Download Quantum Automata Formalism

Quantum
Automata
Formalism
These are general questions related to complexity of quantum
algorithms, combinational and sequential
Models of quantum sequential
circuits
1.
2.
3.
4.
5.
Quantum automata
Quantum state machines
Quantum Turing Machines
Quantum Robots of Benioff
Quantum Cellular Automata (not
quantum dot based).
A new formalism
for classical
(deterministic)
automata
Input state 1
Input state 2
Output state 1
Observe that
this matrix is
not
permutative
and not
unitary
Output state 2
This means that external classical computer has to change the quantum circuit
when a new input in the string comes
A formalism for
classical
non-deterministic
automata
Nondeterminism
for b
Observe that
this matrix is
not
permutative
and not
unitary
There are two paths from
state 1 to state 2, which
have labels sequence bb
Using matrices like
these we can analyze if
certain transitions in
graphs exist and how
many of them exist.
This is used in finding
the languages accepted
be the automata
A FORMALISM
FOR
CLASSICAL
PROBABILISTIC
AUTOMATA
PROBABILISTIC
AUTOMATA
Languages accepted by probabilistic
automata
A FORMALISM
FOR
QUANTUM
AUTOMATA
Quantum Finite
Automata = QFA
Now unitary matrices
Probability that an automaton accepts a string
bra
Unitary matrix
ket
Languages accepted by deterministic
automata
• Review the following:
1. the concept of Rabin-Scott automaton and
language accepted by it.
2. Review the concept of regular expression
3. Show a link between regular expression and
language accepted by an automaton.
4. Language generated by an automaton.
5. Regular languages
Languages accepted by probabilistic
automata
Unitary matrices used here
are only a subset of all
matrices
Model of Quantum Automaton
•
•
Quantum automaton is programmed from
deterministic standard automaton.
It is more similar to FPGA than normal
model of computing like in a processor.
Finite
memory
1. Machine here has a program that
generates pulses that program QA.
2. This is like a memory in FPGA that stores
information about LUT and connections
CLASSICAL AUTOMATON
One pulse for one elementary
rotation in one qubit
Infinite
memory
Quantum Automaton
Quantum Automaton described by a unitary matrix
CLASSICAL
TURING
MACHINES
Classical Turing Machines
Model of calculation of
a standard Turing
Machine
polynomial
Example of Turing Machine
The source of infiniteness is the tape
tape
head
Automaton
control the
head
Finite
State
Machine
This machine has a finite memory,
this is standard automaton.
1.
2.
3.
4.
5.
Move left,
move right,
stop,
write a symbol.
Is the symbol in current cell Xi?
Non-Polynomial, these are tough
problems in real life
Bounded-error probabilistic polynomial
(BPP)
• In computational complexity theory,
bounded-error probabilistic
polynomial time (BPP) is the class of
decision problems that are:
1. solvable by a probabilistic Turing machine
2. in polynomial time,
3. with an error probability of at most 1/3 for all
instances.
Bounded-error probabilistic polynomial
• Informally, a problem is in BPP if there is an
algorithm for it that has the following properties:
1. It is allowed to flip coins and make random decisions
2. It is guaranteed to run in polynomial time
3. On any given run of the algorithm, it has a probability
of at most 1/3 of giving the wrong answer, whether the
answer is YES or NO.
BPP = Bounded-error
Probabilistic
Polynomial
A
complexity
class
Bounded-error
probabilistic
polynomial
QUANTUM
TURING
MACHINES
A sum of two complex
numbers can be a zero
BQP
1. In computational complexity theory BQP (bounded error
quantum polynomial time) is the class of decision problems
solvable by a quantum computer in polynomial time, with an
error probability of at most 1/3 for all instances.
2. It is the quantum analogue of the complexity class BPP.
3. In other words, there is an algorithm for a quantum computer
(a quantum algorithm) that solves the decision problem with
high probability and is guaranteed to run in polynomial time.
4. On any given run of the algorithm, it has a probability of at
most 1/3 that it will give the wrong answer.
BQP (cont)
1. Similarly to other "bounded error" probabilistic classes the
choice of 1/3 in the definition is arbitrary.
2. We can run the algorithm a constant number of times and take
a majority vote to achieve any desired probability of correctness
less than 1, using the Chernoff bound.
3. Detailed analysis shows that the complexity class is
– unchanged by allowing error as high as 1/2 − n−c on the one hand,
– or requiring error as small as 2−nc on the other hand,
• where c is any positive constant,
• and n is the length of input.
Sources:
• Used in 2011.