A Quadratic Lower Bound for Noncommutative Circuits

TL;DR

This paper establishes a quadratic lower bound on the size of noncommutative circuits computing the palindrome polynomial, advancing understanding of circuit complexity in noncommutative algebra.

Contribution

It provides the first quadratic lower bound for fan-in 2 noncommutative circuits computing the palindrome polynomial, refining previous techniques.

Findings

Any fan-in 2 noncommutative circuit computing the palindrome polynomial has size (n d)

When d=n, the lower bound becomes (n^2)

The proof refines earlier methods and builds on previous work by the author.

Abstract

We prove that every fan-in $2$ noncommutative arithmetic circuit computing the palindrome polynomial has size $\Omega(nd)$. In particular, when $d=n$ we obtain an $\Omega(n^2)$ lower bound. The proof builds on and refines a previous work of the author. Key ideas in the proof were generated by Gemini 3.1 Pro.