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Quantum and classical computing

Quantum and classical computing. EECS FER 16.9.2003. Dalibor H RG. How to think?. Review / Classical computing. Classical computing :

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Quantum and classical computing

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  1. Quantum and classical computing EECS FER 16.9.2003. Dalibor HRG

  2. How to think?

  3. Review / Classical computing • Classical computing: • Turing machine (A.Turing,1937.), computability (functions and predicates), Computational Complexity – theory of classical computation. • Bool’s algebra and circuits, today computers,(logic). • Algorithms and complexity classes (P, P/poly, PSPACE, NP, NP-complete, BPP,…) – measuring how efficient is algorithm, can it be useful?

  4. Review / Classical computing • Famous mathematical questions today: • P – predicates which are decidable in polynomial time (head moves of Turing machine) • PSPACE – predicates decidable in polynomial space (cells on Turing machine’s track)

  5. Review / Classical computing • NP – we can check some solution in polynomial time, but finding it, is a difficult problem. • Predicate: • SAT , HC (hamiltonian cycle),TSP (travelling salesman problem), 3-SAT,… • Karp’s reducebility: • NP – complete: each predicate from NPis reducible to 3– SATpredicate.

  6. Review / Classical computing

  7. Review / Classical computing NANOTECHNOLOGY

  8. Review / Quantum computing • (R. Feynman,Caltech,1982.) – impossibility to simulate quantum system! • (D. Deutsch, Oxford, CQC, 1985.) – definition ofQuantum Turing machine, quantum class (BQP) and first quantum algorithm (Deutsch-Jozsa). • Postulates of quantum mechanics, superposition of states, interference, unitary operators on Hilbert space, tensorial calculation,…

  9. Quantum mechanics • Fundamentals: dual picture of wave and particle. • Electron: wave or particle?

  10. Quantum mechanics

  11. Waves!

  12. Secret of the electron Does electron interfere with itself?

  13. Quantum mechanics • Discrete values of energy and momentum. • State represent object (electron’s spin, foton’s polarization, electron’s path,…) and its square amplitude is probability for outcome when measured. • Superposition of states, nothing similar in our life. • Interference of states.

  14. Qubit and classical bit • Bit: in a discrete moment is either “0” (0V) or “1” (5V). • Qubit: vector in two dimensional complex space, infinite possibilities and values. • Physically, what is the qubit?

  15. Qubit

  16. System of N qubits Unitary operators: legal operations on qubit. Unitary operators: holding the lengths of the states. Important!!

  17. Tensors • For representing the state in a quantum register. • Example, system with two qubits: • State in this systems is:

  18. Quantum gates Quantum circuits (one qubit): Pauli-X (UNOT), Hadamard (USRN). (two qubits): CNOT (UCN).

  19. Quantum parallelism All possible values of the n bits argument is encoded in the same time in the n qubits! This is a reason why the quantum algorithmshave efficiency!

  20. Quantum algorithms (1) Initial state Quantum operators Measurement Time

  21. f Quantum algorithms (2) Idea: 1. Make superposition of initial state, all values of argument are in n qubits. 2. Calculate the function in these arguments so we have all results in n qubits. 3. Interference ( Walsh-Hadamard operator on the state of n qubits or register) of all values in the register. We obtain a result.

  22. (No-cloning theorem) Wooters & Zurek 1982 • Unknown quantum state can not be cloned. • Basis for quantum cryptology (or quantum key distribution).

  23. Quantum cryptology (1) Alice Bob Quantum bits Eve

  24. Quantum cryptology (2) Public channel for authentication

  25. Alice Bob Quantum teleportationBennett 1982 • It is possible to send qubit without sending it, with two classical bits as a help. Classical bits. EPR Alice & Bob share EPR (Einstein,Podolsky,Rosen) pair.

  26. Present algorithms? • Deutsch-Josza • Shor - Factoring 1994., • Kitaev - Factoring • Grover - Database searching 1996., • Grover - Estimating median

  27. Who is trying? • Aarhus • Berkeley • Caltech • Cambridge • College Park • Delft • DERA (U.K.) • École normale supérieure • Geneva • HP Labs (Palo Alto and Bristol) • Hitachi • IBM Research (Yorktown Heights and Palo Alto) • Innsbruck • Los Alamos National Labs • McMaster • Max Planck Institute-Munich • Melbourne • MIT • NEC • New South Wales • NIST • NRC • Orsay • Oxford • Paris • Queensland • Santa Barbara • Stanford • Toronto • Vienna • Waterloo • Yale • many others…

  28. Corporations?

  29. Corporations?

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