1. Lower Bounds for Local Search by Quantum Arguments Scott Aaronson (UC Berkeley) http://www.cs.berkeley.edu/~aaronson August 14, 2003

2. Quantum Background Needed for This Talk

3. Outline Problem: Find a local minimum of a function using as few function evaluations (queries) as possible Relational adversary method: A quantum method for proving quantum and classical lower bounds on query complexity (only other example: Kerenidis and de Wolf 2003) Applying the method to L OCAL S EARCH Open problems

4. The L OCAL S EARCH Problem Given: undirected connected graph G=(V,E) and function Task: Find a v V such that for all neighbors w of v 3 3 4 3 3 3 2

5. Motivation Why do local search algorithms work so well in practice? Conventional wisdom: Because finding a local optimum is intrinsically not that hard We show this is falseeven for quantum computers Raises a question: Why do exponentially long chains of descending values, as used for lower bounds, almost never occur in real-world problems?

6. Motivation #2 Quantum adiabatic algorithm (Farhi et al.): Quantum analogue of simulated annealing Can sometimes tunnel through barriers to reach global instead of local optima Further strange feature: For function f(x)=|x| on Boolean hypercube {0,1} n, finds minimum 0 n in O(1) queries, instead of O(n) classically We give first example where adiabatic algorithm is provably only polynomially faster than simulated annealing at finding local optima

7. Motivation #3 Megiddo and Papadimitriou defined a complexity class TFNP, of NP search problems for which we know a solution exists Example: Given a circuit that maps {0,1} n to {0,1} n-1, find two inputs that map to same output Papadimitriou: Are TFNP problems good candidates for fast quantum algorithms? My answer: Probably not –Collision lower bound (A 2002): PPP FBQP relative to an oracle (PPP = Polynomial Pigeonhole Principle, FBQP = Function Bounded-Error Quantum Polytime) –This work: PLS FBQP relative to an oracle (PLS = Polynomial Local Search)

8. FNP TFNP PLSPPP FPFBQP

9. Deterministic Query Complexity of L OCAL S EARCH Depends on graph G For an N-vertex line, (log N) 5479 6 Similar for complete binary tree

10. Deterministic Lower Bound Oracle returns decreasing values of f(v), until the set of queried vertices cuts G into 2 pieces Then oracle restricts the problem to largest piece Cuttability tightly characterizes query complexity 8 7 5 6 24 3 1 Llewellyn, Tovey, Trick : (2 n / n) for Boolean hypercube {0,1} n

11. Randomized Query Complexity for any graph with N vertices and max degree d Steepest descent algorithm: - Choosevertices uniformly and query them - Let v 0 be queried vertex with minimum f - Repeatedly let v t+1 be minimum neighbor of v t, until local min is found Claim: Local min is found when whp Proof: At most vertices have smaller f- value than v 0 whp. In that case distance from v 0 to local min in steepest descent tree is at most

12. Randomized Lower Bound Random walk mixes in n log n steps If you havent yet found a v with f(v)<2 n/2, intuitively the best you can do is continue stabbing in the dark Hard to prove! 12 3 56 8 1213 Aldous 1983: 2 n/2-o(n) for hypercube Idea: Pick random start vertex, then take random walk. Label each vertex with 1 st hitting time

13. Quantum Query Complexity O((Nd) 1/3 ) for any graph with N vertices and max degree d Choose (Nd) 2/3 vertices uniformly at random Use Grovers quantum search algorithm to find the v 0 with minimum f-value in time As before, follow v 0 to local min by steepest descent

14. A: Set of 0-inputsB: Set of 1-inputs Choose a function R(f,g) 0 For all f A, g B, and indices v, let Ambainis Adversary Method Most General Version Then quantum query complexity is (1/ geom ) where

15. Example: ( N) for Inverting a Permutation 315624345621315624345621 Let A = set of permutations of {1,…,N} with 1 on left half, B = set with 1 on right half R(f,g)=1 if g obtained from f by swapping the 1, R(f,g)=0 otherwise fgfg (f,2)=1, but (g,2)=2/N (g,6)=1, but (f,6)=2/N

16. Compare to Relational Adversary Method Let A, B, R(f,g), (f,v), (g,v) be as before Then classical randomized query complexity is (1/ min ) where Example: For inverting a permutation, we get (N) instead of ( N)

17. New Lower Bounds for L OCAL S EARCH On Boolean hypercube {0,1} n : quantum queries randomized queries On d-dimensional cube of N vertices (d3): quantum queries randomized queries

18. Modified Problem Starting from the head, follow a snake of L N descending values to the unique local minimum of f, then return an answer bit found there. Clearly a lower bound for this problem implies an equivalent lower bound for L OCAL S EARCH 6 5 4 3 2 1 7 7 8 8 8 9 9 10 11 9 910 11 12 13 10 (Known) Snake Head Snake Tail (contains binary answer) 10 11 12 G

19. Let D be a distribution over snakes (x 0,…,x L-1 ), with x L-1 =h and x i+1 adjacent to x i for all i We say an X drawn from D is -good if the following holds. Choose j uniformly from {0,…,L-1}, and let D X,j be the distribution over snakes Y=(x 0,…,x L-1 ) drawn from D conditioned on x t =y t for all t>j. Then (1) (2) For all vertices v of G, Good Snakes

20. Theorem: Suppose theres a snake distribution D, such that a snake drawn from D is -good with probability at least 9/10. Then the quantum query complexity of L OCAL S EARCH on G is, and the randomized is

21. Sensitivity 4 3 2 1 5 6 7 8 9 10 11 j x0x0 5 4 3 2 1 6 7 8 9 10 11 y0y0 Large (f X,v) but small (f Y,v) Large (f Y,v) but small (f X,v) x L-1 =y L-1 =h

22. Sources of Trouble 2 1 3 4 5 6 7 8 Bunched-Up Snake 2 1 Snake Tails Intersect 3 4 3 2 1 Idea: Just remove inputs that cause trouble! Lemma: Suppose a graph G has average degree k. Then G has an induced subgraph with minimum degree at least k/2.

23. Instead of Aldous random walk, more convenient to define snake distribution D using a coordinate loop Given v {0,1} n, let v (i) = (v with i th bit flipped) Let x 0 = h, x t+1 = x t with ½ probability, x t+1 = x t (t mod n) with ½ probability Mixes completely in n steps Theorem: A snake drawn from D is n 2 /2 n/2 -good with probability at least 9/10 Boolean Hypercube {0,1} n

24. Drawbacks of random walk become more serious: mixing time is too long, too many self-intersections Instead define D by struts of randomly chosen lengths connected at endpoints d-dimensional cube (d3) Theorem: A snake drawn from D is (logN)/N 1/2-1/d - good with probability at least 9/10

25. Open Problems 2 n/4 vs. 2 n/3 gap for quantum complexity on {0,1} n 2 n/2 /n 2 vs. 2 n/2 n gap for randomized complexity 2D square grid Conjecture: Deterministic, randomized, and quantum query complexities are polynomially related for every family of graphs Apply relational adversary method to other problems