On Exact Space-Depth Trade-Offs in Multi-Controlled Toffoli Decomposition

TL;DR

This paper analyzes the trade-offs between Toffoli depth and ancilla qubits in multi-controlled Toffoli gate decompositions within Clifford+T frameworks, providing formulas, limitations, and optimal bounds for quantum circuit optimization.

Contribution

It explicitly quantifies the space-depth trade-offs, introduces improved techniques for reducing Toffoli depth, and establishes fundamental lower bounds in Clifford+T decompositions.

Findings

Toffoli depth cannot be exactly $oxed{ ext{log}_2 n}$ using conditionally clean ancilla.

T-Depth lower bound is $oxed{ ext{log}_2 n}$, achieved by a binary tree structure.

Various decomposition methods for Toffoli gates are explored for trade-off implications.

Abstract

In this paper, we consider the optimized implementation of Multi Controlled Toffoli (MCT) using the Clifford $+$ T gate sets. While there are several recent works in this direction, here we explicitly quantify the trade-off (with concrete formulae) between the Toffoli depth (this means the depth using the classical 2-controlled Toffoli) of the $n$-controlled Toffoli (hereform we will tell $n$-MCT) and the number of clean ancilla qubits. Additionally, we achieve a reduced Toffoli depth (and consequently, T-depth), which is an extension of the technique introduced by Khattar et al. (2024). In terms of a negative result, we first show that using such conditionally clean ancilla techniques, Toffoli depth can never achieve exactly $\ceil{\log_2 n}$, though it remains of the same order. This highlights the limitation of the techniques exploiting conditionally clean ancilla [Nie et al., 2024, Khattar et al., 2024]. Then we prove that, in a more general setup, the T-Depth in the Clifford + T decomposition, via Toffoli gates, is lower bounded by $\ceil{\log_2 n}$, and this bound is achieved following the complete binary tree structure. Since the ($2$-controlled) Toffoli gate can further be decomposed using Clifford $+$ T, various methodologies are explored too in this regard for trade-off related implications.