Lower Bounds for Collision and Distinctness with Small Range

1. Lower Bounds for Collision and Distinctness with Small Range By: Andris Ambainis {medv, cheskisa}@post.tau.ac.il Mar 17, 2004

2. Agenda • Introduction • Preliminaries • Results • Conclusion

3. Introduction

4. Collision problem • 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

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

6. Preliminaries

7. 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

8. 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.

9. Symmetric function • Definition: is symmetric function if for any

10. Results

11. 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

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

13. Lemma 2 Proof Outline (1  2) • For a giveny set zj = y1j + …+yNjand substitute into Q(z) to obtain P(y) of the same degree

14. 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)

15. 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’.

16. 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’:

17. 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

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

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

20. Conclusion

21. Paper conclusions • Low bound for symmetric function already found for is valid for

22. Related papers • Quantum Algorithm for the Collision Problem Authors:Gilles Brassard , Peter Hoyer , Alain Tapp • Quantum lower bounds for the collision and the element distinctness problems Authors:Yaoyun Shi • Quantum Algorithms for Element Distinctness Authors:Harry Buhrman, Christoph Durr, Mark Heiligman, Peter Hoyer, Frederic Magniez, Miklos Santha, Ronald de Wolf • 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)