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 AchieveIf 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 IssueClassical 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
3m
m
• 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 lognT n O nn
• 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 dr
/ 2 1/ 2 1/2d d
nT nr
3log
log lognT n O nn
• 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/
logdnT 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 #1Can 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 #3Cosmological 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