Exact space-depth trade-offs in multicontrolled Toffoli decomposition

In this paper we consider the optimized implementation of multicontrolled Toffoli decomposition using the $\text{Clifford}+T$ gate set. While there are several recent works in this direction, here we explicitly quantify the trade-off (with concrete formulas) between the Toffoli depth (this means the depth using the classical 2-controlled Toffoli gate) of the $n$-controlled Toffoli decomposition and the number of clean ancilla qubits. Additionally, we achieve a reduced Toffoli (consequently $T$) depth, which is an extension of the technique introduced by Khattar and Gidney [T. Khattar and C. Gidney, arXiv:2407.17966]. In terms of a negative result, we first show that by using such conditionally clean ancilla techniques, the Toffoli depth can never achieve exactly $\ensuremath{\lceil}{log}_{2}n\ensuremath{\rceil}$, though it remains of the same order. This highlights the limitation of the techniques exploiting conditionally clean ancillas [J. Nie et al., arXiv:2402.05053; T. Khattar and C. Gidney, arXiv:2407.17966]. Then we prove that, in a more general setup, the $T$ depth in the $\text{Clifford}+T$ decomposition, via Toffoli gates, is lower bounded by $\ensuremath{\lceil}{log}_{2}n\ensuremath{\rceil}$, and this bound is achieved following the complete binary tree structure. Since the (2-controlled) Toffoli gate can further be decomposed using $\text{Clifford}+T$ gate set, various methodologies are also explored in this regard for trade-off-related implications.