Prefix Sums via Kronecker Products

Aleksandros Sobczyk, Anastasios Zouzias · 2025 · arXiv

| 编号 QA036 | 主题 加法与减法 | 链接 arXiv:2512.16309 |

摘要

摘要保留原文（英文）

摘要取自 arXiv / 出版方元数据，未作翻译，以保证与原文一致。

In this work, we revisit prefix sums through the lens of linear algebra. We describe an identity that decomposes triangular all-ones matrices as a sum of two Kronecker products, and apply it to design recursive prefix sum algorithms and circuits. Notably, the proposed family of circuits is the first one that achieves the following three properties simultaneously: (i) zero-deficiency, (ii) constant fan-out per-level, and (iii) depth that is asymptotically strictly smaller than $2\log (n)$ for input length $n$. As an application, we show how to use these circuits to design quantum adders with $1.893\log (n)+O(1)$ Toffoli depth, $O(n)$ Toffoli gates, and $O(n)$ additional qubits, improving the Toffoli depth and/or Toffoli size of existing constructions.

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