1. 1 Welcome to the presentation on Computational Capabilities with Quantum Computer By Anil Kumar Javali

2. 2 Agenda Introduction Introduction Quantum Parallelism Quantum Parallelism Quantum Algorithms Quantum Algorithms NMR for Quantum Computer ( Q.C. ) NMR for Quantum Computer ( Q.C. ) CNOT Gate for Q. C CNOT Gate for Q. C Obstacles Obstacles References References

3. 3 Introduction What is a Q.C. What is a Q.C.

4. 4 Classical Computer (C.C) vs. Q.C. bit qubit

5. 5 Power & Potential of Q.C A system with 500 qubits => 2 states A system with 500 qubits => 2 states Each state = single list of 500 0’s & 1’s Each state = single list of 500 0’s & 1’s 99 & 100 th qubit 99 & 100 th qubit Best know encryption method RSA, will no longer be the best Best know encryption method RSA, will no longer be the best 500

6. 6 Quantum Interference

7. 7 Quantum Interference (contd)

8. 8 CC vs QC Best know algorithm for classical computer runs in O(exp[(64/3) (ln N) (ln ln N) ]) steps Best know algorithm for classical computer runs in O(exp[(64/3) (ln N) (ln ln N) ]) steps For ex, in 1994, 129 digit number, factorized, 1600 workstations, 8 months. For ex, in 1994, 129 digit number, factorized, 1600 workstations, 8 months. Similarly, 800,000 years to factor a 250 digit number & 10 years to factor a 1000 digit number Similarly, 800,000 years to factor a 250 digit number & 10 years to factor a 1000 digit number 1/32/3 25

9. 9 CC vs QC (cont.) Where as, Q.C takes O((log N) ) steps Where as, Q.C takes O((log N) ) steps 1000 digit number would take only a few million steps. 1000 digit number would take only a few million steps. Public key cryptosystems based on factoring may be breakable Public key cryptosystems based on factoring may be breakable 2+E

10. 10 Quantum Algorithms Shor’s Algorithm Shor’s Algorithm Grover’s Algorithm Grover’s Algorithm

11. 11 Shor’s Algorithm Finds factors of a very large number Finds factors of a very large number For ex; N = 91, For ex; N = 91, Choose a co-prime of 91 which is 729 Choose a co-prime of 91 which is 729 i.e., 729 = 1 (mod 91) i.e., 729 = 1 (mod 91) => 28 x 26 = 0 (mod 91) => 28 x 26 = 0 (mod 91) => either gcd(28,91) or gcd(26,91) will give the factors of 91 => either gcd(28,91) or gcd(26,91) will give the factors of 91 Here, both gives different factors, those are 7 & 13 Here, both gives different factors, those are 7 & 13 91 = 7 x 13 91 = 7 x 13

12. 12 Physical Implementation of Q.C NMR (nuclear magnetic resonance)-Based Q.C NMR (nuclear magnetic resonance)-Based Q.C Heteropolymer-Based Q.C Ion Trap Based Q.C Ion Trap Based Q.C Cavity QED-Based Q.C Cavity QED-Based Q.C

13. 13 NMR for Q.C

14. 14 How NMR works Takes pulse signal as input Takes pulse signal as input Acts on the qubit molecules Acts on the qubit molecules Qubit changes its state Qubit changes its state Measure the density of the qubit to know its new state Measure the density of the qubit to know its new state

15. 15 How NMR works (cont.) Qubits initial state is represented by its density which is represented in the form of matrix ( ‘a’ ) Qubits initial state is represented by its density which is represented in the form of matrix ( ‘a’ ) When input pulse signal ‘x’ acts on ‘a’ When input pulse signal ‘x’ acts on ‘a’ Density of qubit changes to final state ‘b’ Density of qubit changes to final state ‘b’ We can represent the above operation symbolically as ‘a’ X ‘x’ = ‘b’ We can represent the above operation symbolically as ‘a’ X ‘x’ = ‘b’

16. 16 Genetic Algorithm (G.A) to find the pulse signal Using GA, find the pulse signal ‘x’ Using GA, find the pulse signal ‘x’ Train the network using GA for different test cases Train the network using GA for different test cases Test the network for new values Test the network for new values

17. 17 My contribution towards NMR QC Implementing GA to find the pulse signal Implementing GA to find the pulse signal

18. 18 Basic gates for C.C AND, OR, NOT AND, OR, NOT Original 2 inputs can’t be restored Original 2 inputs can’t be restored Electronic circuits are not reversible Electronic circuits are not reversible

19. 19 Basic Gates for Q.C There are AND, OR and NOT gates for Q.C There are AND, OR and NOT gates for Q.C They are not the smallest units for Q.C They are not the smallest units for Q.C Where as CNOT ( controlled NOT ) is Where as CNOT ( controlled NOT ) is CNOT represent AND, OR & NOT operations CNOT represent AND, OR & NOT operations

20. 20 Universal CNOT gate for Q.C 0 1 0 1 CONTROLCONNECTION TARGET X

21. 21 How CNOT works INOUT CONTROLTARGETCONTROLTARGET 000011 1011 10

22. 22 Obstacles There are many obstacles to be resolved to make Q.C a reality like, There are many obstacles to be resolved to make Q.C a reality like, Quantum Entanglement Quantum Entanglement Quantum Teleportation Quantum Teleportation Quantum Error Correction Quantum Error Correction

23. 23 References Isaac L Chuang, M A Nielsen, Quantum Computation and Information, Dec 2000 Isaac L Chuang, M A Nielsen, Quantum Computation and Information, Dec 2000 Center for Quantum Computation, http://www.qubit.com/ Center for Quantum Computation, http://www.qubit.com/ Jacob West, The Quantum Computer, http://www.cs.caltech.edu/~westside/quantum- intro.html April 28, 2000 Jacob West, The Quantum Computer, http://www.cs.caltech.edu/~westside/quantum- intro.html April 28, 2000 Samuel L. Braunstein, Quantum Computation, http://www.informatics.bangor.ac.uk/~schmuel/ comp/comp.html Aug 23, 1995 Samuel L. Braunstein, Quantum Computation, http://www.informatics.bangor.ac.uk/~schmuel/ comp/comp.html Aug 23, 1995

24. 24 Questions ?