Shor's 9-Qubit Code

Overview

Shor's code is the first quantum error correcting code capable of correcting any single-qubit error (X, Y, or Z). It achieves this by combining bit-flip and phase-flip codes.

Structure

CODECopy
Logical qubit → [Phase-flip code] → [Bit-flip code] → 9 physical qubits
                    (3 blocks)       (3 qubits each)

Encoding

The Encoding Circuit

CODECopy
┌───┐     ┌───┐
q_0: ──┤   ├──■──┤ H ├──■────■──
       │   │  │  └───┘┌─┴─┐  │
q_1: ──┤   ├──┼───────┤ X ├──┼──
       │   │  │       └───┘┌─┴─┐
q_2: ──┤   ├──┼────────────┤ X ├
       │ CX│┌─┴─┐┌───┐
q_3: ──┤   ├┤ X ├┤ H ├──■────■──
       │   │└───┘└───┘┌─┴─┐  │
q_4: ──┤   ├──────────┤ X ├──┼──
       │   │          └───┘┌─┴─┐
q_5: ──┤   ├───────────────┤ X ├
       │   │┌───┐┌───┐
q_6: ──┤   ├┤ X ├┤ H ├──■────■──
       └───┘└───┘└───┘┌─┴─┐  │
q_7: ─────────────────┤ X ├──┼──
                      └───┘┌─┴─┐
q_8: ──────────────────────┤ X ├
                           └───┘

Error Correction

Bit-flip errors (X)

• Detected within each 3-qubit block
• Same as 3-qubit repetition code
• Majority vote determines correction

Phase-flip errors (Z)

• Detected across the three blocks
• Check relative phases between blocks
• Sign majority determines correction

Y errors

• Y = iXZ (both bit and phase flip)
• Corrected by detecting both X and Z components

Running the Circuit

PYTHONCopy
from circuit import run_circuit

# No error
result = run_circuit()
print(f"Success: {result['success_rate']:.2%}")

# X error on qubit 4
result = run_circuit(error_type='X', error_qubit=4)
print(f"After X correction: {result['success_rate']:.2%}")

# Y error (both X and Z)
result = run_circuit(error_type='Y', error_qubit=2)
print(f"After Y correction: {result['success_rate']:.2%}")

Code Properties

Historical Significance

• First complete QEC code (1995)
• Proved quantum error correction is possible
• Showed quantum computers can be fault-tolerant
• Foundation for all subsequent codes

Comparison with Other Codes

Learn More

• Shor's Original Paper
• Quantum Error Correction Introduction