Quantum Search of Spatial Regions Scott Aaronson (UC Berkeley) Joint work with Andris Ambainis (IAS...
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Transcript of Quantum Search of Spatial Regions Scott Aaronson (UC Berkeley) Joint work with Andris Ambainis (IAS...
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
Marked item
Robot
n
n
Consider a quantum robot searching a 2D grid:
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
|000
Initial Superposition
|001 |101|100|011|010
|000
Amplitude of Solution State Inverted
|001 |101|100
|011|010
|000
All Amplitudes Inverted About Mean
|001 |101|100|011|010
Talk Outline
• The Physics of Databases
• Algorithm for Space Search
• Application: Disjointness Protocol
• Open Problems
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
We saw Grover search of a 2D grid presented a problem…
Once radius exceeds Schwarzschild bound of (1/), hard disk collapses to form a black hole
Makes things harder to retrieve…
But we care about entropy, not mass
Actually worse—even a 2D hard disk would collapse once radius exceeds (1/)!
1D hard disk would not collapse…
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 1.41069 bits per meter2 of surface area
So Quantum Mechanics and General Relativityboth 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 c3 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
Can we do anything better?
Benioff (2001): Guess we can’t…
Back to the Main Issue
Classical search takes (n) timeQuantum search takes (rn)
(r = maximum radius of region)
REVENGE OF COMPUTER SCIENCE
• We can.
Using amplitude amplification techniques of BHMT’2002, we get:
O(n log3n) for 2D grid
O(n) for 3 and higher dimensions
• Idea: Recursively divide into sub-squares
• 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:
• Alternate query transforms with ‘local’ unitariesWhat does ‘local’ mean? Depends on your religion
, ,i z iv z , 1 ,ix
i iv z v z
What’s the Model?
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
(3) Take U=eiH where H has eigenvalues of absolute value at most , and if (i,j)E then
(1) (2),(3). Upper bounds will work for (1); lower bounds for (2),(3)
, , 0i z j zU
, , 0i z j zH
• 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
Amplitude AmplificationBrassard, Høyer, Mosca, Tapp 2002
2
22 11 2 1
3
mm
• 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 itemElse pick a random subcube and run A on itRepeat n1/11 times using amplitude amplification
• Running time:
1/11 4/5 1/
5/11
dT n n T n O n
O n
In More Detail: d3
• Success probability (unamplified):
• With amplification:
(since is negligible)
• Amplify whole algorithm n1/22 times to get
1/5 4/5P n n P n
d3 (continued)
2/11 1/5 4 /5
1/11
1P n n n P n
n
1/ 22 5/111 ,P n T n O n n O n
• 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:
(best we could get)
d=2
2log
log log
nT n O n
n
• If exactly r marked items:
for d3. Basically optimal:
• If at least r marked items, can use “doubling trick” of BBHT’98 to get same bound for d3. For d=2 we get
Multiple Marked Items
1/ 2 1/ d
nT n O d
r
/ 2 1/ 2 1/2d d
nT n
r
3log
log log
nT n O n
n
• 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
• Main idea: Build tree of subgraphs bottom-up
Search on Irregular Graphs
log
log , 2
2 , 2O n
T n O npoly n d
T n n d
• If G is >2-dimensional, and has h possible marked items (whose locations are known), then
• Intuitively: Worst case is when bits are scattered uniformly in G
Bits Scattered on a Graph
1/
logd
nT n O h poly h
h
• Razborov 2002:
• Problem: Alice has x1…xn{0,1}n, Bob has y1…yn
They want to know if xiyi=1 for some i
Application: Disjointness
• How many qubits must they communicate?
• Buhrman, Cleve, Wigderson 1998: logO n n
• Høyer, de Wolf 2002: log*nO nc
n
A B
, , , ,A Bi z z i A i Bv z v z
State at any time:
Communicating one of 6 directions takes only 3 qubits
Disjointness in O(n) Communication
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 #2Here’s a graph of diameter n that takes (n3/4) time to search (by BBBV’96 hybrid argument):
Does it also take (n3/4) time to decide if every row of a 2D grid has a marked item?
n
Starfish
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?
2D Turing machine
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 nn worktape that it can’t do with a nn worktape?
What about a quantum TM?
2D Turing machine
Related to Feige’s embedding problem: Given n checkers on an nn checkerboard, can we move them to an O(n)O(n) board so that no 2 checkers become farther apart in L1 distance?
• No fundamental obstacle to quantum speedup for search of physical regions
Conclusions
• 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