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=∠r1,...,rk where {r1,r2,...,rk} is an additive
generating set for R and a multilinear polynomial f(x1,...,xm) over R
also accessed as a black-box function f:Rm→R (where we allow the
indeterminates x1,...,xm 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(a1,a2,...,am)=0 for all ai∈R.
We give a quantum algorithm with query complexity O(m(1+α)m/2km+1m) assuming k≥(1+1/α)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 4mmk and constant
success probability and a deterministic test with km query complexity.Comment: 12 page