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Quantum Computing & Algorithms
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1. Quantum Computing & Algorithms Loginov Oleg Department of Computational Physics Saint-Petersburg State University 2004

2. Contents • Fundamentals • Logic • Qubit (short of quantum bit) • Operators • Multi-qubit systems • Entangled states • Quantum Circuits (Gates) • Computational Algorithms • Shor’s Algorithm • Grover’s Algorithm

3. Hilbert Space Inner product: Norm: Dual vector: Outer product:

4. Qubit (short of quantum bit) Measurement non-deterministiccollapse Two possible outputs (constraint) State: Computational basis

5. Operators Unitary: Tensor product For operators

6. Multi-qubit Systems N-qubit quantum computer states 2-qubit QC:

7. Entangled states Entangled state Example: 2-qubitsystem

8. Quantum Computer

9. NOT Gate

12. Control-NOT gate

13. Conclusion • Qubits have probabilistic nature • N-qubit register have 2^N basis functions • Gates that are direct product of other gates do not produce entanglement. • cNOT and one-qubit gates form a universal set of gates. • In principle there is an infinite number unitary operators U.

14. Quantum Algorithms • Shor’s Algorithm (Factorization) • Grover’s Algorithm • Wavelet Q-Search • Extended Search • Root Calculator • Algorithm for Triangle Problem

15. Factorization I non-trivial factors of N

16. Factorization II Example: N = 21

17. Shor #0 t n

18. Shor #1 t n

19. Shor #2 t n

20. Shor #3 t n

21. Discrete Fourier Transform

22. QDFT

23. QDFT 1 0

24. Before Q-DFT Probability distribution (1 register)

25. After QFDT Probability distribution (1 register)

26. Example – Step 1 If x is not coprime to N, then use GCD(x, N), else - Shor

27. Example – Step 2

28. Example – Before QDFT 2-nd register:

29. Example – After QDFT 0 85 171 256 341 427 Probability 0.41e-03 0.25e-03 0.39e-04 j 0 256 341 427 85 171 Probability contribution Prob(j)

30. Continued Fraction Approximation 0 85 171 256 341 427

31. Conclusion • Atomic sizes • Probabilistic character • Speed • Classical machine – • Quantum machine – timesteps timesteps Example: 300 digit code – 1E06 years1000 digit code – 1E25 years several hours

32. Search Task N states: The problem is identify the state Condition:

33. Grover’s Algorithm Take a n-qubit register, where After n-dimension Hadamard Gate: Several iterations of Rotate Phase Operator and Diffusion Operator Measurement

34. Rotate Phase Operator i 0 1 2 3 4 5 6 i 0 1 2 3 4 5 6

35. Diffusion Operator i i 0 1 2 3 4 0 1 2 3 4

36. Measurement Average amplitude: Addition in each step: Exact calculations:

37. Experimental Scheme in Optics H H Rotate Gate H H H Z H

38. Rotate Optical Gate

39. CNOT optical gate control target