Optimal Toffoli-Depth Quantum Adder

Abstract

Efficient quantum arithmetic circuits are commonly found in numerous quantum algorithms of practical significance. To date, the logarithmic-depth quantum adders include a constant coefficient k ≥ 2 while achieving the Toffoli-Depth of k log n + 𝒪(1). In this work, 160 alternative compositions of the carry-propagation structure are comprehensively explored to determine the optimal depth structure for a quantum adder. By extensively studying these structures, it is shown that an exact Toffoli-Depth of log n + 𝒪(1) is achievable. This presents a reduction of Toffoli-Depth by almost 50% compared to the best known quantum adder circuits presented to date. We demonstrate a further possible design by incorporating a different expansion of propagate and generate forms, as well as an extension of the modular framework. Our article elaborates on these designs, supported by detailed theoretical analyses and simulation-based studies, firmly substantiating our claims of optimality within all possible configurations outlined in this work. The results also mirror similar improvements, recently reported in classical adder circuit complexity.