Transcript Quantum Search of Spatial Regions

Quantum Search of Spatial Regions
Scott Aaronson (UC Berkeley)
Joint work with Andris Ambainis (IAS / U. Latvia)
Grover’s O(n) Quantum Search Algorithm:
Great for combinatorial search
But can it help search a physical region?
Why is a computer scientist asking such a thing?
What even a dumb computer scientist knows:
THE SPEED OF LIGHT IS FINITE
Consider a quantum robot searching a 2D grid:
Robot
n
n
Marked item
We need n Grover iterations, each of which
takes n time, so we’re screwed!
Grover’s Algorithm
Unsorted database
of n items
Goal: Find one
“marked” item
• Classically, order n queries to database needed
• Grover 1996: Quantum algorithm using order
n queries
• BBBV 1996: Grover’s algorithm is optimal
Initial Superposition
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Amplitude of Solution State Inverted
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All Amplitudes Inverted About Mean
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Talk Outline
•
The Physics of Databases
•
Algorithm for Space Search
•
Application: Disjointness Protocol
•
Open Problems
We saw Grover search of a 2D grid presented a problem…
So why not pack data in 3 dimensions?
Then the complexity would be n  n1/3 = n5/6
Trouble: Suppose our “hard disk” has mass density 
Once radius exceeds Schwarzschild bound of
(1/), hard disk collapses to form a black hole
Makes things harder to retrieve…
Actually worse—even a 2D hard disk would
collapse once radius exceeds (1/)!
1D hard disk would not collapse…
But we care about entropy, not mass
A ball of radiation of radius r has energy (r) but
entropy (r3/2)
Holographic Principle: A region of space can’t store
more than 1.41069 bits per meter2 of surface area
So Quantum Mechanics and General Relativity
both yield a n lower bound on search
If space had d>3 dimensions, then relativity
bound would be weaker: n1/(d-1)
Is that bound achievable? Apparently not, since
even stronger limit (Bekenstein’s) applies for
weakly-gravitating systems
What We Will Achieve
If n ~ rc bits are scattered in a 3D ball of radius r
(where c3 and bits’ locations are known), search
time is (n1/c+1/6)
(up to polylog factor)
For “radiation disk” (n ~ r3/2): (n5/6) = (r5/4)
For n ~ r2 (saturating holographic bound):
(n2/3) = (r4/3)
To get O(n polylog n), bits would need to be
concentrated on a 2D surface
Objections to the Model
(1)Would need n parallel computing elements to
maintain a quantum database
Response: Might have n “passive elements,” but
many fewer “active elements” (i.e. robots), which
we wish to place in superposition over locations
(2) Must consider effects of time dilation
Response: For upper bounds, will have in mind
weakly-gravitating systems, for which time
dilation is by at most a constant factor
Back to the Main Issue
Classical search takes (n) time
Quantum search takes (rn)
(r = maximum radius of region)
Can we do anything better?
Benioff (2001): Guess we can’t…
REVENGE OF COMPUTER SCIENCE
• We can.
• Idea: Recursively divide into sub-squares
Using amplitude
amplification techniques
of BHMT’2002, we get:
O(n log3n) for 2D grid
O(n) for 3 and higher
dimensions
What’s the Model?
• Undirected connected graph G=(V,E)
• Bit xi at each vertex vi
• Goal: Compute some Boolean f(x1…xn){0,1}
• State can have arbitrary ancilla z:
  i, z vi , z
• Alternate query transforms vi , z   1
with ‘local’ unitaries
xi
What does ‘local’ mean? Depends on your religion
vi , z
Defining Locality: 3 Choices
(1) Unitary must be decomposable into commuting
local operations, each acting on a single edge
(2) Just don’t “send amplitude” between non-adjacent
vertices: if (i,j)E then
U i, z  j ,z  0
(3) Take U=eiH where H has eigenvalues of absolute
value at most , and if (i,j)E then
H i, z  j ,z  0
(1)  (2),(3). Upper bounds will work for (1); lower
bounds for (2),(3)
Amplitude Amplification
Brassard, Høyer, Mosca, Tapp 2002
• Generalization of Grover search
If a quantum algorithm has success probability ,
then by invoking it 2m+1 times (m=O(1/)), we
can make the success probability
  2m  1 
2
   2m  1 
1 


3


2
In More Detail: d3
• Assume there’s a unique marked item
• Divide into n1/5 subcubes, each of size n4/5
• Algorithm A:
If n=1, check whether you’re at a marked item
Else pick a random subcube and run A on it
Repeat n1/11 times using amplitude amplification
• Running time:
T n  n
1/11
T  n   O  n  
 O  n5/11 
4/5
1/ d
d3 (continued)
• Success probability (unamplified):
P n  n
1/ 5
P n
4/5

• With amplification:
P  n   1    n
 n
2/11 1/ 5
1/11
n
P n
4/ 5

 (since  is negligible)
• Amplify whole algorithm n1/22 times to get
P  n    1 , T  n   O  n
1/ 22
n
5/11
  O n
d=2
• Here diameter of grid (n) exactly matches time
for Grover search
• So we have to recurse more, breaking into
squares of size n/log n
• Running time suffers correspondingly:

log n 
T  n  O  n

log log n 

2
(best we could get)
Multiple Marked Items
• If exactly r marked items:

n
T  n   O  d 1/ 21/ d
 r



for d3. Basically optimal:

n
T  n     d / 2 1/ 21/ d
2 r



• If at least r marked items, can use “doubling trick” of
BBHT’98 to get same bound for d3. For d=2 we get

log3 n 
T  n  O  n

log log n 

Search on Irregular Graphs
• Our algorithm can be adapted to any graph with good
expansion properties (not just hypercubes)
• Say G is d-dimensional if for any v, number of vertices
at distance r from v is (min{rd,n})
• Can search in time
T n  O


n poly log n ,
T n  n 2
O

log n

,
d 2
d 2
• Main idea: Build tree of subgraphs bottom-up
Bits Scattered on a Graph
• If G is >2-dimensional, and has h possible
marked items (whose locations are known), then
1/ d


n
T  n   O  h   poly log h 


h




• Intuitively: Worst case is when bits are scattered
uniformly in G
Application: Disjointness
• Problem: Alice has x1…xn{0,1}n, Bob has y1…yn
They want to know if xiyi=1 for some i
• How many qubits must they communicate?
• Buhrman, Cleve, Wigderson 1998: O
• Høyer, de Wolf 2002:
• Razborov 2002:

O
 n

nc log*n


n log n

Disjointness in O(n) Communication
B
A
State at any time:

i , zA , zB
vi , zA  vi , zB
Communicating one of 6 directions takes only 3
qubits
Open Problem #1
Can a quantum walk search a 2D grid efficiently?
(Maybe even n time instead of n log3n?)
Promising numerical evidence (courtesy N. Shenvi)
Open Problem #2
Here’s a graph of diameter n that takes (n3/4) time to
search (by BBBV’96 hybrid argument):
n
Does it also take (n3/4) time to decide if every row of a
2D grid has a marked item?
Open Problem #3
Cosmological constant   10-122 > 0
(type-Ia supernova observations)
Number of bits accessible to any one observer is at
most 3/ (Bousso 2000, Lloyd 2002)
How many of those ~10123 bits could a computer
“use” before they recede past its horizon?
Our result shows a quantum computer could
search more of the bits than a classical one
But what about using them as memory?
Open Problem #3 (con’t)
Consider a “2D Turing machine” with O(n) time, a
square worktape, and a separate input tape
Is there anything it can do with an nn worktape
that it can’t do with a nn worktape?
What about a quantum TM?
Related to Feige’s embedding problem: Given n
checkers on an nn checkerboard, can we move
them to an O(n)O(n) board so that no 2
checkers become farther apart in L1 distance?
Conclusions
• No fundamental obstacle to quantum speedup for
search of physical regions
• We should look for other “pure” CS theory
questions inspired by laws of physics
Quantum computing is just one example
Not all strings have n bits