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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 23 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Lower Bounds for Local Search by Quantum Arguments Scott Aaronson UC Berkeley IAS

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

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

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

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Main Open Problem Are deterministic, randomized, and quantum query complexities of L OCAL S EARCH polynomially related for every family of graphs? Santha & Szegedy, this STOC

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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

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Trivial Observations Complete Graph on N Vertices (N) randomized queries ( N) quantum queries

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

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

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How to find a local minimum in queries (d = maximum degree) Query vertices uniformly at random Quantumly, O(N 1/3 d 1/6 ) queries suffice In the above algorithm, find v using Grover search 43 35 29 13 48 17 4 22 9 30 32 1 Let v be the queried vertex for which f(v) is minimal Follow v to a local minimum by steepest descent

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Ambainis Quantum Adversary Theorem Then the number of quantum queries needed to separate 0- from 1-inputs w.h.p. is (1/p), where 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.

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Example: Inverting a Permutation R(, )=1 if is obtained from by a swap, R(, )=0 otherwise 45172386 but (,x)=2/N Decide whether 1 is on left half (0-input) or right half (1-input) so ( N) quantum queries needed (,x)=1

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

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

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

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Given a 0-input f, how do we create a random 1-input g thats close to f? 1 23 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Starting from the pivot, generate a new tail using (say) a random walk

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

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

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

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What We Get For Boolean hypercube {0,1} n : For d-dimensional cube N 1/d N 1/d (d 3): randomized,quantum randomized,quantum

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Conclusions Local optima arent 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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Chapter 11 Limitations of Algorithm Power Copyright © 2007 Pearson Addison-Wesley. All rights reserved.

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