Quantum Search of Spatial Regions Scott Aaronson UC

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 Consider a quantum robot searching a 2 D grid: Robot n n Marked item We need n Grover iterations, each of which takes n time, so we’re screwed!

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

Defining Locality: 3 Choices (1) Decomposability 1. (2) Zero pattern of U respects graph 0 0 0 1 U is a product of commuting edgewise operations 1. (3) Zero pattern of Hamiltonian H respects graph 2. U = ei. H 3. H has bounded 1 0 0 0 0 1 0 1. (1) (2), (3) 2. Upper bounds work for (1) 3. Lower bounds for (2), (3) 4. Whether they’re

We saw Grover search of a 2 D grid presented a problem… 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

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 2 D surface, 1. 4 1069 bits/meter 2 So Quantum Mechanics and General Relativity both yield a n lower bound on search But can we search a 2 D region in less than n steps? Benioff (2001): Guess we can’t…

REVENGE OF COMPUTER SCIENCE • We can. • 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) ( 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+ ) 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 Diminishing returns Success Probability Better to keep prob low & amplify later # of Iterations

Algorithm for d 3 Dimensions • Assume there’s a unique marked item • Divide into n 1/5 subcubes, each of size n 4/5 • Algorithm A: If n=1, check whether you’re at a marked item Else pick a random subcube and run A on it Amplify n 1/11 times Running Time: T(n) n 1/11(T(n 4/5)+O(n 1/d)) = O(n 5/11) Success Prob: P(n) (1 - )n 2/11 n-1/5 P(n 4/5) = (n-1/11) (we show is negligible) 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 k marked items ( n) ( n / k 1/2 -1/d) O( n log 2 n) O( n log 3 n) Arbitrary graph O( n logcn) n 2 O( log n) Arbitrary graph, h O( h (n/h)1/d logch) n 2 O( log n) possible marked items ( h (n/h)1/d) 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

Application: Disjointness • Problem: Alice has x 1…xn {0, 1}n, Bob has y 1…yn They want to know if xiyi=1 for some i • How many qubits must they communicate? • Buhrman, Cleve, Wigderson 1998: O( n log n) • Høyer, de Wolf 2002: O( n clog*n) • Razborov 2002: ( n)

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

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?