Optimal T depth quantum circuits for implementing arbitrary Boolean functions

TL;DR

This paper introduces a method to construct optimal T depth quantum circuits for any Boolean function, significantly improving efficiency and parallelism in quantum computing implementations.

Contribution

It generalizes existing results by providing a universal construction for minimal T depth circuits based on the algebraic degree of Boolean functions.

Findings

Achieves T depth of log_2 k for any Boolean function with degree k.

Provides a framework for optimal quantum circuit design based on ANF analysis.

Implications for cryptography and quantum algorithm efficiency.

Abstract

In this paper we present a generic construction to obtain an optimal T depth quantum circuit for any arbitrary $n$-input $m$-output Boolean function $f: \{0,1\}^n \rightarrow \{0,1\}^m$ having algebraic degree $k\leq n$, and it achieves an exact Toffoli (and T) depth of $\lceil \log_2 k \rceil$. This is a broader generalization of the recent result establishing the optimal Toffoli (and consequently T) depth for multi-controlled Toffoli decompositions (Dutta et al., Phys. Rev. A, 2025). We achieve this by inspecting the Algebraic Normal Form (ANF) of a Boolean function. Obtaining a benchmark for the minimum T depth of such circuits are of prime importance for efficient implementation of quantum algorithms by enabling greater parallelism, reducing time complexity, and minimizing circuit latency, making them suitable for near-term quantum devices with limited coherence times. The implications of our results are highlighted explaining the provable lower bounds on S-box and block cipher implementations, for example AES.