7 Physical Qubits
Click qubit to inject error:
Qubit indices: 1–7
Syndrome bits:
s1 (X): –
s2 (X): –
s3 (X): –
s4 (Z): –
s5 (Z): –
s6 (Z): –
Ready. Encode a logical state to begin.
Stabilizer Generators
H = [3,5,6,7] X
H = [2,4,6,7] X
H = [1,4,5,7] X
H = [3,5,6,7] Z
H = [2,4,6,7] Z
H = [1,4,5,7] Z
H = [2,4,6,7] X
H = [1,4,5,7] X
H = [3,5,6,7] Z
H = [2,4,6,7] Z
H = [1,4,5,7] Z
Parity check matrix H:
[1 0 1 0 1 0 1]
[0 1 1 0 0 1 1]
[0 0 0 1 1 1 1]
Same H for X and Z errors.
Syndrome (s1,s2,s3) in binary
gives error location (1–7).
Min Hamming distance = 3
→ corrects 1 error
→ detects 2 errors
[1 0 1 0 1 0 1]
[0 1 1 0 0 1 1]
[0 0 0 1 1 1 1]
Same H for X and Z errors.
Syndrome (s1,s2,s3) in binary
gives error location (1–7).
Min Hamming distance = 3
→ corrects 1 error
→ detects 2 errors
Logical operators:
X̄ = X₁X₂X₃X₄X₅X₆X₇
Z̄ = Z₁Z₂Z₃Z₄Z₅Z₆Z₇
X̄ = X₁X₂X₃X₄X₅X₆X₇
Z̄ = Z₁Z₂Z₃Z₄Z₅Z₆Z₇