Novel constructions for the fault-tolerant Toffoli gate

We present two new constructions for the Toffoli gate which substantially reduce resource costs in fault-tolerant quantum computing. The first contribution is a Toffoli gate requiring Clifford operations plus only four $T = \exp(i\pi\sigma^z/8)$gates,whereasconventionalcircuitsrequireseven$T$gates\mathrm{.}Anextensionofthisresultisthatadding$n$ control inputs to a controlled gate requires $4n$ $T$ gates, whereas the best prior result was $8n$. The second contribution is a quantum circuit for the Toffoli gate which can detect a single ${\sigma }^z$ error occurring with probability $p$ in any one of eight $T$ gates required to produce the Toffoli. By post-selecting circuits that did not detect an error, the posterior error probability is suppressed to lowest order from $4p$ (or $7p$, without the first contribution) to $28p^2$ for this enhanced construction. In fault-tolerant quantum computing, this construction can reduce the overhead for producing logical Toffoli gates by an order of magnitude.