Quantum algorithms with polynomial speedups

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Quantum algorithms with polynomial speedups

• Andris Ambainis
• University of Latvia

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Search Grover, 1996
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...

• N objects
• Find an object with a certain property.

Conventional computer N objects Quantum
computer vN objects
Useful for many computational tasks
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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.
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Quantum algorithm
?

• N objects N possible cycles.
• Classical search N steps.
• Quantum search vN steps.

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Search by a quantum walk
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Search by random walk

• Search space with a structure.
• Task find a state with some property.
• Walk randomly, according to a rule that uses the
  structure of search space.

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5
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Quantum walk

• A quantum particle moving around this state
  space.
• Quantum walk with two transition rules
• usual for unmarked vertices
• special for marked.

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Particle drifts toward the marked states.
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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(vT) steps.
• Generalizes Grovers search by using the
  structure of the search space.
• that satisfies some constraints.

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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 10 25 29 25 56 68 12 11
51 23 77 68 72 43 69 46 87 97 45 59
14 30 19 81 81 49 60 85 80 50 61 59
89 67 89 29 86 48 22 15 17 55 36 27
42 55 77 19 45 15 53 22 91 87 17 33
Task find two equal numbers.
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Element distinctness

• 31 40 75 71 93 32 47 11 70 37 78 79
  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

• Classical N steps.
• Quantum N2/3 steps.

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Triangle finding Magniez, Santha, Szegedy, 03

• Graph G with n vertices.
• n2 variables xij xij1 if there is an edge (i,
  j).
• Does G contain a triangle?
• Classically O(n2).
• Quantum O(n1.3).

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Matrix multiplication Buhrman, Špalek, 05

• A, B, C nn matrices.
• Given A, B and C, we can test ABC in
• O(n2) steps by a probabilistic algorithm
• O(n5/3) steps by a quantum algorithm.

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Quantum simulated annealingSomma, 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.

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Quantum speedups are very common
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Evaluating Boolean formulas
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Farhi et al., 07

• AND-OR formula of size M.
• Variables accessed by queries ask i, get xi.

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Motivation

• Vertices chess positions
• Leaves final positions
• xi1 if the 1st player wins
• 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?
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Results

• Full binary tree of depth d.
• N2d leaves.
• Deterministic ?(N).
• Randomized SW,S ?(N.753).
• Quantum?

FGG O(vN) quantum algorithm
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Farhi, Goldstone, Gutmann
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Farhi, Goldstone, Gutmann

• Basis states v?, v vertices of augmented tree.
• Hamiltonian H, H-adjacency matrix of augmented
  tree.

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Farhi, Goldstone, Gutmann

• Starting state ?? on the infinite line left of
  tree.
• Apply Hamiltonian for O(?N) time.
• If T1, the new state is on the right
  (transmission).
• If T0, the new state is on the left
  (reflection).

Proof reflection coefficients of the tree.
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Next steps

• A, Childs, Reichardt, Špalek, Zhang, 2007

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Improvement I
Quantum algorithm for arbitrary formulas
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Our result

• query quantum algorithm
  
• for any size-N formula.

Quantum speedups for anything that can be
expressed by logic formulas
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Improvement II
Farhi, Goldstone, Gutmann
O(?N) time Hamiltonian quantum algorithm
O(N1/2o(1)) query quantum algorithm
We design discrete query algorithm directly
Useful for CS applications
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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.

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Two reflections

• Aharonov, 98 Analysis of Grovers algorithm
• Other applications
• Amplitude amplification
• Quantization of Markov chains.
• Now logic formulas.

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Beyond logic formulas Reichardt, Špalek, 2008

• Input x1, ..., xN ? vectors v1, ..., vM.
• Output F(x1, ..., xN) 1 if there are vi1,vi2,
  ..., vik
• vvi1vi2...vik.

Span program with witness size T
O(vT) query quantum algorithm
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Span programs Reichardt, Špalek, 2008
Logic formula of size T
Span program with witness size T
O(vT) query quantum algorithm
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Span programs Reichardt, 2009
Span program with witness size T
?
O(vT) query quantum algorithm
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Adversary bound A, 2001, Hoyer, Lee, Špalek,
2007

• Boolean function f(x1, ..., xN)
• Inputs x (x1, ..., xN)
• Theorem If there is a matrix A Ax, y?0 only if
  f(x) ? f(y), then computing f requires
• quantum queries

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Span programs Reichardt, 2009
Optimal span program
Semidefinite program (SDP)
Dual SDP
Optimal adversary bound
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Big question 1

• What other problems have quantum speedups?

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Big question 2

• What properties of a problem imply a quantum
  speedup?

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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.

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Open question

• Other classes of problems for which speedup is at
  most polynomial?