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Lower Bounds for Local Search by Quantum Arguments

Scott Aaronson

UC Berkeley IAS

Quantum Generosity…

Giving back because we careTM

Can quantum ideas help us prove new classical results?

Examples:Kerenidis & de Wolf 2003Aharonov & Regev 2004

LOCAL SEARCHGiven 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

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ResultsFirst 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}n

Previous bound: 2n/2-o(n) (Aldous 1983)Upper bound: O(2n/2n)

First randomized or quantum lower bounds for LOCAL SEARCH on constant-dimensional hypercubes

Main Open Problem

Are deterministic, randomized, and quantum query complexities of LOCAL SEARCH polynomially related for every family of graphs?

Santha & Szegedy, this STOC

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

Trivial Observations

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

Trivial Observations

So interesting graphs are of intermediate connectedness…

Line Graph on N VerticesO(log N) deterministic queries suffice

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

How to find a local minimum in queries (d = maximum degree)

O Nd

Query vertices uniformly at random

Nd

Quantumly, O(N1/3d1/6) queries suffice

In the above algorithm, find v using Grover search

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Let v be the queried vertex for which f(v) is minimal

Follow v to a local minimum by steepest descent

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.

, , : , 0,max , ,

A B x R A B A x B xp A x B x

Example: Inverting a Permutation

R(,)=1 if is obtained from by a swap, R(,)=0 otherwise

4 5 1 7 2 3 8 6but (,x)=2/N

Decide whether ‘1’ is on left half (0-input) or right half (1-input)

2max , ,x x

N

so (N) quantum queries needed

(,x)=1

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

min , , ,A x B x , ,A x B x

Yields up to quadratically better bound—e.g. (N) instead of (N) for permutation problem

0-in

pu

ts

1-inp

uts

To apply the lower bound theorems to LOCAL SEARCH, we use “snakes”

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Known “head” vertex

Unique local minimum of fAll 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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Given a 0-input f, how do we create a random 1-input g that’s “close” to f?

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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?

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Starting from the pivot, generate a new “tail” using (say) a random walk

Handwaving ArgumentFor all vertices vG, either (f,v) or (g,v) should be at most ~1/N (as in the permutation problem)

Quantum lower bound:

Randomized lower bound: 1

min , , ,N

f v g v

1/ 41

, ,N

f v g v

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g

(f,v)=1 but (g,v)1/N

(g,v)=1 but (f,v)1/N

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:

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:

What We Get

For Boolean hypercube {0,1}n:

For d-dimensional cube N1/dN1/d (d3):

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2n

n

/ 42n

n

randomized, quantum

1/ 2 1/

log

dN

N

1/ 2 1/

log

dN

N

randomized, quantum

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”