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Quantum Computation and Algorithms

Quantum Computation and Algorithms. Debasis Sadhukhan M.Sc. Physics, IIT Bombay. Basics of Quantum Computation . Quantum Circuits Quantum Fourier Transform and it’s applications. Quantum Search Algorithm. Plan of Talk. WHAT WE NEED TO KNOW Basic Quantum Mechanics &

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Quantum Computation and Algorithms

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  1. Quantum Computation and Algorithms DebasisSadhukhan M.Sc. Physics, IIT Bombay

  2. Basics of Quantum Computation. • Quantum Circuits • Quantum Fourier Transform and it’s applications. • Quantum Search Algorithm Plan of Talk

  3. WHAT WE NEED TO KNOW • Basic Quantum Mechanics & • A little Background of Computer Science BACKGROUND

  4. Important Concepts

  5. Quantum Superposition

  6. The Qubit in Bloch Sphere Representation

  7. Product State

  8. So, if the state can’t be written in the product state form, then they are Entangled. They are called to be Entangled State. • Classical Analogy: No classical analog exists. But you can think of : Harry Potter and Voldemort Quantum EntanglementThe Greatest Love Story Ever Told

  9. Examples: Bell states or EPR pairs Some of the very important applications are : • Super-dense coding • Quantum Teleportation • Quantum Cryptography • Quantum Games Applications

  10. Density Operator

  11. Represent a quantum state as a triangle with attached wires & do operation on quantum states just manipulating this picture Graphical Tensor Notation

  12. Execution of an classical algorithm require hardware, consist of many electrical circuits containing wires and logic gates. • These logic gates are the basic building block of a classical computer. • Similarly, to execute a quantum algorithm we must have a quantum computer where the building blocks are quantum gates. • So, What are the Quantum Gates…? • As the name suggests, the gates are quantum, the laws of quantum mechanics must be applicable here. • So, they must be unitary operator and can be made reversible. Quantum Gates

  13. Quantum Gates

  14. Controlled Operations

  15. Note: The target and control qubit are not basis independent i.e. our target and control qubit may change if we use a different basis . • In Classical Computation, we have seen NAND and NOR gate as universal quantum gate. A similar universality is true for quantum computation also. • Every classical gates can be created using unitary quantum gates. In that sense quantum circuits include all the classical circuits. • So, universality of quantum gates is obvious. Uni1versal Quantum Gates

  16. An algorithm is a well defined procedure or a set of instructions to perform an information processing task. • Turing-Church Thesis: Any algorithmic process can be simulated efficiently using a probabilistic Turing machine. • Complexity Classes: P , NP • Quantum algorithms are those that uses quantum mechanical principles at the time of it’s execution. Hard to design ! Quantum Algorithms

  17. Shor’s Algorithms

  18. Quantum Fourier Transform

  19. Phase Estimation

  20. The final state of the 1st register: Now, apply Inverse Fourier Transform on the 1st register. Final state: Overall Circuit: Phase Estimation

  21. The major applications are • Order finding • Prime factorizationThese can be used to break the cryptosystem used in classical computer • Period Finding etc. Applications of QFT

  22. Grover’s Algorithm

  23. Mathematical Description of the Search Problem

  24. The Operations

  25. Grover’s Iteration Operator

  26. Geometric Visualization

  27. Summary

  28. Examples: C:\Users\DEBASIS\Desktop\GroversQuantumSearchAlgorithm.cdf C:\Users\DEBASIS\Desktop\SimulatedQuantumComputerAlgorithmForDatabaseSearching.cdf • Drawback: • Still, the problem remains in NP class. • If we don’t know the exact no of solution, we may not reach to our solution as no of iteration explicitly depends on M. Examples and Drawback

  29. Future Directions

  30. References: • [1] Michael A. Nielsen and Isaac I. Chuang, Quantum Computation and Quantum Information, Cambridge University Press(2002). • [2] Phillip Kaye, Raymond Laflamme and Michele Mosca, An Introduction to Quantum Computing, Oxford University Press(2007). • [3] Jamie Smith and Michele Mosca, arXiv:1001.0767v2 [quant-ph] • [4] Lecture notes of John Preskill, California Institute of Technology: http://theory.caltech.edu/~preskill/ph229/ Thank You

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