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