Counting Abelian Squares for a Problem in Quantum Computing

TL;DR

This paper links the problem of counting abelian squares over large alphabets to the expressiveness of parameterized quantum circuits, providing an efficient calculation method for this combinatorial problem relevant to quantum computing.

Contribution

It introduces a novel connection between abelian square counting and quantum circuit expressiveness, utilizing a new formula for efficient computation over large alphabets.

Findings

Efficient calculation of abelian squares for large alphabets

Reduced quantum circuit expressiveness analysis to combinatorial counting

Potential applications in quantum circuit design and analysis

Abstract

In a recent work I developed a formula for efficiently calculating the number of abelian squares of length $t+t$ over an alphabet of size $d$, where $d$ may be very large. Here I show how the expressiveness of a certain class of parameterized quantum circuits can be reduced to the problem of counting abelian squares over a large alphabet, and use the recently developed formula to efficiently calculate this quantity.