Steane's 7-Qubit Code

Overview

Steane's code is a [[7,1,3]] CSS code — it encodes 1 logical qubit into 7 physical qubits with distance 3, correcting any single-qubit error. Built from the classical Hamming [7,4,3] code, it is more efficient than Shor's 9-qubit code (rate 1/7 vs 1/9) and has the remarkable property that H, CX, and S gates can be applied transversally (bitwise), making them automatically fault-tolerant.

CSS Code Construction

Steane's code is a Calderbank-Shor-Steane (CSS) code, meaning its X and Z stabilizers come from independent classical codes:

Start with the Hamming [7,4,3] code (classical, corrects 1 bit error)
The Hamming parity check matrix defines Z stabilizers (detect X errors)
The same parity check matrix defines X stabilizers (detect Z errors)
Because Hamming [7,4,3] is self-dual (C contains C-perp), this works!

This self-dual property is what makes transversal H possible — applying H to all 7 qubits simply swaps the X and Z stabilizers, which have the same structure.

Stabilizers

X Stabilizers (detect Z errors)

Stabilizer	Qubits	Operator
g1	3, 4, 5, 6	X_3 X_4 X_5 X_6
g2	1, 2, 5, 6	X_1 X_2 X_5 X_6
g3	0, 2, 4, 6	X_0 X_2 X_4 X_6

Z Stabilizers (detect X errors)

Stabilizer	Qubits	Operator
g4	3, 4, 5, 6	Z_3 Z_4 Z_5 Z_6
g5	1, 2, 5, 6	Z_1 Z_2 Z_5 Z_6
g6	0, 2, 4, 6	Z_0 Z_2 Z_4 Z_6

Logical Operators

X_L = X_0 X_1 X_2 X_3 X_4 X_5 X_6 (X on all qubits)
Z_L = Z_0 Z_1 Z_2 Z_3 Z_4 Z_5 Z_6 (Z on all qubits)

Syndrome Table

The syndrome is a 3-bit number equal to (qubit_index + 1) in binary:

Syndrome	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

The Z syndrome identifies X errors; the X syndrome identifies Z errors. For a Y error, both syndromes are non-trivial (same location).

The Encoding Circuit

CODE
     ┌───┐
q_0: ┤ H ├──■──────■────────────■──
     ├───┤  │      │            │
q_1: ┤ H ├──┼──────┼────■───■──┼──
     ├───┤  │      │    │   │  │
q_2: ┤ H ├──┼──────┼────┼───┼──┼──
     └───┘┌─┴─┐  ┌─┴─┐  │   │  │
q_3: ─────┤ X ├──┤ X ├──┼───┼──┼──  (from q0, q1)
          └───┘  └───┘┌─┴─┐ │  │
q_4: ──────────────────┤ X ├─┼──┼──  (from q0, q2)
                       └───┘┌┴──┴┐
q_5: ───────────────────────┤ X  ├─  (from q1, q2)
                            └────┘
q_6: ──── (from q0, q1, q2) ──────

Transversal Gates — The Key Advantage

Steane's code supports transversal (bitwise) application of several gates:

Gate	Application	Fault-Tolerant?
H	Apply H to all 7 qubits	Yes (swaps X <-> Z stabilizers)
CX	Apply CX bitwise between two code blocks	Yes
S	Apply S to all 7 qubits	Yes
X, Y, Z	Apply bitwise	Yes
T	NOT transversal	No (needs magic state distillation)

Why this matters: Transversal gates cannot spread errors between qubits within a code block. A single-qubit error stays a single-qubit error, so the code can still correct it. This is the simplest route to fault tolerance.

Running the Circuit

PYTHON
from circuit import run_circuit, verify_steane_code

# 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%}")

# Z error on qubit 5
result = run_circuit(error_type='Z', error_qubit=5)
print(f"After correction: {result['success_rate']:.2%}")

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

Code Properties

Property	Value
Notation	[[7, 1, 3]]
Physical qubits	7
Logical qubits	1
Code distance	3
Correctable errors	t = 1 (any type)
Rate	1/7 ~ 0.143
Syndrome qubits	6 (3 for X, 3 for Z)
Total qubits	13
Code family	CSS (self-dual Hamming)

Comparison with Shor Code

Feature	Shor [[9,1,3]]	Steane [[7,1,3]]
Physical qubits	9	7
Rate	1/9	1/7
Structure	Concatenated	CSS (algebraic)
Syndrome qubits	8	6
Transversal H	No	Yes
Transversal CX	No	Yes
Total qubits	17	13