Magic-state distillation with the four-qubit code

The distillation of magic states is an often-cited technique for enabling universal quantum computing once the error probability for a special subset of gates has been made negligible by other means. We present a routine for magic-state distillation that reduces the required overhead for a range of parameters of practical interest. Each iteration of the routine uses a four-qubit error-detecting code to distill the $+1$ eigenstate of the Hadamard gate at a cost of ten input states per two improved output states. Use of this routine in combination with the $15$-to-$1$ distillation routine described by Bravyi and Kitaev allows for further improvements in overhead.