [0][  0]( 0.4472 0.0000i): 0.2000
// [1][  1]( 0.8944 0.0000i): 0.8000
The amplitudes 0.4472 and 0.8944 match the original state exactly. The bit-flip error has been corrected and the logical qubit is fully restored.

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Complete example

qsim := q.New()

q0 := qsim.New(1, 2) // (0.2, 0.8)

// encode
q1 := qsim.Zero()
q2 := qsim.Zero()
qsim.CNOT(q0, q1)
qsim.CNOT(q0, q2)

// error: first qubit is flipped
qsim.X(q0)

// add ancilla qubit
q3 := qsim.Zero()
q4 := qsim.Zero()

// error correction
qsim.CNOT(q0, q3)
qsim.CNOT(q1, q3)
qsim.CNOT(q1, q4)
qsim.CNOT(q2, q4)

m3 := qsim.Measure(q3)
m4 := qsim.Measure(q4)

qsim.CondX(m3.IsOne() && m4.IsZero(), q0)
qsim.CondX(m3.IsOne() && m4.IsOne(), q1)
qsim.CondX(m3.IsZero() && m4.IsOne(), q2)

// decode
qsim.CNOT(q0, q2)
qsim.CNOT(q0, q1)

for _, s := range qsim.State(q0) {
  fmt.Println(s)
}

// [0][  0]( 0.4472 0.0000i): 0.2000
// [1][  1]( 0.8944 0.0000i): 0.8000

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