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…
quantum-error-correction
Stabilizer formalism: codespace = simultaneous +1 eigenspace of abelian subgroup S ⊂ P_n
Gottesman 1997 stabilizer formalism. The Pauli group P_n on n qubits has 4·4^n elements; an abelian subgroup S ⊂ P_n with n − k…
CSS construction: dual classical codes C_1 ⊃ C_2^⊥ → stabilizer code correcting X + Z errors separately
Calderbank-Shor-Steane 1996 construction. Start from two classical binary linear codes C_1 ⊃ C_2^⊥ (i.e. every codeword of C_2^⊥ is also a…
Surface code: stabilizers are local 4-body plaquettes + vertices on a 2D lattice
Kitaev 1997 toric / surface code. Qubits live on the edges of a 2D square lattice. Star operators S_v = ∏_{edges e∋v} X_e at every vertex…
Knill-Laflamme: ⟨i|E_a†E_b|j⟩ = α_ab δ_ij ⇔ code corrects error set {E_a}
Knill-Laflamme 1997 necessary-and-sufficient conditions for exact error correction. An error set {E_a} acting on a quantum code C =…
Threshold theorem: p < p_th ⇒ concatenated encoding drives logical error to 0 exponentially
Aharonov-Ben-Or / Kitaev / Knill / Zurek / Preskill 1997 threshold theorem. For a code correcting t errors, the logical error rate at…
Surface code: p_L ~ A (p/p_th)^⌊(d+1)/2⌋ — suppression ratio d^(1/2) per extra code distance
Fowler-Mariantoni-Martinis-Cleland 2012 / Wang-Fowler-Hollenberg 2011 analysis: a distance-d surface code corrects ⌊(d−1)/2⌋ errors and —…