Optimal T-depth Quantum Circuits for Implementing Arbitrary Boolean Functions

Abstract In this paper, we present a generic construction for synthesizing an optimal T-depth quantum circuit for any arbitrary n -input, m -output Boolean function f : {0, 1 } n → {0, 1} m with algebraic degree k ≤ n , achieving an exact Toffoli (consequently T) depth of ⌈log 2 k ⌉. This broadly generalizes the recent result establishing the optimal Toffoli (and T) depth for multi-controlled Toffoli decompositions (Dutta et al., Phys. Rev. A, 2025). The optimality of T-depth in this initiative is considered in the context of implementing an n -MCT, assuming the decomposition via Clifford plus Toffoli gates. The key technique involves inspecting the Algebraic Normal Form (ANF) of the Boolean function. Obtaining a benchmark for the minimum T-depth of such circuits is crucial for the 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 broader implications of our results include a provable lower bounds on T-depth for S-box and block cipher implementations, such as AES. Finally, we also explain the impact of our result in identifying the T-depth for the generic cryptanalysis of block ciphers using Grover’s algorithm.