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 $O(m(1+α)^{m/2} k^{\frac{m}{m+1}})$ assuming $k\geq (1+1/α)^{m+1}$. 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 $4^mmk$ and constant success probability and a deterministic test with $k^m$ query complexity.
We give deterministic black-box polynomial identity testing algorithms for multilinear read-once oblivious algebraic branching programs (ROABPs), in n^(lg^2 n) time. Further, our algorithm is oblivious to the order of the variables. This is the first sub-exponential time algorithm for this model. Furthermore, our result has no known analogue in the model of read-once oblivious boolean branching programs with unknown order, as despite recent work there is no known pseudorandom generator for this model with sub-polynomial seed-length (for unbounded-width branching programs). This result extends …
A read-once oblivious arithmetic branching program (ROABP) is an arithmetic branching program (ABP) where each variable occurs in at most one layer. We give the first polynomial time whitebox identity test for a polynomial computed by a sum of constantly many ROABPs. We also give a corresponding blackbox algorithm with quasi-polynomial time complexity $n^{O(\log n)}$. In both the cases, our time complexity is double exponential in the number of ROABPs. ROABPs are a generalization of set-multilinear depth-$3$ circuits. The prior results for the sum of constantly many set-multilinear depth-$3$ c…
We show an equivalence between 1-query quantum algorithms and representations by degree-2 polynomials. Namely, a partial Boolean function $f$ is computable by a 1-query quantum algorithm with error bounded by $ε<1/2$ iff $f$ can be approximated by a degree-2 polynomial with error bounded by $ε'<1/2$. This result holds for two different notions of approximation by a polynomial: the standard definition of Nisan and Szegedy and the approximation by block-multilinear polynomials recently introduced by Aaronson and Ambainis (STOC'2015, arxiv:1411.5729). We also show two results for polynomials of h…
The Aaronson-Ambainis conjecture (Theory of Computing '14) says that every low-degree bounded polynomial on the Boolean hypercube has an influential variable. This conjecture, if true, would imply that the acceptance probability of every $d$-query quantum algorithm can be well-approximated almost everywhere (i.e., on almost all inputs) by a $\mathrm{poly}(d)$-query classical algorithm. We prove a special case of the conjecture: in every completely bounded degree-$d$ block-multilinear form with constant variance, there always exists a variable with influence at least $1/\mathrm{poly}(d)$. In a …
Motivated by the Hadamard product of matrices we define the Hadamard product of multivariate polynomials and study its arithmetic circuit and branching program complexity. We also give applications and connections to polynomial identity testing. Our main results are the following. 1. We show that noncommutative polynomial identity testing for algebraic branching programs over rationals is complete for the logspace counting class $\ceql$, and over fields of characteristic $p$ the problem is in $\ModpL/\Poly$. 2.We show an exponential lower bound for expressing the Raz-Yehudayoff polynomial as t…
Read-$k$ oblivious algebraic branching programs are a natural generalization of the well-studied model of read-once oblivious algebraic branching program (ROABPs). In this work, we give an exponential lower bound of $\exp(n/k^{O(k)})$ on the width of any read-$k$ oblivious ABP computing some explicit multilinear polynomial $f$ that is computed by a polynomial size depth-$3$ circuit. We also study the polynomial identity testing (PIT) problem for this model and obtain a white-box subexponential-time PIT algorithm. The algorithm runs in time $2^{\tilde{O}(n^{1-1/2^{k-1}})}$ and needs white box a…
In (Kabanets, Impagliazzo, 2004) it is shown how to decide the circuit polynomial identity testing problem (CPIT) in deterministic subexponential time, assuming hardness of some explicit multilinear polynomial family for arithmetical circuits. In this paper, a special case of CPIT is considered, namely low-degree non-singular matrix completion (NSMC). For this subclass of problems it is shown how to obtain the same deterministic time bound, using a weaker assumption in terms of determinantal complexity. Hardness-randomness tradeoffs will also be shown in the converse direction, in an effort to…
This paper is our second step towards developing a theory of testing monomials in multivariate polynomials. The central question is to ask whether a polynomial represented by an arithmetic circuit has some types of monomials in its sum-product expansion. The complexity aspects of this problem and its variants have been investigated in our first paper by Chen and Fu (2010), laying a foundation for further study. In this paper, we present two pairs of algorithms. First, we prove that there is a randomized $O^*(p^k)$ time algorithm for testing $p$-monomials in an $n$-variate polynomial of degree …
We present a deterministic algorithm which computes the multilinear factors of multivariate lacunary polynomials over number fields. Its complexity is polynomial in $\ell^n$ where $\ell$ is the lacunary size of the input polynomial and $n$ its number of variables, that is in particular polynomial in the logarithm of its degree. We also provide a randomized algorithm for the same problem of complexity polynomial in $\ell$ and $n$. Over other fields of characteristic zero and finite fields of large characteristic, our algorithms compute the multilinear factors having at least three monomials of …
We study the problem of obtaining deterministic black-box polynomial identity testing algorithms (PIT) for algebraic branching programs (ABPs) that are read-once and oblivious. This class has an deterministic white-box polynomial identity testing algorithm (due to Raz and Shpilka), but prior to this work there was no known such black-box algorithm. The main result of this work gives the first quasi-polynomial sized hitting sets for size S circuits from this class, when the order of the variables is known. As our hitting set is of size exp(lg^2 S), this is analogous (in the terminology of boole…
We present a single, common tool to strictly subsume all known cases of polynomial time blackbox polynomial identity testing (PIT) that have been hitherto solved using diverse tools and techniques. In particular, we show that polynomial time hitting-set generators for identity testing of the two seemingly different and well studied models - depth-3 circuits with bounded top fanin, and constant-depth constant-read multilinear formulas - can be constructed using one common algebraic-geometry theme: Jacobian captures algebraic independence. By exploiting the Jacobian, we design the first efficien…
Using ideas from automata theory we design a new efficient (deterministic) identity test for the \emph{noncommutative} polynomial identity testing problem (first introduced and studied in \cite{RS05,BW05}). We also apply this idea to the reconstruction of black-box noncommuting algebraic branching programs. Assuming the black-box model allows us to query the ABP for the output at any given gate, we can reconstruct an (equivalent) ABP in deterministic polynomial time. Finally, we explore commutative identity testing when the coefficients of the input polynomial come from an arbitrary finite com…
We consider the problem of exact identification for read-once functions over arbitrary Boolean bases. We introduce a new type of queries (subcube identity ones), discuss its connection to previously known ones, and study the complexity of the problem in question. Besides these new queries, learning algorithms are allowed to use classic membership ones. We present a technique of modeling an equivalence query with a polynomial number of membership and subcube identity ones, thus establishing (under certain conditions) a polynomial upper bound on the complexity of the problem. We show that in som…
In this paper we study algebraic branching programs (ABPs) with restrictions on the order and the number of reads of variables in the program. Given a permutation $π$ of $n$ variables, for a $π$-ordered ABP ($π$-OABP), for any directed path $p$ from source to sink, a variable can appear at most once on $p$, and the order in which variables appear on $p$ must respect $π$. An ABP $A$ is said to be of read $r$, if any variable appears at most $r$ times in $A$. Our main result pertains to the identity testing problem. Over any field $F$ and in the black-box model, i.e. given only query access to t…
We investigate the computational complexity of the discrete logarithm, the computational Diffie-Hellman and the decisional Diffie-Hellman problems in some identity black-box groups G_{p,t}, where p is a prime number and t is a positive integer. These are defined as quotient groups of vector space Z_p^{t+1} by a hyperplane H given through an identity oracle. While in general black-box groups with unique encoding these computational problems are classically all hard and quantumly all easy, we find that in the groups G_{p,t} the situation is more contrasted. We prove that while there is a polynom…
We prove a characterization of $t$-query quantum algorithms in terms of the unit ball of a space of degree-$2t$ polynomials. Based on this, we obtain a refined notion of approximate polynomial degree that equals the quantum query complexity, answering a question of Aaronson et al. (CCC'16). Our proof is based on a fundamental result of Christensen and Sinclair (J. Funct. Anal., 1987) that generalizes the well-known Stinespring representation for quantum channels to multilinear forms. Using our characterization, we show that many polynomials of degree four are far from those coming from two-que…
Let X = (x_0,...,x_{n-1})$ be a sequence of n numbers. For ε> 0, we say that x_i is an ε-approximate median if the number of elements strictly less than x_i, and the number of elements strictly greater than x_i are each less than (1+ε)n/2. We consider the quantum query complexity of computing an ε-approximate median, given the sequence X as an oracle. We prove a lower bound of Ω(\min{{1/ε},n}) queries for any quantum algorithm that computes an ε-approximate median with any constant probability greater than 1/2. We also show how an ε-approximate median may be computed with O({1/ε}\log({1\/ε}) \…
We study the problem of obtaining efficient, deterministic, black-box polynomial identity testing algorithms for depth-3 set-multilinear circuits (over arbitrary fields). This class of circuits has an efficient, deterministic, white-box polynomial identity testing algorithm (due to Raz and Shpilka), but has no known such black-box algorithm. We recast this problem as a question of finding a low-dimensional subspace H, spanned by rank 1 tensors, such that any non-zero tensor in the dual space ker(H) has high rank. We obtain explicit constructions of essentially optimal-size hitting sets for ten…
Cornell University（康奈尔大学）；Stanford University（斯坦福大学）；Columbia University（哥伦比亚大学）；Simons Institute for the Theory of Computing（西蒙斯计算理论研究所）
A recent model for property testing of probability distributions (Chakraborty et al., ITCS 2013, Canonne et al., SICOMP 2015) enables tremendous savings in the sample complexity of testing algorithms, by allowing them to condition the sampling on subsets of the domain. In particular, Canonne, Ron, and Servedio (SICOMP 2015) showed that, in this setting, testing identity of an unknown distribution $D$ (whether $D=D^\ast$ for an explicitly known $D^\ast$) can be done with a constant number of queries, independent of the support size $n$ -- in contrast to the required $Ω(\sqrt{n})$ in the standar…
We consider the problem of $\textit{subgroup testing}$ for a quantum circuit $C$: given access to $C$, determine whether it implements a unitary that is $a$-close or $b$-far from a subgroup $\mathcal{G}$ of the unitary group. It encompasses the problem of exact testing, property testing and tolerant testing. In this work, we study these problems with the group $\mathcal{G}$ as the trivial subgroup (i.e. identity testing) or the Pauli or Clifford group and their $q$-ary extension, and a $\textit{promise}$ version of these problems where $C$ is promised to be in some subgroup of the unitaries th…
We prove the following conjecture, raised by Aaronson and Ambainis in 2008: Let $f:\{-1,1\}^n \rightarrow [-1,1]$ be a multilinear polynomial of degree $d$. Then there exists a variable $x_i$ whose influence on $f$ is at least $\mathrm{poly}(\mathrm{Var}(f)/d)$. As was shown by Aaronson and Ambainis, this result implies the following well-known conjecture on the power of quantum computing, dating back to 1999: Let $Q$ be a quantum algorithm that makes $T$ queries to a Boolean input and let $ε,δ> 0$. Then there exists a deterministic classical algorithm that makes $\mathrm{poly}(T,1/ε,1/δ)$ que…
