Optimal Toffoli-Depth Quantum Adder

TL;DR

This work addresses the longstanding challenge of depth-efficient quantum addition by exhaustively exploring carry-propagation designs and establishing a quantum prefix-tree carry-lookahead architecture that attains Toffoli-Depth $\log n + O(1)$. By leveraging a Sklansky (Slansky) prefix tree and a repeat gate to circumvent qubit no-cloning, the authors demonstrate a substantial depth reduction (roughly 50%) over prior Brent-Kung-based adders, with two evaluation strategies showing the optimal result at $\log n + 1$ Toffoli-Depth. They further extend the framework with Ling-based propagation/generation and a modular (VBE-based) addition extension, outlining depth-versus-resource tradeoffs and identifying cases where Ling can be detrimental due to added OR logic. Overall, the paper delivers a concrete path to near-optimal depth quantum adders and lays groundwork for scalable arithmetic in quantum algorithms, including factoring and error-corrected implementations.

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.

Figures (8)

• Figure 1: Design Choices.
• Figure 2: Quantum Prefix Tree. Gray nodes in the prefix trees represent CNOT operations, with numbers in circles indicating the times of CNOT operations.
• Figure 8: 4 bit Quantum Adder Examples. The dashed lines divide the entire circuit into five parts according to steps 1 to 4. In the first step, purple represents the calculation for initial propagation and generation. In the second step, red represents the copy operation, orange represents the first layer of propagation, and green and blue represent the first and the rest layers of generation, respectively. In the third step, orange represents propagation uncomputation, and red represents the uncomputation of the initial propagation copy $p_i$. Next, pink represents the operations of the fourth step. In the final step of Kogge-Stone Adder, we perform uncomputation on all the initially copied $g_i$, represented in red.
• Figure 9: Gidney's Logical-And structure.
• Figure 10: Comparative Cost Analysis of Quantum Optimal Toffoli-Depth Adder and Other top 3 Prominent Quantum CLA Adders.