Suggestion for Optical Implementation of Hadamard Gate

1. Suggestion for Optical Implementation of Hadamard Gate Amir Feizpour Physics Department Sharif University of Technology

2. Contents • Motivation • Implementation Methods • Optical Implementations • The Main Problem • Solution • Model Proposed • Results Contents of my talk • Motivation • Implementation Methods • Optical Implementations • The main Problem • Solution • Model Proposed • Results

3. Contents • Motivation • Implementation Methods • Optical Implementations • The Main Problem • Solution • Model Proposed • Results Why QI & QC? • Quantum Computation • Reduces the needed steps to accomplish a certain job • Quantum Information • Reduces the amount of data needed to transmit a certain amount of Information

4. Contents • Motivation • Implementation Methods • Optical Implementations • The Main Problem • Solution • Model Proposed • Results Quantum Computer • DiVincenzo’s Criteria • A scalable physical system with well characterized qubits. • The ability to initialize the state of the qubits to a simple fiducial state. • A universal set of quantum gates such as generic one-qubit gates and a two-qubit gate. • A qubit-specific measurement capability. • Long relevant decoherence times, much longer than the gate operation time.

5. Contents • Motivation • Implementation Methods • Optical Implementations • The Main Problem • Solution • Model Proposed • Results Implementation Candidates • NMR • Ion trap and neutral atom trap • Schemes based on solid state physics • Quantum dot qubits • Superconducting qubits • Schemes based on quantum optics

6. Contents • Motivation • Implementation Methods • Optical Implementations • The Main Problem • Solution • Model Proposed • Results Why Photons? • Advantages of using photons as qubit • Quantum optics is a well developed field. • Photons decohere slowly. • Photons travel well. • Photons can be experimented with at room temperature.

7. Contents • Motivation • Implementation Methods • Optical Implementations • The Main Problem • Solution • Model Proposed • Results How to use optics? • From the view point of qubit • Single photon, • Coherent states. • From the view point of gates • Linear optics, • Non-linear optics.

8. Contents • Motivation • Implementation Methods • Optical Implementations • The Main Problem • Solution • Model Proposed • Results Optical Schemes • Early optical quantum computer based on non-linearities • Qantum optical Fredkin gate (Milburn 1989) • N- port interferometers and optical circuits • Decomposition of unitary (Zielinger et. al, 1998) • Optical Simulation of Quantum Logic (Cerf et. al, 1998)

9. Contents • Motivation • Implementation Methods • Optical Implementations • The Main Problem • Solution • Model Proposed • Results Optical Schemes (Continued) • KLM theory (Knill et al, 2001) • Linear optics (beam splitter and phase shifter) • Probabilistic gates • Teleported gates • Schemes based on coherent state • Non-linear optics • Linear optics

10. Contents • Motivation • Implementation Methods • Optical Implementations • The Main Problem • Solution • Model Proposed • Results What’s the problem? • Single photon • Photons do not interact directly, making two qubit gates very difficult • Coherent State • Producing superposition states is a hard to accomplish

11. Contents • Motivation • Implementation Methods • Optical Implementations • The Main Problem • Solution • Model Proposed • Results What’s the way out? • Pay more to get what you want • KLM theory: ancila bits and post-measurement • Using a intermediate medium: optical non-linearities • But optical non-linearities are usually weak

12. Contents • Motivation • Implementation Methods • Optical Implementations • The Main Problem • Solution • Model Proposed • Results There’s yet another way • Enhance the effective non-linearity of the medium • Trapping the photons in the medium • Thus: Increasing the interaction time • How to do that? • Micro-resonator • Photonic crystal

13. Contents • Motivation • Implementation Methods • Optical Implementations • The Main Problem • Solution • Model Proposed • Results Qubit • Coherent state with a - phase difference and the same average number of photons • Larger values of make the chosen basis more nearly orthogonal

14. Contents • Motivation • Implementation Methods • Optical Implementations • The Main Problem • Solution • Model Proposed • Results Semi-Hadamard Gate • Consider this example: The fidelity can be used as a proper criteria This transformation is possible using a Kerr media which produces a phase change.

15. Contents • Motivation • Implementation Methods • Optical Implementations • The Main Problem • Solution • Model Proposed • Results Gate: CROW • A coupled resonator optical waveguide made up of micro-rings with large Kerr coefficient

16. Contents • Motivation • Implementation Methods • Optical Implementations • The Main Problem • Solution • Model Proposed • Results Dispersion Relation • Transfer Matrix Method

17. Contents • Motivation • Implementation Methods • Optical Implementations • The Main Problem • Solution • Model Proposed • Results Unitary Evolution • Effective unitary evolution • where

18. Contents • Motivation • Implementation Methods • Optical Implementations • The Main Problem • Solution • Model Proposed • Results Fidelity • Fidelity of obtained output to the desired output for

19. Contents • Motivation • Implementation Methods • Optical Implementations • The Main Problem • Solution • Model Proposed • Results Size Sensitivity

20. Acknowledgement At the end, I must thank my advisors Prof. A. R. Bahrampour and Prof. V.Karimipour, and all members of Quantum Information Group and Optics Group at Sharif University of Technology.