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Prefix Sums via Kronecker Products

Aleksandros Sobczyk, Anastasios Zouzias

TL;DR

This work recasts prefix sums as linear-algebraic objects and derives a two-term Kronecker decomposition of triangular all-ones matrices to design recursive, zero-deficiency prefix circuits. By parameterizing the construction with a block size $s$, the authors obtain a family of circuits with depth $D(n)\le s\lceil\log_s(n)\rceil-2$ and LOGTIME-uniformity; in particular, $s=2$ yields depth $2\log(n)+O(1)$, while $s=3$ achieves about $1.893\log(n)+O(1)$. The framework not only reproduces classic circuits (e.g., Brent–Kung, Ladner–Fischer) as special cases but also eliminates deficiency, providing near-optimal depth for zero-deficiency prefixing. As an application, the approach yields quantum adders with improved Toffoli depth and gate counts, specifically $1.893\log(n)+O(1)$ depth for $s=3$, $O(n)$ Toffoli gates, and $O(n)$ qubits. Overall, the paper advances prefix-circuit design and offers practical gains for quantum arithmetic via a clean algebraic lens.

Abstract

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.

Prefix Sums via Kronecker Products

TL;DR

This work recasts prefix sums as linear-algebraic objects and derives a two-term Kronecker decomposition of triangular all-ones matrices to design recursive, zero-deficiency prefix circuits. By parameterizing the construction with a block size , the authors obtain a family of circuits with depth and LOGTIME-uniformity; in particular, yields depth , while achieves about . The framework not only reproduces classic circuits (e.g., Brent–Kung, Ladner–Fischer) as special cases but also eliminates deficiency, providing near-optimal depth for zero-deficiency prefixing. As an application, the approach yields quantum adders with improved Toffoli depth and gate counts, specifically depth for , Toffoli gates, and qubits. Overall, the paper advances prefix-circuit design and offers practical gains for quantum arithmetic via a clean algebraic lens.

Abstract

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 for input length n. As an application, we show how to use these circuits to design quantum adders with Toffoli depth, Toffoli gates, and additional qubits, improving the Toffoli depth and/or Toffoli size of existing constructions.

Paper Structure

This paper contains 10 sections, 3 theorems, 11 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Notation.
  4. Kronecker products
  5. Constructing prefix circuits
  6. Identifying the deficiency.
  7. A new zero-deficiency circuit family.
  8. Optimizing the block size.
  9. Quantum adders
  10. Conclusion

Key Result

Theorem 1

Fix three integers $n,n_1,n_2>1$, such that $n=n_1n_2$. The lower triangular all-ones matrix $\bm{L}_n$ can be decomposed as follows: Equivalently, $\bm{U}_n=\bm{U}_{n_1}\otimes \bm{U}_{n_2}+\bm{U}^{-}_{n_1}\otimes \bm{L}^{-}_{n_2}$.

Figures (4)

  • Figure 1: Illustration of Kronecker decomposition (Theorem \ref{['theorem:triu_kron']}).
  • Figure 2: Zero-deficiency Kronecker prefix circuit.
  • Figure 3: Toffoli gates for carry propagation/generation.
  • Figure 4: Algorithm to construct quantum adder. The lines that are numbered indicate the corresponding Toffoli layer (layers marked as $*:$ do not contain Toffoli gates).

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof