SLIDE 1
TQC 2008 2
Space-efficient quantum Space-efficient quantum automata automata - - PowerPoint PPT Presentation
Space-efficient quantum Space-efficient quantum automata automata - - PowerPoint PPT Presentation
Space-efficient quantum Space-efficient quantum automata automata Andris Ambainis Nikolay Nahimov Department of Computer Science University of Latvia TQC 2008 Quantum Finite Automata Mathematical model for quantum computers with
SLIDE 2
SLIDE 3
TQC 2008 3
DFA requires O(p) states
Computational problem
} , ) (mod | { P p p j a L
j p
∈ ≡ =
QFA requires O(log(p)) states
...
p states
SLIDE 4
TQC 2008 4
Known results
- A. Ambainis, R. Freivalds “1-way quantum
finite automata: strengths, weaknesses and generalizations”
- A. Ambainis, N. Nahimovs “Improved
constructions of quantum automata”
SLIDE 5
TQC 2008 5
Improvements
Simpler construction with better constant in
front of log (p) and with a much simpler analysis.
A simple rule for derandomization of
automata construction.
SLIDE 6
TQC 2008 6
Define simple 2-state QFA Uk depending on
single parameter k
Take c·log (p) automata Uk with different k
Construction steps
SLIDE 7
TQC 2008 7
Q = {q0, q1}, Qacc = {q0}, Qrej = {q1} Starting state q0 Left and right endmarker leaves state
unchanged
Construction: building blocks
q1 q0
SLIDE 8
TQC 2008 8
Construction: building blocks
Reading 'a' performs Uk state “rotation”
1 1 1
cos sin sin cos q q q q q q
k k k k
φ φ φ φ + − → + →
p k
k
π φ 2 = where
Parameter k represents a rotation frequency
SLIDE 9
TQC 2008 9
Construction: building blocks
After reading aj the state of Uk is
1
2 sin 2 cos q p j k q p j k π π +
Accepts aj with probability
⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ p j k π 2 cos2
SLIDE 10
TQC 2008 10
Construction: building blocks
After reading aj the state of Uk is
1
2 sin 2 cos q p j k q p j k π π +
Accepts aj with probability If accepts aj with probability 1
p j
L a ∈
⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ p j k π 2 cos2
SLIDE 11
TQC 2008 11
Construction
...
Uk1 Uk2 Ukd q1,1 q1,0 q2,1 q2,0 qd,1 qd,0
Take d = c·log(p) automata Uk with different k
d=c·log (p)
SLIDE 12
TQC 2008 12
Construction
Start at equal state superposition
( )
, , 2 , 1
... 1
d start
q q q d + + + = ϕ
, i
q U
… q1,1 q1,0 q2,1 q2,0 qd,1 qd,0
SLIDE 13
TQC 2008 13
Construction
Start at equal state superposition
, i
q
Alternative view: start at and on the left
endmarker perform a transformation
( )
, , 2 , 1 , 1
... 1
d
q q q d q + + + →
, 1
q
( )
, , 2 , 1
... 1
d start
q q q d + + + = ϕ
SLIDE 14
TQC 2008 14
Construction
Transformation for 'a' is as before
1 , , 1 , 1 , , ,
cos sin sin cos
i k i k i i k i k i
q q q q q q
i i i i
φ φ φ φ + − → + → p k i
k i
π φ 2 =
where U
… q1,1 q1,0 q2,1 q2,0 qd,1 qd,0
SLIDE 15
TQC 2008 15
Construction
Transformation for 'a' is as before
1 , , 1 , 1 , , ,
cos sin sin cos
i k i k i i k i k i
q q q q q q
i i i i
φ φ φ φ + − → + → p k i
k i
π φ 2 =
where
After reading aj the state of U is
∑
⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ +
i i i i i
q p j k q p j k d
1 , ,
2 sin 2 cos 1 π π
SLIDE 16
TQC 2008 16
Construction
Measure difference from initial state
( )
, , 2 , 1
... 1
d start
q q q d + + + = ϕ
U
… q1,1 q1,0 q2,1 q2,0 qd,1 qd,0
SLIDE 17
TQC 2008 17
Construction
Alternative view: on the right endmarker
perform transformation
∑
+ →
j j j i
q q d q
, , 1 ,
1 α
Measure difference from initial state
( )
, , 2 , 1
... 1
d start
q q q d + + + = ϕ
and measure if state is
, 1
q
SLIDE 18
TQC 2008 18
Construction
Accepts aj with probability
2 2 1 2
2 cos ... 2 cos 2 cos 1 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + + p j k p j k p j k d
d
π π π
If accepts with probability 1 If we want probability to be small
p j
L a ∈
j
a
p j
L a ∉
SLIDE 19
TQC 2008 19
Theorem
For any ε > 0 and p ∈ P, there exist k1 ,k2 ,...,kd , that for all j ∈ {1, ..., p-1}
⎥ ⎥ ⎤ ⎢ ⎢ ⎡ ⋅ = ε ) 2 ( log 2 p d
ε π π π < ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + +
2 2 1 2
2 cos ... 2 cos 2 cos 1 p j k p j k p j k d
d
SLIDE 20
TQC 2008 20
Notes on the theorem proof
Proof is by a probabilistic argument. It applies
sequence of parameters k1,k2,...,kd that are chosen at random.
Proof does not give an explicit parameter
sequence.
A deterministic sequence construction rule is
required.
SLIDE 21
TQC 2008 21
If g is a primitive root modulo p ∈ P*, then sequence for all d and all j ∈ {1, ..., p-1} satisfies
Derandomization hypothesis
d i i i g
p g k S
1
} mod {
=
≡ =
ε π π π < ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + +
2 2 1 2
2 cos ... 2 cos 2 cos 1 p j k p j k p j k d
d
* All gi (mod p), i ∈ {0, ..., p-1}, are different
SLIDE 22
TQC 2008 22
Notation
We will refer Sg = {gi (mod p)} as cyclic
sequence and g as the sequence generator.
We will also use Srand to denote a random
sequence.
SLIDE 23
TQC 2008 23
Hypothesis check
We have checked all p ∈ {2, …, 9973} For each p all generators g For each p and g all sequence lengths d < p
choosing a corresponding ε value
We haven't found any counterexample.
SLIDE 24
TQC 2008 24
Random vs. cyclic sequences
In 99.98% - 99.99% of our experiments,
random sequences achieved the bound of theorem.
For randomly selected p ∈ P, ε > 0 and
generator g, a cyclic sequence Sg gives a better result than a random sequence Srand in 98.29% of cases.
SLIDE 25
TQC 2008 25
Random vs. cyclic sequences
Probability of error for different p, ε and g
p e g e_rand e_g 1523 0.1 948 0.03635 0.01517 2689 0.1 656 0.03767 0.0195 3671 0.1 2134 0.03803 0.02122 4093 0.1 772 0.03822 0.01803 5861 0.1 2190 0.03898 0.01825 6247 0.1 406 0.03922 0.02006 7481 0.1 6978 0.03932 0.01691 8581 0.1 5567 0.03942 0.02057 9883 0.1 1260 0.04011 0.01905
SLIDE 26
TQC 2008 26
Different generators
Every p ∈ P might have multiple generators. Different generators give QFA with different
probabilities of error. One with a minimal error will be referred as a minimal generator.
Typically, the minimal generator give a QFA
with a substantially smaller probability of error.
SLIDE 27
TQC 2008 27
Different generators
Minimal generators for different p
p e g e_g g_min e_g_min 1523 0.1 948 0.01517 624 0.00919 2689 0.1 656 0.0195 1088 0.0106 3671 0.1 2134 0.02122 1243 0.01121 4093 0.1 772 0.01803 1063 0.01154 5861 0.1 2190 0.01825 5732 0.01133 6247 0.1 406 0.02006 97 0.01182 7481 0.1 6978 0.01691 2865 0.01205 8581 0.1 5567 0.02057 4362 0.01335 9883 0.1 1260 0.01905 5675 0.01319
SLIDE 28
TQC 2008 28
Open questions
Strict mathematical proof of derandomization
hypothesis.
Is it possible to find a minimal generator
without an exhaustive search of all generators ?
Can other functions be represented in a
similar way ?
SLIDE 29