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We present magic state factory constructions for producing $|CCZ\rangle$states and $|T\rangle$ states. For the $|CCZ\rangle$ factory we apply thesurface code lattice surgery construction techniques described by Fowler et al.to the fault-tolerant Toffoli. The resulting factory has a footprint of $12d\times 6d$ (where $d$ is the code distance) and produces one $|CCZ\rangle$every $5.5d$ surface code cycles. Our $|T\rangle$ state factory uses the$|CCZ\rangle$ factory's output and a catalyst $|T\rangle$ state to exactlytransform one $|CCZ\rangle$ state into two $|T\rangle$ states. It has afootprint 25% smaller than the factory of Fowler et al. but outputs $|T\rangle$states twice as quickly. We show how to generalize the catalyzed transformationto arbitrary phase angles, and note that the case $\theta=22.5^\circ$ producesa particularly efficient circuit for producing $|\sqrt{T}\rangle$ states.Compared to using the $12d \times 8d \times 6.5d$ $|T\rangle$ factory of Fowleret al., our $|CCZ\rangle$ factory can quintuple the speed of algorithms thatare dominated by the cost of applying Toffoli gates, including Shor's algorithmand the chemistry algorithm of Babbush et al.. Assuming a physical gate errorrate of $10^{-3}$, our CCZ factory can produce $\sim 10^{10}$ states on averagebefore an error occurs. This is sufficient for classically intractableinstantiations of the chemistry algorithm, but for more demanding algorithmssuch as Shor's algorithm the mean number of states until failure can beincreased to $\sim 10^{12}$ by increasing the factory footprint ~20%.

Efficient magic state factories with a catalyzed |CCZ> to 2|T> transformation