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

Fault-tolerant fidelity based on few-qubit codes: Parity-check circuits for biased error channels