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Quantum algorithms with polynomial speedups

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Quantum algorithms with polynomial speedups Powered By Docstoc
					Quantum algorithms with
 polynomial speedups
       Andris Ambainis
      University of Latvia
      Search [Grover, 1996]

 ?        ?       ?       ?      ...
                                       ?

 N objects;
 Find an object with a certain property.
       Conventional computer: N objects
       Quantum computer: √N objects
     Useful for many computational tasks
Hamiltonian cycles
              Does this graph have a
               cycle that contains every
               vertex?

              Hamiltonian cycles
               are:
                 Easy to verify;

                 Hard to find (too

                  many possibilities).

NP-complete problem.
            Quantum algorithm

      ?        =


   N objects = N possible cycles.
   Classical search: N steps.
   Quantum search: √N steps.
Search by a quantum
        walk
            Search by random walk
                        Search space with a
        1                structure.
                        Task: find a state with
2                3
                         some property.
                        Walk randomly,
    4            5       according to a rule that
                         uses the structure of
             6
                         search space.
                    Quantum walk
                        A quantum particle moving around
        1                this state space.

2               3       Quantum walk with two transition
                         rules:
                            “usual” for unmarked vertices;
    4           5
                            “special” for marked.

            6
                               Particle “drifts” toward
                               the marked states.
      Random vs. quantum walks
   [Szegedy, 2004] If a classical random walk* finds
    a marked state in T steps, a quantum walk finds
    it in O(√T) steps.
   Generalizes Grover’s search by using the
    structure of the search space.



*that satisfies some constraints.
     Element distinctness (A, 2004)
28   12   18 76 96 82 94 99 21 78   88   93   39   44   64
32   99   70 18 94 82 92 64 95 46   53   16   35   42   72
31   40   75 71 93 32 47 11 70 37   78   79   36   63   40
69   92   71 28 85 41 80 10 52 63   88   57   43   84   67
57   31   98 39 65 74 24 90 26 83   60   91   27   96   35
20   26   52 95 65 66 97 54 30 62   79   33   84   50   38
49   20   47 24 54 48 98 23 41 16   66   75   38   13   58
56   86   34 73 61 73 21 44 62 34   14   51   74   76   83
37   90   58 13 find 29 equal
          Task: 10 25 two25 56 68   12   11   51   23   77
68   72   43 69 46 87 97 45 59 14   30   19   81   81   49
60   85   numbers. 89 67 89 29
          80 50 61 59               86   48   22   15   17
55   36   27 42 55 77 19 45 15 53   22   91   87   17   33
       Element distinctness
31 40 75 71 93 32      Classical: N steps.
47 11 70 37 78 79      Quantum: ~N2/3 steps.
36 63 40 48 98 23
41 16 66 75 38 27
42 55 77 19 45 15
53 22 91 37 90 58
13 10 25 29 25 56
68 12 11 51 23 77
15 17
Triangle finding [Magniez,
   Santha, Szegedy, 03]
           Graph G with n vertices.
           n2 variables xij; xij=1 if there is
            an edge (i, j).
           Does G contain a triangle?
           Classically: O(n2).
           Quantum: O(n1.3).
Matrix multiplication [Buhrman,
          Špalek, 05]
   A, B, C – n*n matrices.
   Given A, B and C, we can test AB=C in:
     O(n2) steps by a probabilistic algorithm;
     O(n5/3) steps by a quantum algorithm.
     Quantum simulated annealing
     [Somma, Boixo, Barnum, 2008]
   Simulated annealing is a general heuristic for
    solving optimization problems.
   [Somma, Boixo, Barnum, 2008]: quantum
    version of simulated annealing, with a quadratic
    speedup.
Quantum speedups are
   very common
Evaluating Boolean
     formulas
                  [Farhi et al., 07]
                               AND-OR formula of size M.
                               Variables accessed by queries: ask
                                i, get xi.
           AND


     OR               OR


x1        x2     x3        x4
                   Motivation
                  Vertices = chess positions;
                  Leaves = final positions;
     OR
                  xi=1 if the 1st player wins;
x1        x2      At internal vertices, AND/OR
                   evaluates whether the player who
                   makes the move can win.


How well can we play chess if we only
know the position tree?
                            Results
                               Full binary tree of depth d.
                               N=2d leaves.
                               Deterministic: (N).
           AND                 Randomized [SW,S]: (N.753…).
                               Quantum?
     OR               OR
                                        FGG: O(√N)
x1        x2     x3        x4           quantum algorithm
    [Farhi, Goldstone, Gutmann]:


           0   1   1    0




…         …                 …                  …

                       Infinite line in two directions
[Farhi, Goldstone, Gutmann]:
               Basis states |v, v – vertices
                of augmented tree.
                Hamiltonian H, H-
                adjacency matrix of
                augmented tree.


…          …
    [Farhi, Goldstone, Gutmann]:
                   Starting state | on the infinite
                    line left of tree.
                   Apply Hamiltonian for O(N)
                    time.
                   If T=1, the new state is on the
                    right (transmission).
                   If T=0, the new state is on the left
…                … (reflection).

    Proof: reflection coefficients of the tree.
       Next steps
[A, Childs, Reichardt, Špalek, Zhang,
                2007]
                Improvement I

                      AND

                            OR
      OR

 x1        x2         AND             AND

                 x3         x4   x5    x6   x7


Quantum algorithm for arbitrary formulas
              Our result
              query quantum algorithm
               for any size-N formula.



Quantum speedups for anything
that can be expressed by logic formulas
          Improvement II

 [Farhi, Goldstone, Gutmann]:

O(N) time Hamiltonian quantum algorithm

 O(N1/2+o(1)) query quantum algorithm

We design discrete query algorithm directly

            Useful for CS applications
                Loose end III
   FGG algorithm uses scattering theory and looks
    very different from the previous quantum
    algorithms.
   Our work: relations to search, amplitude
    amplification.
   New understanding of FGG.
Two reflections
       [Aharonov, 98] Analysis of
        Grover’s algorithm;
       Other applications:
         Amplitude amplification;
         Quantization of Markov
          chains.
         Now: logic formulas.
    Beyond logic formulas [Reichardt,
             Špalek, 2008]
   Input x1, ..., xN  vectors v1, ..., vM.
   Output F(x1, ..., xN) = 1 if there are vi1,vi2, ..., vik
    v=vi1+vi2+...+vik.


            Span program with witness size T


            O(√T) query quantum algorithm
Span programs [Reichardt, Špalek,
             2008]

      Logic formula of size T


     Span program with witness size T


     O(√T) query quantum algorithm
Span programs [Reichardt, 2009]

  Span program with witness size T



             
  O(√T) query quantum algorithm
    Adversary bound [A, 2001, Hoyer,
           Lee, Špalek, 2007]
   Boolean function f(x1, ..., xN);
   Inputs x = (x1, ..., xN);
   Theorem If there is a matrix A: A[x, y]≠0 only if
    f(x) ≠ f(y), then computing f requires



    quantum queries
Span programs [Reichardt, 2009]

     Optimal span program

   Semidefinite program (SDP)

          Dual SDP

       Optimal adversary bound
 Big question #1
What other problems have quantum
            speedups?
  Big question #2
What properties of a problem imply a
        quantum speedup?
              Structural results
   [Beals et al., 1998] Let f(x1, ..., xN) – total
    Boolean function. Then,
                       D(f) ≤ Q6(f).
   [Aaronson, A, 2008] Let f(x1, ..., xN) – symmetric
    function. Then,
                       R(f) ≤ Q9(f).
   R(f), Q(f) – number of variables that should be
    evaluated by classical/quantum algorithm.
               Open question
   Other classes of problems for which speedup is
    at most polynomial?

				
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