1. Quantum Cost Calculation of Reversible Circuit Sajib Mitra MS/2008-09 Department of Computer Science and Engineering University of Dhaka sajibmitra.csedu@yahoo.com

2. O VERVIEW Reversible Logic Quantum Computing Quantum Gates Realization of Quantum NOT Quantum wire and Special Cases Quantum Cost Calculation of RC Conclusion Assignment References

3. Reversible Logic Equal number of input and output vectors Preserves an unique mapping between input and output vectors of the particular circuit One or more operation can implement in a single unit called Reversible Gate ( N x N ) Reversible Gate has N number of inputs and N number of outputs where N= {1, 2, 3, …}

4. Reversible Logic (cont…) Advantage Recovers bit-loss as well as production of heat Adaptable for Quantum Computing Multiple operations in a single cycle Uses low power CMOS technology

5. Reversible Logic (cont…) Limitation Feedback is strictly restricted Maximum and minimum Fan-out is always one

6. Reversible Logic (cont…) Most Popular reversible gates are as follows: Fig. 3x3 Dimensional Reversible gates

7. Reversible Logic (cont…) Most Popular reversible gates are as follows: Fig. 4x4 Dimensional Reversible gates

8. Quantum Computing First proposed in the 1970s, quantum computing relies on quantum physics by taking advantage of certain quantum physics properties of atoms or nuclei that allow them to work together as quantum bits, or qubits, to be the computer's processor and memory. Qubits can perform certain calculations exponentially faster than conventional computers. Quantum computers encode information as a series of quantum-mechanical states such as spin directions of electrons or polarization orientations of a photon that might represent as or might represent a superposition of the two values.

9. Quantum Computing (cont…) Quantum Computation uses matrix multiplication rather than conventional Boolean operations and the information measurement is realized using qubits rather than bits The matrix operations over qubits are simply specifies by using quantum primitives as follows:

10. Quantum Computing (cont…) InputOutput ABPQ 0000 0101 1011 1110 Input/output Pattern Symbol 00a 01b 10c 11d

11. Quantum Computing (cont…)

12. InputOutput ABPQ 0000 0101 1011 1110

13. Quantum Gates Fig: Quantum Gates are used for realizing Reversible Circuit

14. Quantum Gates (cont…) What is SRN? But

15. Quantum Gates (cont…) What is SRN? But NOT But How?

16. Realization of Quantum NOT Basic operator for single input line: 1. NOT 2. Coin Flip 3. Quantum Coin Flip

17. Realization of Quantum NOT (cont…)

18. 1 01 1010 1/2 1/4 Probability of 0 or 1 based on Coin Flip:

19. Realization of Quantum NOT (cont…) 1 01 1010 1/2 1/4 Probability of 0 or 1 based on Coin Flip: So the Probability of P(0)=1/2 P(1)=1/2

20. Realization of Quantum NOT (cont…) Probability of |0> or |1> based on Quantum Coin Flip: |1 > |0 > |1 > |0 > |1 >

21. Realization of Quantum NOT (cont…) Probability of |0> or |1> based on Quantum Coin Flip: |1 > |0 > |1 > |0 > |1 > So the Probability of P(|0>)=1 P(|1>)=0

22. Realization of Quantum NOT (cont…) NOT operation can be divided into to SRN matrix production NO T 10

23. Quantum Cost ( QC ) of any reversible circuit is the total number of 2x2 quantum primitives which are used to form equivalent quantum circuit.

24. Quantum Wire and Special Cases (cont…) Quantum XOR gate, cost is 1

25. Quantum Wire and Special Cases (cont…) Two Quantum XOR gates, but cost is 0

26. Quantum Wire and Special Cases (cont…) Quantum Wire

27. Quantum Wire and Special Cases (cont…) SRN and its Hermitian Matrix on same line. VV + = Identity and the total cost = 0 Quantum Cost of V and V + are same, equal to one.

28. Quantum Wire and Special Cases (cont…) SRN and its Hermitian Matrix on same line. VV + = Identity and the total cost = 0

29. Quantum Wire and Special Cases (cont…) The attachment of SRN (Hermitian Matrix of SRN) and EX-OR gate on the same line generates symmetric gate pattern has a cost of 1. Here T= V or V+

30. Quantum Wire and Special Cases (cont…) The cost of all 4x4 Unitary Matrices (b, c, d) and the symmetric gate pattern (e, f, g, h) are unit.

31. Quantum Cost of F2G

32. Quantum Cost of Toffoli Gate But How?

33. Quantum Cost of Toffoli Gate InputOutput ABR 00C 01C 10C 11C’

34. Quantum Cost of Toffoli Gate NOT InputOutput ABR 00C 01C 10C 11C’ InputOutput ABR 00C 01C 10C 11C’

35. Quantum Cost of Toffoli Gate InputOutput ABR 00C 01C 10C 11C’ InputOutput ABR 00C 01C 10C 11C’

36. Now

37. Quantum Cost of Toffoli Gate InputOutput ABR 00C 01C 10C 11C’ Have anything wrong?

38. Quantum Cost of Toffoli Gate InputOutput ABR 00C 01C 10C 11C’ Ok

39. Quantum Cost of Toffoli Gate (cont…) Alternate representation of Quantum circuit of TG…

40. Quantum Cost of Fredkin Gate But How?

41. Quantum Cost of Fredkin Gate (cont…)

47. Quantum Cost of Peres Gate

48. Quantum Cost of NFT Gate

49. Quantum Cost of MIG Gate

50. Assignment Find out cost

51. About Author Sajib Kumar Mitra is an MS student of Dept. of Computer Science and Engineering, University of Dhaka, Dhaka, Bangladesh. His research interests include Electronics, Digital Circuit Design, Logic Design, and Reversible Logic Synthesis.

52. T HANKS T O A LL