Quantum Search of Spatial Regions Scott Aaronson (UC Berkeley) Joint work with Andris Ambainis (U. Latvia)

Grovers 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: Grovers algorithm is optimal Great for combinatorial searchbut 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 were screwed!

Undirected connected graph G=(V,E) Bit x i at each vertex v i Goal: Compute some Boolean f(x 1 …x n ) {0,1} State can have arbitrary workspace z: | = i,z |v i,z Alternate query transforms |v i,z (-1) x(i) |v i,z with local unitaries U What does local mean? Depends on your religion Whats the Model?

Defining Locality: 3 Choices (1)Decomposability U is a product of commuting edgewise operations (2) Zero pattern of U respects graph (3) Zero pattern of Hamiltonian H respects graph U = e iH H has bounded eigenvalues (1) (2),(3) Upper bounds work for (1) Lower bounds for (2),(3) Whether theyre equivalent is open

So why not pack data in 3 dimensions? Then the complexity would be n n 1/3 = n 5/6 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, bits/meter 2 So Quantum Mechanics and General Relativity both yield a n lower bound on search But can we search a 2D region in less than n steps? Benioff (2001): Guess we cant…

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(n 1/2+ ) Example: Take a classical subroutine that searches a square of size n in n steps Run n copies in superposition and use Grover O(n 3/4 ) Revenge of computer science Can we do better? Say n?

Amplitude Amplification Brassard, 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-m 2 p/3)m 2 p # of Iterations Success Probability Diminishing returns Better to keep prob low & amplify later

Assume theres a unique marked item Divide into n 1/5 subcubes, each of size n 4/5 Algorithm A: If n=1, check whether youre at a marked item Else pick a random subcube and run A on it Amplify n 1/11 times Algorithm for d 3 Dimensions T(n) n 1/11 (T(n 4/5 )+O(n 1/d )) = O(n 5/11 ) P(n) (1- )n 2/11 n -1/5 P(n 4/5 ) = (n -1/11 ) (we show is negligible) Running Time: Success Prob: Amplify whole algorithm n 1/22 times to get T(n) = O(n 1/22 n 5/11 ) = O( n),P(n) = (1)

Summary of Bounds d 3 d=2 Unique marked item ( n)O( n log 2 n) k marked items ( n / k 1/2-1/d )O( n log 3 n) Arbitrary graph O( n log c n) n 2 O( log n) Arbitrary graph, h possible marked items O( h (n/h) 1/d log c h) ( h (n/h) 1/d ) n 2 O( 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{r d,n}) When there are h possible marked items with known locations, the worst case is that theyre evenly scattered

Razborov 2002: ( n) Problem: Alice has x 1 …x n {0,1} n, Bob has y 1 …y n They want to know if x i y i =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 c log*n )

A B State at any time: Communicating one of 6 directions takes only 3 qubits Disjointness in O( n) Communication i,z(A),z(B) |v i,z A |v i,z B

Recent Progress Childs-Goldstone: Spatial search by quantum walk O(n 5/6 ) 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 d 3 Connection to Dirac equation?