Prefix Sums via Kronecker Products
Fuente:
arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912784857432064 |
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| author | Sobczyk, Aleksandros Zouzias, Anastasios |
| author_facet | Sobczyk, Aleksandros Zouzias, Anastasios |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_16309 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Prefix Sums via Kronecker Products Sobczyk, Aleksandros Zouzias, Anastasios Quantum Physics Data Structures and Algorithms 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. |
| title | Prefix Sums via Kronecker Products |
| topic | Quantum Physics Data Structures and Algorithms |
| url | https://arxiv.org/abs/2512.16309 |