TR08-086 Authors: Vikraman Arvind, Partha Mukhopadhyay

Publication: 17th September 2008 21:12

Downloads: 2137

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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 $R=\angle{r_1,\cdots,r_k}$ where

$\{r_1,r_2,\cdots,r_k\}$ is an additive generating set for $R$ and a

multilinear polynomial $f(x_1,\cdots,x_m)$ over $R$ also accessed as

a black-box function $f:R^m\rightarrow R$ (where we allow the

indeterminates $x_1,\cdots,x_m$ 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

$f(a_1,a_2,\cdots,a_m)=0$ for all $a_i\in R$.

1. 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.

2. We also observe a randomized test with query complexity $4^mmk$

and constant success probability and a deterministic test with $k^m$

query complexity.