On Lifting Lower Bounds for Noncommutative Circuits using Automata

We revisit the main result of Carmosino et al \cite{CILM18} which shows that an $Ω(n^{ω/2+ε})$ size noncommutative arithmetic circuit size lower bound (where $ω$ is the matrix multiplication exponent) for a constant-degree $n$-variate polynomial family $(g_n)_n$, where each $g_n$ is a noncommutative polynomial, can be ``lifted'' to an exponential size circuit size lower bound for another polynomial family $(f_n)$ obtained from $(g_n)$ by a lifting process. In this paper, we present a simpler and more conceptual automata-theoretic proof of their result.