Download Quantum algorithm

Quantum algorithms with
polynomial speedups
Andris Ambainis
University of Latvia
Search [Grover, 1996]
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 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

1
3
2
4
5
6


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.
Quantum walk

A quantum particle moving around
this state space.

Quantum walk with two transition
rules:
1
3
2
4
5
6


“usual” for unmarked vertices;
“special” for marked.
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
32
31
69
57
20
49
56
37
68
60
55
12
99
40
92
31
26
20
86
90
72
85
36
18 76 96 82 94 99 21 78
70 18 94 82 92 64 95 46
75 71 93 32 47 11 70 37
71 28 85 41 80 10 52 63
98 39 65 74 24 90 26 83
52 95 65 66 97 54 30 62
47 24 54 48 98 23 41 16
34 73 61 73 21 44 62 34
58 13 10
25 two
29 25
56 68
Task:
find
equal
43 69 46 87 97 45 59 14
numbers.
80 50 61 59 89 67 89 29
27 42 55 77 19 45 15 53
88
53
78
88
60
79
66
14
12
30
86
22
93
16
79
57
91
33
75
51
11
19
48
91
39
35
36
43
27
84
38
74
51
81
22
87
44
42
63
84
96
50
13
76
23
81
15
17
64
72
40
67
35
38
58
83
77
49
17
33
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.
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
x1
OR
x2
x3
x4
AND-OR formula of size M.
Variables accessed by queries: ask
i, get xi.
Motivation


OR
x1

x2

Vertices = chess positions;
Leaves = final positions;
xi=1 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?
Results



AND

OR
x1
OR
x2
x3

x4
Full binary tree of depth d.
N=2d leaves.
Deterministic: (N).
Randomized [SW,S]: (N.753…).
Quantum?
FGG: O(√N)
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, Hadjacency 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
x3
AND
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?