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Quantum Search of Spatial Regions

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.

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Quantum Search of Spatial Regions

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

  2. 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?

  3. Consider a quantum robot searching a 2D grid: n Robot Marked item n We need n Grover iterations, each of which takes n time, so we’re screwed! What even a dumb computer scientist knows: THE SPEED OF LIGHT IS FINITE

  4. What’s the Model? • 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,z • with ‘local’ unitaries U • What does ‘local’ mean? Depends on your religion

  5. Defining Locality: 3 Choices (2) Zero pattern of U respects graph • Decomposability U is a product of commuting edgewise operations (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

  6. Trouble: Suppose our “hard disk” has mass density  We saw Grover search of a 2D grid presented a problem… So why not pack data in 3 dimensions? Then the complexity would be n  n1/3 = n5/6

  7. 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, 1.41069 bits/meter2 So Quantum Mechanics and General Relativity bothyield a n lower bound on search But can we search a 2D region in less than n steps? Benioff (2001): Guess we can’t…

  8. Revenge of computer science • 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) (n time to move across grid is needed for subroutine anyway) By adding more levels of recursion, can make running time O(n1/2+) REVENGE OF COMPUTER SCIENCE • We can. Can we do better? Say n?

  9. 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 Diminishing returns Success Probability Better to keep prob low & amplify later # of Iterations

  10. Algorithm for d3 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) • 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 item • Else pick a random subcube and run A on it • Amplify n1/11 times

  11. Summary of Bounds 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

  12. Application: Disjointness • Problem: Alice has x1…xn{0,1}n, Bob has y1…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)

  13. Disjointness in O(n) Communication B A State at any time:  i,z(A),z(B) |vi,zA  |vi,zB Communicating one of 6 directions takes only 3 qubits

  14. Recent Progress Childs-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 d3 Connection to Dirac equation?

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