Fault tolerance

Fault-tolerant quantum computation is the project of running a logical circuit on noisy hardware so that the logical error stays small. The early result this glossary cites is Shor's 1995 code for protecting quantum information in memory: errors on physical qubits need not destroy a stored logical state, if the code and the recovery are matched to the noise. Memory correction is not the whole threshold theorem, and this entry does not state a numerical threshold. Those theorems have their own papers. Cai and Young (2026) prove a threshold for a different object: relative noise on the small controlled rotations inside Shor's Fourier transform. The digest of 12 Oct 2026 states that cutoff. Their remark on whether the threshold theorem answers it is recorded there as the authors' belief, not as a theorem of this library.

The connection to Shor's algorithm is practical rather than historical. The 1997 factoring bound counts gates of an ideal gate array. Shor's own practicality section already says precision and decoherence dominate any attempt to run it. Resource estimates such as Gidney and Ekerå's count physical qubits under an error-correcting code. NISQ devices are the complementary regime: interesting, and not yet fault tolerant. A qubit count that ignores that distinction is not a count of runnable logical gates.