1. Classical and Quantum Automata Abuzer Yakaryılmaz Advisor: Prof. Cem Say

2. Outline Deterministic Finite Automa (DFA) 2-Way Deterministic Finite Automata (2-DFA) Probabilistic Finite Automata (PFA) 2-Way Probabilistic Finite Automata Quantum Physics Quantum Computation Quantum Automata 2-Way Quantum Finite Automata

3. Deterministic Finite Automa (DFA) Basic computation model Read-only machines Finite alphabet Finite state (memory) Transition function Regular languages

4. 2-Way Deterministic Finite Automata Similar to DFA But, 2-way move head (right,left,stationary) More powerful than DFA?  Yes, fewer state, simple computation  No, just recognize regular languages

5. Probabilistic Finite Automata Transitions are probabilistic not deterministic Bounded error – ε Strings (tapes) are accepted or rejected with 1-ε probability Any PFA with ε is equivalent to a DFA

6. 2-Way Probabilistic Finite Automata Transitions are probabilistic not deterministic 2-way move head Is equivalent to DFA?  2PFA-POLYTIME = REGULAR  Freivalds machine [1981] recognizes some non- regular (and non-context-free languages)

7. Quantum Physics 20. century [Plank, Einstein, Bohr, Pauli, Heisenberg, Shrondinger,...] New paradigm Possibilities (complex numbers) Multi-universe Teleportation Entanglement

8. Quantum Computation Feynman: Quantum events cannot be simulated in classical computers in feasible time David Deutsch: Formalize the quantum computers (Quantum Turing Machine) Hilbert Space, Unitary matrix Superpositions of qubits, quantum gates Shor’s factorization algorithm

9. Quantum Automata Quantum model of automata Finite alphabet, finite states Transitions are unitary matrix Superpositon of states Bounded error – ε 1QFA recognize a proper subset of regular languages

10. 2-Way Quantum Finite Automata Recognize some non-regular and non-context free languages in polynomial time [Kondacs & Watrous, 1997] MS Thesis (Fatih Mehmet Atak, 2006)  QFA simulator (java)  New methods MS Thesis (Abuzer Yakaryılmaz, progress)  Formal analysis of FMA  (Maybe) PFA simulator  New methods fewer state, fewer step, better bound error Different approaches  New languages (practical or theoritical)

11. COMMENTS & QUESTION