Concatenated codes and fault-tolerant computation threshold p_th ≈ 1%
Fault-tolerant quantum computation relies on an error threshold p_th. Below p_th, each additional
level of code concatenation exponentially suppresses the logical error rate. For a [[d,1,d]] code,
the logical error rate at level k is p_L^(k) ≈ (p/p_th)^(d^k) · p_th. Above threshold,
concatenation makes things worse — errors cascade faster than they can be corrected.
The threshold p_th ≈ 1/A where A ~ d^2/2 is the number of faults that must conspire.