Optimal Toffoli-Depth Quantum Adder

TL;DR

This paper introduces an optimal quantum adder with nearly half the Toffoli-Depth of previous designs, achieved through exploring diverse carry-propagation structures and theoretical analysis.

Contribution

It presents the first quantum adder with a Toffoli-Depth of log n + O(1), significantly improving over prior circuits and demonstrating optimality through comprehensive analysis.

Findings

Achieved Toffoli-Depth of log n + O(1) for quantum addition

Reduced Toffoli-Depth by nearly 50% compared to previous circuits

Validated optimality through theoretical and simulation studies

Abstract

Efficient quantum arithmetic circuits are commonly found in numerous quantum algorithms of practical significance. Till date, the logarithmic-depth quantum adders includes a constant coefficient k >= 2 while achieving the Toffoli-Depth of klog 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 till 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 paper elaborates on these designs, supported by detailed theoretical analyses and simulation-based studies, firmly substantiating our claims of optimality. The results also mirror similar improvements, recently reported in classical adder circuit complexity.