Lower Bounds for Collision and Distinctness with

Small Range

By: Andris Ambainis

{medv, cheskisa}@post.tau.ac.ilMar 17, 2004

Agenda

Introduction Preliminaries Results Conclusion

Introduction

Given a function Check if its one-to-one or two-to

one Classical solution is queries Quantum upper bound [1] is Quantum low bound [2] is if Quantum low bound [2] is if

Collision problem:{1, } {1, }f N M

12( )N

13( )O N

13( )N 3

2NM

14( )N M N

Given a function Check if there are

Quant. low bound [2] is if Quant. low bound [3] is if

Distinctness problem:{1, } {1, }f N M

23( )N 2( )M N12( )N M N

, : , ( ) ( )i j i j f i f j

Preliminaries

Polynomial lower bounds We can describe

by NxM Boolean variables which are 1 if and 0 otherwise

We say that a polynomial P approximates the function if

:{1, } {1, }f N M,i jy

ix j

( ) 1 1 ( ) 1f P y ( ) 0 0 ( )f P y ( ) 0 ( ) 1f P y

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(y11,…,yNM) of degree at most 2T that approximates φ.

Definition: is symmetric function if for any

Symmetric function

, ; ( ) ( )N MS S f f

Results

New polynomial representation

A new representation of function f: z =(z1,…,zM); zj = #i [N] s.t. f(i)=j We say that a polynomial Q

approximates the function if ( ) 1 1 ( ) 1f Q z ( ) 0 0 ( )f Q z ( ) 0 ( ) 1f Q z

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

1 2, ... Mz z z

1,1 1,2 ,, ... N My y y

Lemma 2 Proof Outline (1 2)

For a given y set zj = y1j + …+yNj 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 = (y11,…,yNM) consistent with z = (z1,…,zM) (i.e., zj = ∑yij )

It can be shown that Q is a polynomial of the same degree in z1,…,zM

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(y11,…,yNM) approximating φ is equal to the minimum degree of P’(y11,…,yNN) 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

1 1,...,[ ]

' l

l l

j

ccc c i i

i N

Q z z

1. Consider input function f2. In at most N are

nonzero3. Consider permutation

4. Such that only the first N elements are non-zero

Q approximates φ

1 2, ... Mz z z

, ; ( ) ( )MS f f f f

Q approximates φ – cont. By construction,

Hence Q approximates φ, Q.E.D.

1 2 1 2( , ,... ,0,0...0) ( , ,... )N NQ z z z Q z z z

Conclusion

Low bound for symmetric function already found for is valid for

Paper conclusions

M NM N

Related papers

1. Quantum Algorithm for the Collision Problem Authors: Gilles Brassard , Peter Hoyer , Alain Tapp

2. Quantum lower bounds for the collision and the element distinctness problems Authors: Yaoyun Shi

3. Quantum Algorithms for Element Distinctness Authors: Harry Buhrman, Christoph Durr, Mark Heiligman, Peter Hoyer, Frederic Magniez, Miklos Santha, Ronald 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)