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

Quantum Background Needed for This Talk Quantum Background Needed for This Talk

Outline • Problem: Find a local minimum of a function using as few function 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 LOCAL SEARCH • Open problems

The LOCAL SEARCH Problem • Given: undirected connected graph G=(V, E) and function • The LOCAL SEARCH Problem • Given: undirected connected graph G=(V, E) and function • Task: Find a v V such that for all neighbors w of v 3 2 3 3 4

Motivation • Why do local search algorithms work so well in practice? • Conventional 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 false—even 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?

Motivation #2 • Quantum adiabatic algorithm (Farhi et al. ): Quantum analogue of simulated 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

Motivation #3 • Megiddo and Papadimitriou defined a complexity class TFNP, of NP search 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)

FNP TFNP PLS PPP FP FBQP FNP TFNP PLS PPP FP FBQP

Deterministic Query Complexity of LOCAL SEARCH • Depends on graph G • For an Deterministic Query Complexity of LOCAL SEARCH • Depends on graph G • For an N-vertex line, (log N) 5 4 6 • Similar for complete binary tree 7 9

Deterministic Lower Bound • Llewellyn, Tovey, Trick : (2 n/ n) for Boolean hypercube Deterministic Lower Bound • Llewellyn, Tovey, Trick : (2 n/ n) for Boolean hypercube {0, 1}n 4 5 2 3 7 6 1 8 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

Randomized Query Complexity • for any graph with N vertices and max degree d Randomized Query Complexity • for any graph with N vertices and max degree d • Steepest descent algorithm: - Choose vertices uniformly and query them - Let v 0 be queried vertex with minimum f - Repeatedly let vt+1 be minimum neighbor of vt, 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

Randomized Lower Bound • Aldous 1983: 2 n/2 -o(n) for hypercube • Idea: Pick Randomized Lower Bound • 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 8 13 3 12 1 6 2 5 Random walk mixes in n log n steps If you haven’t 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!

Quantum Query Complexity • O((Nd)1/3) for any graph with N vertices and max degree 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 Grover’s 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

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

Example: ( N) for Inverting a Permutation Let A = set of permutations of Example: ( N) for Inverting a Permutation 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 f 3 1 5 6 2 4 g 3 4 5 6 2 1 (f, 2)=1, but (g, 2)=2/N (g, 6)=1, but (f, 6)=2/N

Relational Adversary Method Let A, B, R(f, g), (f, v), (g, v) be as 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 Compare to Example: For inverting a permutation, we get (N) instead of ( N)

New Lower Bounds for LOCAL SEARCH On Boolean hypercube {0, 1}n: • quantum queries New Lower Bounds for LOCAL SEARCH On Boolean hypercube {0, 1}n: • quantum queries • randomized queries On d-dimensional cube of N vertices (d≥ 3): • quantum queries • randomized queries

Modified Problem 8 (Known) Snake Head 7 9 10 3 11 G 12 8 Modified Problem 8 (Known) Snake Head 7 9 10 3 11 G 12 8 9 4 6 10 11 2 13 5 7 9 10 11 12 1 Snake Tail 8 (contains 9 10 11 binary answer) 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 LOCAL SEARCH

Good Snakes Let D be a distribution over snakes (x 0, …, x. L-1), Good Snakes Let D be a distribution over snakes (x 0, …, x. L-1), with x. L -1=h and xi+1 adjacent to xi 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 DX, j be the distribution over snakes Y=(x 0, …, x. L-1) drawn from D conditioned on xt=yt for all t>j. Then (1) (2) For all vertices v of G,

Theorem: Suppose there’s a snake distribution D, such that a snake drawn from D Theorem: Suppose there’s 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 LOCAL SEARCH on G is , and the randomized is

1 x 0 3 Large (f. X, v) but small (f. Y, v) 2 1 x 0 3 Large (f. X, v) but small (f. Y, v) 2 5 4 7 4 3 j 6 5 Large (f. Y, v) but small (f. X, v) 8 9 1 y 0 2 10 11 x. L-1=y. L-1=h

Sources of Trouble Bunched-Up Snake 1 3 7 4 8 1 6 2 5 Sources of Trouble Bunched-Up Snake 1 3 7 4 8 1 6 2 5 Snake Tails Intersect 1 2 3 3 4 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.

Boolean Hypercube {0, 1}n Instead of Aldous’ random walk, more convenient to define snake Boolean Hypercube {0, 1}n 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 bit flipped) Let x 0 = h, xt+1 = xt with ½ probability, xt+1 = xt(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

d-dimensional cube (d≥ 3) Drawbacks of random walk become more serious: mixing time is d-dimensional cube (d≥ 3) 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 Theorem: A snake drawn from D is (log. N)/N 1/2 -1/dgood with probability at least 9/10

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