Quantum Algorithms II

1. Quantum Algorithms II Andrew C. Yao Tsinghua University & Chinese U. of Hong Kong

2. Outline of Talk I. Introduction II. Element Distinctness & Sorting III. Quantum Algorithm -- element distinctness IV. Lower Bounds -- sorting problems V. Conclusions

4. Quantum Mechanics More generally • A quantum state is a unit vector u in CN • A measurement is a family of orthogonal subspaces (V1, V2 , …,Vk): u = u1+u2 +…+uk will be measured as uiwith prob |ui|2 • A computation step is a unitary operator (a ‘rotation’) • A Quantum Algorithm specifies > an initial quantum state > a sequence of unitary operators > a final measurement.

5. Classical algorithm Quantum Algorithm

6. I. Introduction • Shor (‘94): Fast factorization of integers • Grover (‘96): Searching n-item list in time Exactly one is 1, all other are 0 Problem: Find j

7. I. Introduction • Hilbert space with base vectors -- Start with -- Use a unitary operator to compute (where ) • Grover’s Theorem:

8. I. Introduction • Initially, • After t steps, • Take measurement in base the probability of seeing |j> is

9. Random Walk Hitting j by classical random walks takes nsteps

10. Quantum Random Walk Hitting j by quantum random walks takes steps

11. I. Introduction Optimality Theorem(Bernstein et al ‘97): Any quantum algorithm must use steps to locate j Question:What other problems can be speeded up with quantum algorithms? This talk: Progress on sorting-like problems -- Element Distinctness Problem -- Sorting Problem

12. I. Introduction • Element Distinctness: Given decide whether thereexist such that • Sorting: Given distinct determine such that • For conventional computers, essentially same problem (solvable in time ). • For quantum computers, very different...

13. II. Element Distinctness & Sorting • Theorem 1. Element Distinctness can be solved in quantum steps. *Buhrman et al ‘00 * Ambainis ‘03 • Theorem 2. Any quantum algorithm for Element Distinctness must use time. * Aaronson & Shi ‘04

14. II. Element Distinctness & Sorting • Sortingcan be done in time classically. • Unlike Element Distinctness, it cannot be speeded up by quantum algorithms. Theorem 3. Any quantum algorithm for Sorting must use time. * Hoyer, Neerbek and Shi ’02 • Will illustrate Element Distinctness upper bound & Sorting lower bound.

15. F1 F2 Fj Fn III. Element Distinctness Quantum Algorithm for Element Distinctness • First, Grover’s list-searching implies: 0 0 1 1 0 Computing takes only evals. • Def: Element x in S is called a repeater if x = some x’ Question:Any repeater in S ?

16. III. Element Distinctness Quantum Algorithm for Element Distinctness 1. Choose k, divide Sintogroups of size k 2. Design Repeateri: Output 1 iff some x in Si is a repeater

17. III. Element Distinctness Repeater1: Any repeater in ? 1. Sort ; if there are equal elements, halt and return 1. 2. Use Grover to decide if some outside = some in ; if yes, return 1 * Phase 1 takes time * Phase 2 takes quantum time to compute

18. III. Element Distinctness Quantum algorithm for Element Distinctness: Use Grover to evaluate

19. IV. Sorting for Partial Order P P: partial order on n objects • Given inputconsistent with P Determine such that • The standard sorting problem is the special case P = empty

20. IV. Sorting Problem for Partial Order P • e(P): # of linear extensions consistent with P Information bound: Upper bound:[Fredman ‘75] [Kahn & Saks ‘91] • Quantum Complexity Theorem 4 (Yao ‘04) constants

21. Proof Outline A.Quantum Partial Order Problem B.P=empty: Review of Proof C.ReducingA to a Combinatorial Problem D.Graph Entropy Chain Polytope

22. A. Quantum Partial Order Problem • = Linear extensions consistent withP Thus e(P) = | | • Quantum algorithm S for partial order P computes the function • Note is a “partial function”

23. Quantum Decision Tree • Initial state • Unitary Operators • Final measurement M With prob >,Mapplied togives the correct

24. B. Review of Case • An entropy type argument (extending Ambainis) • At any time, the current state has an entropy (uncertainty). Initially, it has L=n log nbits of entropy. After the sorting, it should have 0 entropy. If each operation can reduce that entropy by at most , then the number of operations must be at least

25. B. Review of Case[Hoyer, Neerbek, Shi ‘02] • For input x, after step j, the quantum state is • Initially all inputs x have states • In the end, every pair x, y have nearly orthogonal states • Define weight functionw(x, y) forinputs x, y • Find lower bound to and upper bound to for any algorithm

26. B. Review of Case • Let Lemma 1 Lemma 2

27. Extension to • Use the natural extension of their approach • Need to showa lower boundL to

28. C. Reduced to a Combinatorial Problem (*) • This is a purely combinatorial problem -- recall • (*) can be proved by using combinatorial and information-theoretic tools developed by Korner (‘73), Stanley (‘86), Kahn & Kim (‘95).

29. D. Graph Entropy, Chain Polytope, etc. Proof outline for • The natural entropy for a partial order P would be simply log e(P). • Kahn and Kim (‘95) definedH(P),an alternative entropy (based on Korner’s graph entropy), and showedH(P) = (log e(P)) • Our proof uses Stanley’schain polytope theory to show that

30. V. Conclusions • Quantum Complexity -- many open questions • Integer factorization has classical polynomial time algorithm? • Graph isomorphism has quantum polynomial time algorithm? • Rich source of problems: eg quantum complexity for computing graph properties.