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Leibniz international proceedings in informatics

Quantum Query Complexity of Multilinear Identity Testing

V. Arvind, Partha Mukhopadhyay

Published Jan 1, 2009• 3 citations
Publisher

Abstract

Motivated by the quantum algorithm for testing commutativity of black-box groups (Magniez and Nayak, 2007), we study the following problem: Given a black-box finite ring by an additive generating set and a multilinear polynomial over that ring, also accessed as a black-box function (we allow the indeterminates of the polynomial to be commuting or noncommuting), we study the problem of testing if the polynomial is an \emph{identity} for the given ring. We give a quantum algorithm with query complexity sub-linear in the number of generators for the ring, when the number of indeterminates of the input polynomial is small (ideally a constant). Towards a lower bound, we also show a reduction from a version of the collision problem (which is well studied in quantum computation) to a variant of this problem.

DOI
10.4230/lipics.stacs.2009.1801

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Motivated by the quantum algorithm for testing commutativity of black-box groups (Magniez and Nayak, 2007), we study the following problem: Given a black-box finite ring by an additive generating set and a multilinear polynomial over that ring, also accessed as a black-box function (we allow the indeterminates of the polynomial to be commuting or noncommuting), we study the problem of testing if the polynomial is an \emph{identity} for the given ring. We give a quantum algorithm with query complexity sub-linear in the number of generators for the ring, when the number of indeterminates of the input polynomial is small (ideally a constant). Towards a lower bound, we also show a reduction from a version of the collision problem (which is well studied in quantum computation) to a variant of this problem.
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}R=∠r1​,...,rk​ where {r1,r2,...,rk}\{r_1,r_2,...,r_k\}{r1​,r2​,...,rk​} is an additive generating set for RRR and a multilinear polynomial f(x1,...,xm)f(x_1,...,x_m)f(x1​,...,xm​) over RRR also accessed as a black-box function f:Rm→Rf:R^m\to Rf:Rm→R (where we allow the indeterminates x1,...,xmx_1,...,x_mx1​,...,xm​ to be commuting or noncommuting), we study the problem of testing if fff is an \emph{identity} for the ring RRR. More precisely, the problem is to test if f(a1,a2,...,am)=0f(a_1,a_2,...,a_m)=0f(a1​,a2​,...,am​)=0 for all ai∈Ra_i\in Rai​∈R. We give a quantum algorithm with query complexity O(m(1+α)m/2kmm+1)O(m(1+α)^{m/2} k^{\frac{m}{m+1}})O(m(1+α)m/2km+1m​) assuming k≥(1+1/α)m+1k\geq (1+1/α)^{m+1}k≥(1+1/α)m+1. Towards a lower bound, we also discuss a reduction from a version of mmm-collision to this problem. We also observe a randomized test with query complexity 4mmk4^mmk4mmk and constant success probability and a deterministic test with kmk^mkm query complexity.
We present quantum query complexity bounds for testing algebraic properties. For a set S and a binary operation on S, we consider the decision problem whether SSS is a semigroup or has an identity element. If S is a monoid, we want to decide whether S is a group. We present quantum algorithms for these problems that improve the best known classical complexity bounds. In particular, we give the first application of the new quantum random walk technique by Magniez, Nayak, Roland, and Santha that improves the previous bounds by Ambainis and Szegedy. We also present several lower bounds for testing algebraic properties.
We obtain the strongest separation between quantum and classical query complexity known to date -- specifically, we define a black-box problem that requires exponentially many queries in the classical bounded-error case, but can be solved exactly in the quantum case with a single query (and a polynomial number of auxiliary operations). The problem is simple to define and the quantum algorithm solving it is also simple when described in terms of certain quantum Fourier transforms (QFTs) that have natural properties with respect to the algebraic structures of finite fields. These QFTs may be of independent interest, and we also investigate generalizations of them to noncommutative finite rings.
We examine the number T of queries that a quantum network requires to compute several Boolean functions on {0,1}^N in the black-box model. We show that, in the black-box model, the exponential quantum speed-up obtained for partial functions (i.e. problems involving a promise on the input) by Deutsch and Jozsa and by Simon cannot be obtained for any total function: if a quantum algorithm computes some total Boolean function f with bounded-error using T black-box queries then there is a classical deterministic algorithm that computes f exactly with O(T^6) queries. We also give asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings. Finally, we give new precise bounds for AND, OR, and PARITY. Our results are a quantum extension of the so-called polynomial method, which has been successfully applied in classical complexity theory, and also a quantum extension of results by Nisan about a polynomial relationship between randomized and deterministic decision tree complexity.
We consider the problem of testing the commutativity of a black-box group specified by its k generators. The complexity (in terms of k) of this problem was first considered by Pak, who gave a randomized algorithm involving O(k) group operations. We construct a quite optimal quantum algorithm for this problem whose complexity is in O (k^{2/3}). The algorithm uses and highlights the power of the quantization method of Szegedy. For the lower bound of Omega(k^{2/3}), we give a reduction from a special case of Element Distinctness to our problem. Along the way, we prove the optimality of the algorithm of Pak for the randomized model.
We examine the number T of queries that a quantum network requires to compute several Boolean functions on {0,1}/sup N/ in the black-box model. We show that, in the black-box model, the exponential quantum speed-up obtained for partial functions (i.e. problems involving a promise on the input) by Deutsch and Jozsa and by Simon cannot be obtained for any total function: if a quantum algorithm computes some total Boolean function f with bounded-error using T black-box queries then there is a classical deterministic algorithm that computes f exactly with O(T/sup 6/) queries. We also give asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings. Finally, we give new precise bounds for AND, OR, and PARITY. Our results are a quantum extension of the so-called polynomial method, which has been successfully applied in classical complexity theory, and also a quantum extension of results by Nisan about a polynomial relationship between randomized and deterministic decision tree complexity.