Multi-qubit Toffoli with exponentially fewer T gates

Abstract

Prior work of Beverland et al. has shown that any exact Clifford+$T$ implementation of the $n$-qubit Toffoli gate must use at least $n$ $T$ gates. Here we show how to get away with exponentially fewer $T$ gates, at the cost of incurring a tiny $1/\mathrm{poly}(n)$ error that can be neglected in most practical situations. More precisely, the $n$-qubit Toffoli gate can be implemented to within error $ε$ in the diamond distance by a randomly chosen Clifford+$T$ circuit with at most $O(\log(1/ε))$ $T$ gates. We also give a matching $Ω(\log(1/ε))$ lower bound that establishes optimality, and we show that any purely unitary implementation achieving even constant error must use $Ω(n)$ $T$ gates. We also extend our sampling technique to implement other Boolean functions. Finally, we describe upper and lower bounds on the $T$-count of Boolean functions in terms of non-adaptive parity decision tree complexity and its randomized analogue.