  In the shallow sub-threshold regime, fault-tolerant quantum computation
requires a tremendous amount of qubits. In this paper, we study the error
correction in the deep sub-threshold regime. We estimate the physical error
rate for achieving the logical error rates of $10^{-6} - 10^{-15}$ using
few-qubit codes, i.e. short repetition codes, small surface codes and the
Steane code. Error correction circuits that are efficient for biased error
channels are identified. Using the Steane code, when error channels are biased
with a ratio of $10^{-3}$, the logical error rate of $10^{-15}$ can be achieved
with the physical error rate of $10^{-5}$, which is much higher than the
physical error rate of $10^{-9}$ for depolarising errors.
