Quantum Search of Spatial Regions

Scott Aaronson (UC Berkeley)

Joint work with Andris Ambainis (U. Latvia)

Grover’s Search Algorithm

Unsorted database of n items

Goal: Find one “marked” item

• Classically, (n) queries to database needed

• Grover 1996: O(n) queries quantumly

• BBBV 1996: Grover’s algorithm is optimal

Great for combinatorial search—but can it help with a physical database?

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!

• 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 workspace z:

| = i,z |vi,z

• Alternate query transforms |vi,z (-1)x(i) |vi,zwith ‘local’ unitaries UWhat does ‘local’ mean? Depends on your religion

What’s the Model?

Defining Locality: 3 Choices(1) Decomposability

U is a product of commuting edgewise operations

(2) Zero pattern of U respects graph

0 1 0 0

0 0 1 0

0 0 0 1

1 0 0 0

(3) Zero pattern of Hamiltonian H respects graph

U = eiH

H has bounded eigenvalues

(1) (2),(3)

Upper bounds work for (1)

Lower bounds for (2),(3)

Whether they’re equivalent is open

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/), database collapses to form a black hole

Makes things harder to retrieve…

Holographic Principle: Best one can do asymptotically is store data on a 2D surface, 1.41069 bits/meter2

So Quantum Mechanics and General Relativityboth yield a n lower bound on search

But can we search a 2D region in less than n steps?Benioff (2001): Guess we can’t…

REVENGE OF COMPUTER SCIENCE• We can.

(n time to move across grid is needed for subroutine anyway)

By adding more levels of recursion, can make running time O(n1/2+)

• Example: Take a classical subroutine that searches a square of size n in n stepsRun n copies in superposition and use Grover O(n3/4)

Can we do better? Say n?

Amplitude AmplificationBrassard, Høyer, Mosca, Tapp 2002

Theorem: If a quantum algorithm has success probability p and returns a certificate, then by invoking it m times, m=O(1/p), we can amplify success probability to (1-m2p/3)m2p

# of Iterations

Success Probability

Diminishing returns

Better to keep prob low & amplify later

• 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 itAmplify n1/11 times

Algorithm for d3 Dimensions

T(n) n1/11(T(n4/5)+O(n1/d)) = O(n5/11)P(n) (1-)n2/11n-1/5P(n4/5) = (n-1/11)

(we show is negligible)

Running Time: Success Prob:

Amplify whole algorithm n1/22 times to get T(n) = O(n1/22n5/11) = O(n), P(n) = (1)

Summary of Boundsd3 d=2

Unique marked item (n) O(n log2n)

k marked items (n / k1/2-1/d) O(n log3n)

Arbitrary graph O(n logcn) n 2O(log n)

Arbitrary graph, h possible marked items

O(h (n/h)1/d logch)

(h (n/h)1/d)

n 2O(log n)

When d=2, time for Grover search matches radius of grid

An arbitrary graph is d-dimensional if for any vertex v, number of vertices at distance r from v is (min{rd,n})

When there are h possible marked items with known locations, the worst case is that they’re evenly scattered

• Razborov 2002: (n)

• 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: O(n log n)

• Høyer, de Wolf 2002: O(n clog*n)

A BState at any time:

Communicating one of 6 directions takes only 3 qubits

Disjointness in O(n) Communication

i,z(A),z(B) |vi,zA |vi,zB

Recent ProgressChilds-Goldstone: Spatial search by quantum walk

O(n5/6) for d=3, O(n log n) for d=4, O(n) for d>4

Running time not competitive with ours in low dimensions, but less memory needed

Ambainis-Kempe: Discrete walk with 2-bit coin

O(n log n) for d=2, O(n) for d3

Connection to Dirac equation?