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

2. Computational Algorithms
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)
Computational basis
State:
Measurement
non-deterministic collapse
Two possible outputs
(constraint)

5. Operators
Unitary:
Tensor product
For operators

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

7. Entangled states
2-qubit system
Entangled state
Example:

8. Quantum Computer

9. NOT Gate

10. One-Qubit Hadamard Gate

11. Multi-Qubit Hadamard 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. non-trivial factors of N
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

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

25. Probability distribution
After QFDT
Probability distribution
(1 register)

26. If x is not coprime to N, then use GCD(x, N), else - Shor
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 j Probability 0.41e-03 85 171 256 341 42785
171
256
341
427
0.25e-03
0.39e-04 j 85
171
256
341
427
Probability contribution Prob(j)

30. Continued Fraction Approximation85
171
256
341
427

31. Conclusion Atomic sizes Probabilistic character Speed
Classical machine –
Quantum machine –
timesteps
timesteps
Example:
300 digit code – 1E06 years 1000 digit code – 1E25 years
several hours

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

33. Several iterations of Rotate Phase Operator and Diffusion Operator
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 1 2 3 4 5 6 i 1 2 3 4 5 6

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

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

37. Experimental Scheme in OpticsZ
Rotate
Gate

38. Rotate Optical Gate

39. CNOT optical gate
control
target

40. Conclusion Advantage: Disadvantages: Speed instead of
Difficulty of assigning data

41. Acknowledgement Prof. A.V. Tsiganov Prof. S.Y. Slavyanov
My mom and all my friends…