Lower Bounds for Collision and Distinctness with Small Range By: Andris Ambainis {medv, Mar 17, 2004

Agenda Introduction Preliminaries Results Conclusion

Introduction

l Given a function l Check if its one-to-one or two-to one l Classical solution is queries l Quantum upper bound [1] is l Quantum low bound [2] is if Collision problem

l Given a function l Check if there are l Quant. low bound [2] is if l Quant. low bound [3] is if Distinctness problem

Preliminaries

Polynomial lower bounds l We can describe by NxM Boolean variables which are 1 if and 0 otherwise l We say that a polynomial P approximates the function if

Polynomial degree – Lemma 1 Lemma 1 [4]: If a quantum algorithm computes φ with bounded error using T queries then there is a polynomial P(y 11,…,y NM ) of degree at most 2T that approximates φ.

l Definition: is symmetric function if for any Symmetric function

Results

New polynomial representation l A new representation of function f: l z =(z 1,…,z M ); z j = #i [N] s.t. f(i)=j l We say that a polynomial Q approximates the function if

l The following two statements are equivalent: 1. There is exists a polynomial Q of degree at most k in approximating 2. There is exists a polynomial P of degree at most k in approximating Lemma 2

Lemma 2 Proof Outline (1  2) For a given y set z j = y 1j + … +y Nj and substitute into Q(z) to obtain P(y) of the same degree

Lemma 2 Proof Outline (2  1) For a given P(y), define Q(z) = E[P(y)] for a random y = (y 11, …,y NM ) consistent with z = (z 1, …,z M ) (i.e., z j = ∑y ij ) It can be shown that Q is a polynomial of the same degree in z 1, …,z M Since φ is symmetric, φ(f) is the same for any f with same z; thus if P(y)≈ φ(f) then Q(z)≈ φ(f)

Theorem 2 (main result) Let φ be symmetric. Let φ’ be restriction of φ to f: [N]  [N]. Then the minimum degree of polynomial P(y 11,…,y NM ) approximating φ is equal to the minimum degree of P’(y 11,…,y NN ) approximating φ’.

Theorem 2 Proof Outline - 1 Obviously, deg(P’ ) ≤ deg(P) For a given P’(y’) construct Q’(z’), then construct Q(z) from Q’(z’), and P(y) from Q(z) Constructing Q from Q’:

Constructing Q from Q’ Since Q ’ is symmetric, it is a sum of symmetric polynomials Q will be the sum of same symmetric polynomials in variables z1, …,zM

1. Consider input function f 2. In at most N are nonzero 3. Consider permutation 4. Such that only the first N elements are non-zero Q approximates φ

Q approximates φ – cont. By construction, Hence Q approximates φ, Q.E.D.

Conclusion

l Low bound for symmetric function already found for is valid for Paper conclusions

Related papers 1. Quantum Algorithm for the Collision Problem Authors: Gilles Brassard, Peter Hoyer, Alain TappGilles BrassardPeter HoyerAlain Tapp 2. Quantum lower bounds for the collision and the element distinctness problems Authors: Yaoyun ShiYaoyun Shi 3. Quantum Algorithms for Element Distinctness Authors: Harry Buhrman, Christoph Durr, Mark Heiligman, Peter Hoyer, Frederic Magniez, Miklos Santha, Ronald de WolfHarry BuhrmanChristoph DurrMark Heiligman Peter HoyerFrederic MagniezMiklos SanthaRonald de Wolf 4. Quantum Lower Bounds by Polynomials Authors: Robert Beals (U of Arizona), Harry Buhrman (CWI), Richard Cleve (U of Calgary), Michele Mosca (U of Oxford), Ronald de Wolf (CWI and U of Amsterdam)Robert BealsHarry BuhrmanRichard CleveMichele MoscaRonald de Wolf