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Um ponto de vista simbólico sobre a Simulação de Algoritmos Quânticos

Um ponto de vista simbólico sobre a Simulação de Algoritmos Quânticos. António Pereira & Rosália Rodrigues CEOC-UA – CIMA-UE 2006. Quantum Computation. Research in Quantum Computation: building quantum devices designing algorithms for quantum devices. How to Simulate it ?.

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Um ponto de vista simbólico sobre a Simulação de Algoritmos Quânticos

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  1. Um ponto de vista simbólico sobre a Simulação de Algoritmos Quânticos António Pereira & Rosália Rodrigues CEOC-UA – CIMA-UE2006

  2. Quantum Computation Research in Quantum Computation: • building quantum devices • designing algorithms for quantum devices How to Simulate it ? CEOC – CIMA - 2006

  3. Simulating Quantum Computation Vectorial approach: state  vector on a Hilbert space evolution  matrix products advantage: easy to implement and trace drawback: exponential growth in space and time Symbolic approach: state  linear expression evolution  algebraic rules advantage: control over complexity drawback: “convince” Mathematica not to evaluate... yet Symbolic Quantum Computer Simulator CEOC – CIMA - 2006

  4. qudits & qubits CEOC – CIMA - 2006

  5. kets in SQCS Basis qudit state Object with head ket General qudit state Linear expression of ket objects CEOC – CIMA - 2006

  6. bras in SQCS Riesz Theorem: CEOC – CIMA - 2006

  7. braKets in SQCS braKet • conjugate linear • in the first argument • linear in the second CEOC – CIMA - 2006

  8. Qudit Systems ………… 1 2 3 n ………… CEOC – CIMA - 2006

  9. The Kronecker product in SQCS Properties of the Tensor Product (Kronecker Product): • Associative • Noncommutative • Distributive with respect • to linear combinations CEOC – CIMA - 2006

  10. Operators in SQCS The discrete time evolution of a closed quantum system is described by the action of a unitary operator Quantum Algorithm Initial state + Sequence of unitary operators + Measurement Every linear operator is represented in SQCS by an object op[name_,n_,f_] • where: • name ― label for the operator • n ― number of qudits on which the operator acts • f ― function that defines the action of the operatoron the basis qudits states (set of rules) CEOC – CIMA - 2006

  11. Operators in SQCS The Hadamard operator • Creates a uniform superposition • Is its own inverse CEOC – CIMA - 2006

  12. Operators in SQCS The Walsh-Hadamard operator CEOC – CIMA - 2006

  13. Operators in SQCS The Outer Product operator Completeness Relation: CEOC – CIMA - 2006

  14. Simulating Grover’s Algorithm CEOC – CIMA - 2006

  15. Simulating Grover’s Algorithm Classical Database Case: Quantum Computer Classical Database f(x) CEOC – CIMA - 2006

  16. Simulating Grover’s Algorithm Quantum Database Case: Quantum Computer Quantum Database CEOC – CIMA - 2006

  17. Simulating Grover’s Algorithm Step by step: Database of size 25=32 Index of the element to be searched for Number of steps The Oracle Grover’s operator CEOC – CIMA - 2006

  18. Simulating Grover’s Algorithm Step by step: CEOC – CIMA - 2006

  19. Simulating Grover’s Algorithm Step by step: Probability distribution CEOC – CIMA - 2006

  20. Simulating Grover’s Algorithm Step by step: Probability distribution CEOC – CIMA - 2006

  21. Simulating Grover’s Algorithm Step by step: Probability distribution CEOC – CIMA - 2006

  22. Simulating Grover’s Algorithm Step by step: Probability distribution CEOC – CIMA - 2006

  23. Grover’s Algorithm – Simulation Times Classical Database Case: Time × Number of qubits Time × Database size Mathematica 5, Pentium IV, 3.0 GHz, 1GB RAM CEOC – CIMA - 2006

  24. Grover’s Algorithm – Simulation Times Quantum Database Case: Time × Number of qubits Time × Database size Mathematica 5, Pentium IV, 3.0 GHz, 1GB RAM CEOC – CIMA - 2006

  25. Conclusions & Further work Conclusions: • Symbolic Approach to Quantum Computation: • Provides a suitable environment for testing quantum algorithms. • Allows for larger problem instances. • Algorithms can be programmed at high-level. • Useful tool for the teaching of Quantum Computation. Further work: • Measuring Operators. • A quantum register address manager. • Simulate other quantum algorithms: Deutsch-Jozsa, Shor, … • Use SQCS as a tool for the development of new quantum algorithms. CEOC – CIMA - 2006

  26. References 1. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press (2000) 2. Kitaev, A.Y., Shen, A., Vyalyi, M.: Classical and quantum computation. Volume 47 of Graduate Studies in Mathematics. American Mathematical Society (2002) 3. Wolfram, S.: The Mathematica Book, Fifth Edition. Wolfram Media, Inc. (2003) 4. Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proc. 28th Annual ACM Symposium on the Theory of Computing. (1996) 212-219 5. Biham, E., Biham, O., Biron, D., Grassl, M., Lidar, D.A.: Grover's quantum search algorithm for an arbitrary initial amplitude distribution. Physical Review A 60 (1999) 27-42 6. Pereira, António, Rodrigues, Rosália: A Symbolic Approach to Quantum Computation Simulation. Lecture Notes in Computer Science (2006) Vol. 3992. 454 – 461 7. Pereira, António, Rodrigues, Rosália: Symbolic Quantum Computation Simulation with Mathematica. Cadernos de Matemática. Universidade de Aveiro. CM05/I-44 (2005) Thank You CEOC – CIMA - 2006

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