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    Quantum Query Complexity of Multilinear Identity Testing

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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=∠r1,...,rkR=\angle{r_1,...,r_k} where {r1,r2,...,rk}\{r_1,r_2,...,r_k\} is an additive generating set for RR and a multilinear polynomial f(x1,...,xm)f(x_1,...,x_m) over RR also accessed as a black-box function f:Rmβ†’Rf:R^m\to R (where we allow the indeterminates x1,...,xmx_1,...,x_m to be commuting or noncommuting), we study the problem of testing if ff is an \emph{identity} for the ring RR. More precisely, the problem is to test if f(a1,a2,...,am)=0f(a_1,a_2,...,a_m)=0 for all ai∈Ra_i\in R. We give a quantum algorithm with query complexity O(m(1+Ξ±)m/2kmm+1)O(m(1+\alpha)^{m/2} k^{\frac{m}{m+1}}) assuming kβ‰₯(1+1/Ξ±)m+1k\geq (1+1/\alpha)^{m+1}. Towards a lower bound, we also discuss a reduction from a version of mm-collision to this problem. We also observe a randomized test with query complexity 4mmk4^mmk and constant success probability and a deterministic test with kmk^m query complexity.Comment: 12 page
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