Prefix Sums via Kronecker Products

TL;DR

This paper introduces a novel linear algebra approach to prefix sums using Kronecker products, resulting in circuits with optimal properties and improved quantum adder designs with reduced depth and size.

Contribution

It presents a new identity for decomposing matrices, enabling recursive prefix sum algorithms and circuits that are zero-deficiency, have constant fan-out, and smaller depth than previous methods.

Findings

First circuits with zero-deficiency, constant fan-out, and sub-$2\log(n)$ depth.

Quantum adders with $1.893\log(n)+O(1)$ Toffoli depth.

Improved Toffoli gate complexity and qubit usage in quantum addition.

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.