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

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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 the surface 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 exactly transform one |CCZ\rangle state into two |T\rangle states. It has a footprint 25% smaller than the factory of Fowler et al. but outputs |T\rangle states twice as quickly. We show how to generalize the catalyzed transformation to arbitrary phase angles, and note that the case \theta=22.5^\circ produces a particularly efficient circuit for producing |\sqrt{T}\rangle states. Compared to using the 12d \times 8d \times 6.5d |T\rangle factory of Fowler et al., our |CCZ\rangle factory can quintuple the speed of algorithms that are dominated by the cost of applying Toffoli gates, including Shor's algorithm and the chemistry algorithm of Babbush et al.. Assuming a physical gate error rate of 10^{-3}, our CCZ factory can produce \sim 10^{10} states on average before an error occurs. This is sufficient for classically intractable instantiations of the chemistry algorithm, but for more demanding algorithms such as Shor's algorithm the mean number of states until failure can be increased to \sim 10^{12} by increasing the factory footprint ~20%.