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Lower Bounds for Local Search by Quantum Arguments

11. 12. 10. 13. 9. 14. 18. 15. 8. 17. 16. 2. 3. 1. 4. 7. 3. 2. 5. 4. 1. 6. 14. 6. 5. 18. 15. 13. 17. 16. 7. 12. 8. 11. 9. 10. Lower Bounds for Local Search by Quantum Arguments. Scott Aaronson UC Berkeley  IAS.

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Lower Bounds for Local Search by Quantum Arguments

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  1. 11 12 10 13 9 14 18 15 8 17 16 2 3 1 4 7 3 2 5 4 1 6 14 6 5 18 15 13 17 16 7 12 8 11 9 10 Lower Bounds for Local Search by Quantum Arguments Scott Aaronson UC Berkeley  IAS

  2. Can quantum ideas help us prove new classical results? Quantum Generosity…Giving back because we careTM Examples:Kerenidis & de Wolf 2003Aharonov & Regev 2004

  3. LOCAL SEARCH Given a graph G=(V,E) and oracle access to a function f:V{0,1,2,…}, find a local minimum of f—a vertex v such that f(v)f(w) for all neighbors w of v. Use as few queries to f as possible 4 4 2 3 5

  4. Results First quantum lower bound for LOCAL SEARCH:On Boolean hypercube {0,1}n,any quantum algorithm needs (2n/4/n) queries to find a local min Better classical lower bound via a quantum argument: Any randomized algorithm needs (2n/2/n2) queries to find a local min on {0,1}nPrevious bound: 2n/2-o(n) (Aldous 1983)Upper bound: O(2n/2n) First randomized or quantum lower bounds for LOCAL SEARCH on constant-dimensional hypercubes

  5. Main Open Problem Santha & Szegedy, this STOC Are deterministic, randomized, and quantum query complexities of LOCAL SEARCH polynomially related for every family of graphs?

  6. Motivation • Why is optimization hard? Are local optima the only reason? • Quantum adiabatic algorithm (Farhi et al. 2000): What are its limitations? • Papadimitriou 2003: Can quantum computers help solve total function problems? PPADS PODN PPP PLS

  7. Trivial Observations Complete Graph on N Vertices(N) randomized queries(N) quantum queries

  8. Trivial Observations Line Graph on N VerticesO(log N) deterministic queries suffice So interesting graphs are of intermediate connectedness…

  9. Boolean Hypercube {0,1}n Aldous 1983: Any randomized algorithm needs 2n/2-o(n) queries to find local min Proof uses complicated random walk analysis

  10. Query vertices uniformly at random 30 22 17 13 43 29 1 2 3 9 35 48 4 How to find a local minimum in queries (d = maximum degree) Let v be the queried vertex for which f(v) is minimal Follow v to a local minimum by steepest descent Quantumly, O(N1/3d1/6) queries suffice In the above algorithm, find v using Grover search

  11. Ambainis’ Quantum Adversary Theorem Given: 0-inputs, 1-inputs, and function R(A,B)0 that measures the “closeness” of 0-input A to 1-input B For all 0-inputs A and query locations x, let (A,x) be probability that A(x)B(x), where B is a 1-input chosen with probability proportional to R(A,B).Define (B,x) similarly. Then the number of quantum queries needed to separate 0- from 1-inputs w.h.p. is (1/p), where

  12. so (N) quantum queries needed Decide whether ‘1’ is on left half (0-input) or right half (1-input) Example: Inverting a Permutation (,x)=1 4 5 1 7 2 3 8 6 but (,x)=2/N R(,)=1 if  is obtained from  by a swap, R(,)=0 otherwise

  13. We prove an analogue of the quantum adversary theorem for classical randomized query complexity Statement is identical, except is replaced by Yields up to quadratically better bound—e.g. (N) instead of (N) for permutation problem Proof Idea: Show that each query can separate only so many input pairs 0-inputs 1-inputs

  14. b{0,1} All vertices of G not in the snake just lead to the head To apply the lower bound theorems to LOCAL SEARCH, we use “snakes” 11 12 10 13 Known “head” vertex 9 14 Length  N 18 15 8 17 16 7 3 2 4 1 6 Unique local minimum of f 5 To get a decision problem, we put an “answer bit” at the local minimum

  15. Choose a “pivot” vertex uniformly at random on the snake Given a 0-input f, how do we create a random 1-input g that’s “close” to f? 11 12 10 13 9 14 18 15 8 17 16 7 3 2 4 1 6 5

  16. Given a 0-input f, how do we create a random 1-input g that’s “close” to f? 2 3 1 4 5 14 6 18 15 13 17 16 7 12 8 11 9 10 Starting from the pivot, generate a new “tail” using (say) a random walk

  17. (f,v)=1 but (g,v)1/N (g,v)=1 but (f,v)1/N Randomized lower bound: Quantum lower bound: Handwaving Argument For all vertices vG, either (f,v) or (g,v) should be at most ~1/N (as in the permutation problem) f g

  18. The Nontrivial Stuff Need to prevent snake tails from intersecting, spending too much time in one part of the graph, … Solutions: (1) Generalize quantum adversary method to work with most inputs instead of all

  19. The Nontrivial Stuff Need to prevent snake tails from intersecting, spending too much time in one part of the graph, … Solutions: (2) Use a “coordinate loop” instead of a random walk.It mixes faster and has fewer self-intersections

  20. What We Get For Boolean hypercube {0,1}n: randomized, quantum For d-dimensional cube N1/dN1/d (d3): randomized, quantum

  21. Conclusions • Local optima aren’t the only reason optimization is hard • Total function problems: below NP, but still too hard for quantum computers? • “The Unreasonable Effectiveness of Quantum Lower Bound Methods”

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