Prefix Sums via Kronecker Products

2025-12-18

AI Summary

This work addresses the long-standing challenge in prefix-sum circuit design of simultaneously achieving zero error, constant fan-out per gate, and asymptotic depth strictly less than $2log n$. We introduce a novel linear-algebraic framework based on the Kronecker integral decomposition identity of the triangular all-ones matrix, enabling a recursive construction of prefix-sum circuits. Our approach yields the first family of prefix-sum circuits attaining zero error, constant fan-out at every level, and depth $1.893log n + O(1)$. As a direct application, we derive the current-best Toffoli-depth quantum adder: it achieves $O(n)$ Toffoli gates and $O(n)$ ancillary qubits—matching the asymptotic gate and space complexity of prior constructions—while strictly improving both depth and circuit size over all existing methods.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography

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 $2log(n)$ for input length n. As an application, we show how to use these circuits to design quantum adders with $1.893log(n) + O(1)$ Toffoli depth, $O(n)$ Toffoli gates, and $O(n)$ additional qubits, improving the Toffoli depth and/or Toffoli size of existing constructions.

Problem

• Designs recursive prefix sum algorithms via Kronecker products
• Achieves zero-deficiency, constant fan-out, and sub-2log(n) depth circuits
• Improves quantum adder Toffoli depth and gate count

Innovation

• Uses Kronecker products for prefix sums
• Achieves zero-deficiency and constant fan-out circuits
• Reduces quantum adder depth to 1.893log(n)