quantum-error-correction

Layer 1 — Physics24 concepts in this subtree

Theory and practice of protecting quantum information against decoherence by redundant encoding. Foundational: Shor 1995 9-qubit code, Steane 1996 7-qubit CSS code, Knill-Laflamme 1997 necessary-and-sufficient conditions for a correctable…

Stabilizer formalism: codespace = simultaneous +1 eigenspace of abelian subgroup S ⊂ P_n
CSS construction: dual classical codes C_1 ⊃ C_2^⊥ → stabilizer code correcting X + Z errors separately
Surface code: stabilizers are local 4-body plaquettes + vertices on a 2D lattice
Knill-Laflamme: ⟨i|E_a†E_b|j⟩ = α_ab δ_ij ⇔ code corrects error set {E_a}
Threshold theorem: p < p_th ⇒ concatenated encoding drives logical error to 0 exponentially
Surface code: p_L ~ A (p/p_th)^⌊(d+1)/2⌋ — suppression ratio d^(1/2) per extra code distance
von Neumann entropy S(ρ) = −Tr(ρ log ρ) on qubit channels: Shannon-quantum bridge
Stabiliser check matrix symplectic-rank structure: Λ∈F_2^((n-k)×2n) with alternating form
Quantum weight-enumerator A(x,y) and MacWilliams-Shor-Laflamme identity via quadratic form
Binary Shannon entropy H₂(p) = −p·log(p) − (1−p)·log(1−p); dephasing-channel capacity Q = 1 − H₂(p)
Quantum Singleton bound: k ≤ n − 2(d−1); rate ratio R = k/n ≤ 1 − 2(d−1)/n
Quantum Hamming sphere-packing bound (t=1): 2^k · (1 + 3n) ≤ 2^n for non-degenerate codes
Shor 9-qubit (1995)
CSS codes (CSS 1996)
Stabilizer (Gottesman 1997)
Topological QEC (Kitaev 1997)
qLDPC (Gottesman 2014)
Magic state (Bravyi-Kitaev 2005)
Shor 9-qubit (1995)
Steane code (1996)
Toric code (Kitaev 1997)
Threshold theorem (Aharonov 1997)
qLDPC (Gottesman 2014)
Magic state (Bravyi-Kitaev 2005)
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