3-Qubit Bit-Flip Code

Overview

The 3-qubit bit-flip code is the simplest quantum error correcting code. It protects against single X (bit-flip) errors by encoding one logical qubit into three physical qubits.

Encoding

The Circuit

Encoding

CODECopy
┌───┐
q_0: ┤ X ├──■────■──  (logical |1⟩)
     └───┘┌─┴─┐  │
q_1: ─────┤ X ├──┼──
          └───┘┌─┴─┐
q_2: ──────────┤ X ├
               └───┘

Syndrome Measurement

CODECopy
┌───┐
data_0: ─┤   ├─────
         │ Z₁Z₂│
data_1: ─┤   ├──■──
         └───┘┌─┴─┐
data_2: ──────┤ Z₂Z₃├
              └───┘

Syndrome Table

Running the Circuit

PYTHONCopy
from circuit import run_circuit

# No error
result = run_circuit(error_qubit=None)
print(f"Success: {result['success_rate']:.2%}")

# Inject error on qubit 1
result = run_circuit(error_qubit=1)
print(f"Corrected: {result['success_rate']:.2%}")

Expected Output

Limitations

• Only corrects X errors: Z (phase-flip) errors not corrected
• Single error only: Two or more errors cause failure
• Overhead: 3 physical qubits per logical qubit

Why It Works

The code uses redundancy and parity checks:

1. Redundancy: Information spread across 3 qubits
2. Parity: Syndrome measures parity of pairs
3. Majority vote: Effective classical interpretation

Applications

• Educational: Foundation for understanding QEC
• Building block: Combined with phase-flip code for Shor code
• Repetition code: Extendable to 5, 7, 9 qubits

Learn More

• Introduction to QEC
• Qiskit Textbook: Error Correction