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.
For any finite ring (R, +, •) the ring R[x 1 , x 2 , • • • , x m ] is the ring of polynomials in commuting variables x 1 , x 2 , • • • , x m and coefficients in R. The ring R{x 1 , x 2 , • • • , x m } is the ring of polynomials where the indeterminates x i are noncommuting. By noncommuting variables, we mean x i x j -x j x i = 0 for i = j.
For the algorithmic problem we study in this paper, we assume that that the elements of the ring (R, +, •) are uniformly encoded by binary strings of length n and R = r 1 , r 2 , • • • , r k is given by an additive generating set {r 1 , r 2 , • • • , r k }. That is,
Also, the ring operations of R are performed by black-box oracles for addition and multiplication that take as input two strings encoding ring elements and output their sum or product (as the case may be). Additionally, we assume that the zero element of R is encoded by some fixed string. We now define the problem which we study in this paper.
The input to the problem is a black-box ring R = r 1 , • • • , r k given by an additive generating set, and a multilinear polynomial f (x 1 , • • • , x m ) (in the ring R[x 1 , • • • , x m ] or the ring R{x 1 , • • • , x m }) that is also given by black-box access. The problem is to test if f is an 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 ∈ R.
A natural example of an instance of this problem is the bivariate polynomial f (x 1 , x 2 ) = x 1 x 2 -x 2 x 1 over the ring R{x 1 , x 2 }. This is an identity for R precisely when R is a commutative ring. Clearly, it suffices to check if the generators commute with each other which gives a naive algorithm that makes O(k 2 ) queries to the ring oracles.
Given a polynomial f (x 1 , • • • , x m ) and a black-box ring R by generators, we briefly recall some facts about the complexity of checking if f = 0 is an identity for R. The problem can be NP-hard when the number of indeterminates m is unbounded, even when R is a fixed ring. To see this, notice that a 3-CNF formula F (x 1 , • • • , x n ) can be expressed as a O(n) degree multilinear polynomial f (x 1 , x 2 , • • • , x n ) over F 2 , by writing F in terms of addition and multiplication over F 2 . It follows that f = 0 is an identity for F 2 if and only if F is an unsatisfiable formula.
We remark that a closely related problem is Polynomial Identity Testing (PIT). For PIT we ask whether the polynomial f (x 1 , • • • , x m ) is the zero polynomial, which is a stronger property. To see the difference, consider a standard example: For a prime p, notice that x p -x = 0 is an identity for F p but x p -x is not a zero polynomial in F p [x]. However, when the ring R is a field F and the degree of f is smaller than the size of the field F then the two problems coincide as a consequence of the Schwartz-Zippel lemma [Sch80,Zip79]. More precisely, f = 0 is an identity for F if and only if f is the zero polynomial.
When f is given by an arithmetic circuit then PIT is known to be in randomized polynomial time over fields [Sch80,Zip79] and even finite commutative rings with unity [AB03,AMS08]. This is quite unlike MIT which can be NP-hard for polynomials over small fields as already observed above.
On the other hand, when f is given by black-box access as a function f : R m → R then there is no way to distinguish between the problems PIT and MIT. Algorithmically, they coincide.
Over the years, Polynomial Identity Testing has emerged as an important algorithmic problem [AB03,KI03]. Due to its significance in complexity theory, PIT has been actively studied in recent years [DS06,KS07,RS05].
In this paper we focus on the query complexity of multilinear identity testing (MIT). In our query model, each ring operation, which is performed by a query to the ring oracle, is of unit cost. Furthermore, we consider each evaluation of f (a 1 , • • • , a m ) to be of unit cost for a given input (a 1 , • • • , a m ) ∈ R m . This model is reasonable because we consider m as a parameter that is much smaller than k.
Our goal is to find upper and lower bounds on the query complexity for the problem. We are interested in the query complexity for both classical and quantum computation. The main motivation for our study is a result of Magniez and Nayak in [MN05], where the authors study the quantum query complexity of group commutativity testing: Let G be a finite black-box group given by a generating set g 1 , g 2 , • • • , g k and group operations are performed by a group oracle. The algorithmic task is to check if G is commutative. For this problem the authors in [MN05] give a quantum algorithm with query complexity O(k 2/3 log k) and time complexity O(k 2/3 log 2 k). Furthermore a Ω(k 2/3 ) lower bound for the quantum query complexity is also shown. The main technical tool for their upper bound result was a method of quantization of random walks first showed by Szegedy [Sze04]. More recently, Magniez et al in [MNRS07] discovered a s
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