Exact space-depth trade-offs in multicontrolled Toffoli decomposition

Article Type

Research Article

Publication Title

Physical Review A

Abstract

In this paper we consider the optimized implementation of multicontrolled Toffoli decomposition using the 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 ⌈log2n⌉, though it remains of the same order. This highlights the limitation of the techniques exploiting conditionally clean ancillas [J. Nie, 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 Clifford+T decomposition, via Toffoli gates, is lower bounded by ⌈log2n⌉, and this bound is achieved following the complete binary tree structure. Since the (2-controlled) Toffoli gate can further be decomposed using Clifford+T gate set, various methodologies are also explored in this regard for trade-off-related implications.

DOI

10.1103/PhysRevA.111.052611

Publication Date

5-1-2025

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