Quantum Automata Formalism These are general questions related

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 Output state 2 Observe that this matrix is not permutative and not unitary 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 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, 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) 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.