Multi-qubit Toffoli with exponentially fewer T gates
David Gosset, Robin Kothari, Chenyi Zhang
TL;DR
This work shows that exact exponential-to-precision barriers for multi-qubit Toffoli gates in the Clifford+$T$ framework can be overcome by allowing a tiny diamond-distance error, achieving $\mathcal{T}^{\mathrm{mixed}}_\epsilon(\mathrm{Toff}_n) = O(\log(1/\epsilon))$ gates. The core technique reduces Toffoli to a mixture of small, Clifford+$T$ circuits via random parity (XOR) sampling to approximate OR, yielding an efficient, scalable construction that generalizes to other Boolean functions. The authors develop a Fourier-analytic framework (Fourier 1-norm) and connect $T$-count to non-adaptive/paritized decision-tree complexities, providing both upper and lower bounds and demonstrating broad applicability to many functions (OR, HW, CW, MEQ, RankOne, GT, INC, ADD, MAJ). These results offer practical reductions in magic-state overhead for fault-tolerant quantum computation and deepen the understanding of when low-$T$-count approximations are viable. The work also establishes limits via stabilizer-nullity and-adaptive lower bounds, and discusses concurrent work and potential applications in learning and simulation of low-$T$ circuits.
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.