Steane's 7-Qubit Code

Overview

Steane's code is a [[7,1,3]] quantum error correcting code, meaning it encodes 1 logical qubit into 7 physical qubits with distance 3. It's more efficient than Shor's 9-qubit code and has elegant mathematical structure.

CSS Codes

Steane's code is a CSS code (Calderbank-Shor-Steane), built from two classical codes:

X stabilizers from classical code C₁
Z stabilizers from classical code C₂ ⊆ C₁

Here, both are based on the [7,4,3] Hamming code.

Code Properties

Property	Value
Notation	[[7,1,3]]
Physical qubits	7
Logical qubits	1
Distance	3
Correctable errors	1
Rate	1/7 ≈ 0.143

Stabilizers

X Stabilizers (detect Z errors)

X₀X₂X₄X₆
X₁X₂X₅X₆
X₃X₄X₅X₆

Z Stabilizers (detect X errors)

Z₀Z₂Z₄Z₆
Z₁Z₂Z₅Z₆
Z₃Z₄Z₅Z₆

Logical Operators

X_L: X₀X₁X₂X₃X₄X₅X₆ (all X)
Z_L: Z₀Z₁Z₂Z₃Z₄Z₅Z₆ (all Z)

The Encoding Circuit

CODE
┌───┐
q_0: ┤ H ├──■────■───────■──
     ├───┤  │    │       │
q_1: ┤ H ├──┼────┼────■──┼──
     ├───┤  │    │    │  │
q_2: ┤ H ├──┼────┼────┼──┼──
     └───┘┌─┴─┐  │  ┌─┴─┐│
q_3: ─────┤ X ├──┼──┤ X ├┼──
          └───┘┌─┴─┐└───┘│
q_4: ──────────┤ X ├─────┼──
               └───┘   ┌─┴─┐
q_5: ──────────────────┤ X ├
                       └───┘
q_6: [connected to q0,q1,q2]

Running the Circuit

PYTHON
from circuit import run_circuit

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

# X error on qubit 3
result = run_circuit(error_type='X', error_qubit=3)
print(f"After correction: {result['success_rate']:.2%}")

Syndrome Interpretation

Z Syndrome	X Error Location
000	No error
001	Qubit 0
010	Qubit 1
011	Qubit 2
100	Qubit 3
101	Qubit 4
110	Qubit 5
111	Qubit 6

(Similar table for X syndrome → Z error)

Comparison with Shor Code

Feature	Shor	Steane
Qubits	9	7
Rate	1/9	1/7
Structure	Concatenated	CSS
Syndrome qubits	8	6

Transversal Gates

Steane code supports transversal:

X, Y, Z gates
H gate
CNOT gate
S gate

This makes it attractive for fault-tolerant computation.