Lower Bounds for Collision and Distinctness with Small Range By: Andris Ambainis {medv,...
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Transcript of Lower Bounds for Collision and Distinctness with Small Range By: Andris Ambainis {medv,...
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Lower Bounds for Collision and Distinctness with
Small Range
By: Andris Ambainis
{medv, cheskisa}@post.tau.ac.ilMar 17, 2004
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Agenda
Introduction Preliminaries Results Conclusion
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Introduction
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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
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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
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Preliminaries
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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
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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.
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Definition: is symmetric function if for any
Symmetric function
, ; ( ) ( )N MS S f f
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Results
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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
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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
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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
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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)
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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’.
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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’:
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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
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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
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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
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Conclusion
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Low bound for symmetric function already found for is valid for
Paper conclusions
M NM N
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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)