Shor's 9-Qubit Code

Overview

Shor's code is the first quantum error correcting code (1995), proving that quantum information can be protected despite the no-cloning theorem. It encodes 1 logical qubit into 9 physical qubits and corrects any single-qubit error — X, Y, or Z — using a concatenated structure of bit-flip and phase-flip codes.

Historical Significance

• First complete QEC code — proved quantum error correction is possible
• Overcame the no-cloning barrier — you can protect quantum information without copying it
• Foundation for fault tolerance — showed quantum computers can work reliably
• Led directly to the threshold theorem: if physical error rates are low enough, arbitrary-length quantum computation is possible

Concatenated Structure

The code has two levels:

CODECopy
Logical qubit
    │
    ▼
┌────────────────────────────────────────┐
│  Outer Phase-Flip Code (3 blocks)      │
│                                        │
│  Block 0        Block 1        Block 2 │
│  ┌──────┐      ┌──────┐      ┌──────┐ │
│  │ q0   │      │ q3   │      │ q6   │ │
│  │ q1   │      │ q4   │      │ q7   │ │
│  │ q2   │      │ q5   │      │ q8   │ │
│  │      │      │      │      │      │ │
│  │ Inner │      │ Inner │      │ Inner │ │
│  │Bit-Flip│    │Bit-Flip│    │Bit-Flip│ │
│  └──────┘      └──────┘      └──────┘ │
└────────────────────────────────────────┘

Encoding

Encoding Circuit

CODECopy
           Phase-flip          Bit-flip (per block)
         ┌───────────┐       ┌──────────────────┐
q_0: ──■────■──┤ H ├───────■────■──
       │    │  └───┘      ┌─┴─┐  │
q_1: ──┼────┼─────────────┤ X ├──┼──
       │    │             └───┘┌─┴─┐
q_2: ──┼────┼──────────────────┤ X ├
     ┌─┴─┐  │  ┌───┐
q_3: ┤ X ├──┼──┤ H ├─────■────■──
     └───┘  │  └───┘    ┌─┴─┐  │
q_4: ───────┼───────────┤ X ├──┼──
            │           └───┘┌─┴─┐
q_5: ───────┼────────────────┤ X ├
          ┌─┴─┐ ┌───┐
q_6: ─────┤ X ├─┤ H ├───■────■──
          └───┘ └───┘  ┌─┴─┐  │
q_7: ──────────────────┤ X ├──┼──
                       └───┘┌─┴─┐
q_8: ───────────────────────┤ X ├
                            └───┘

How Each Error Type Is Caught

X (bit-flip) errors

• Detected within each 3-qubit block by the inner bit-flip code
• Same mechanism as the standalone 3-qubit bit-flip code
• Corrected with conditional X gate on the affected qubit

Z (phase-flip) errors

• Changes the relative sign between |000> and |111> within a block
• Detected across blocks by the outer phase-flip code
• Corrected with conditional Z gate on any qubit in the affected block

Y errors

• Y = iXZ — both a bit-flip AND a phase-flip simultaneously
• The X component is caught by the inner code
• The Z component is caught by the outer code
• Both corrections applied independently

Syndrome Extraction

Bit-Flip Syndromes (6 ancillae, 2 per block)

Phase-Flip Syndromes (2 ancillae)

Running the Circuit

PYTHONCopy
from circuit import run_circuit, verify_shor_code

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

# X error on qubit 4 (middle of block 1)
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%}")

# Full verification
v = verify_shor_code()
for check in v["checks"]:
    print(f"[{'PASS' if check['passed'] else 'FAIL'}] {check['name']}")

Code Properties

Comparison with Other Codes

Learn More

• Shor's Original Paper (1995)
• Devitt et al. — QEC for Beginners
• Nielsen & Chuang, Section 10.4