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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
f
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):
/ 2
2
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”