Quantum Threshold Theorem
fault tolerance · error correction · p < p_th
Physics: Quantum threshold theorem (Aharonov-Ben-Or 1997, Kitaev 1997): if physical error rate p < p_th, then by adding levels of error correction, logical error rate → 0 with only polynomial resource overhead. p_th ≈ 1% for surface code under depolarizing noise. Concatenated codes: level-k logical error rate p_k = (p/p_th)^(2^k). Physical qubit overhead: ~1000–10,000 physical per logical qubit. Fault-tolerant gates: transversal Clifford gates + magic state distillation for T gate. Current IBM: ~0.3% two-qubit gate error rate. Google 2023: demonstrated logical qubit below break-even point (surface code distance 7 vs 5).