  We analyse a model for fault-tolerant quantum computation with low overhead
suitable for situations where the noise is biased. The basis for this scheme is
a gadget for the fault-tolerant preparation of magic states that enable
universal fault-tolerant quantum computation using only Clifford gates that
preserve the noise bias. We analyse the distillation of $|T\rangle$-type magic
states using this gadget at the physical level, followed by concatenation with
the 15-qubit quantum Reed-Muller code, and comparing our results with standard
constructions. In the regime where the noise bias (rate of Pauli $Z$ errors
relative to other single-qubit errors) is greater than a factor of 10, our
scheme has lower overhead across a broad range of relevant noise rates.
