Quantum Query Complexity of Multilinear Identity Testing

TL;DR

This paper investigates the quantum query complexity of testing multilinear identities in finite rings, providing new quantum algorithms and bounds for identity testing problems in algebraic structures.

Contribution

It introduces a quantum algorithm with specific query complexity for multilinear identity testing and discusses lower bounds and classical tests, advancing understanding of quantum advantages in algebraic property testing.

Findings

Quantum algorithm with query complexity O(m(1+α)^{m/2} k^{m/(m+1)})

Classical randomized test with query complexity 4^mmk

Deterministic test with query complexity k^m

Abstract

Motivated by the quantum algorithm in \cite{MN05} for testing commutativity of black-box groups, we study the following problem: Given a black-box finite ring $R=\angle{r_1,...,r_k}$ where $\{r_1,r_2,...,r_k\}$ is an additive generating set for $R$ and a multilinear polynomial $f(x_1,...,x_m)$ over $R$ also accessed as a black-box function $f:R^m\to R$ (where we allow the indeterminates $x_1,...,x_m$ to be commuting or noncommuting), we study the problem of testing if $f$ is an \emph{identity} for the ring $R$. More precisely, the problem is to test if $f(a_1,a_2,...,a_m)=0$ for all $a_i\in R$. We give a quantum algorithm with query complexity $O(m(1+\alpha)^{m/2} k^{\frac{m}{m+1}})$ assuming $k\geq (1+1/\alpha)^{m+1}$. Towards a lower bound, we also discuss a reduction from a version of $m$-collision to this problem. We also observe a randomized test with query complexity $4^mmk$ and constant success probability and a deterministic test with $k^m$ query complexity.