“ Implementing quantum algorithms using quantum logic gates made from optical components .”

1. “Implementing quantum algorithms using quantum logic gates made from optical components.” Mark Tame

2. Promising Technologies for Quantum Computing • Ion Traps • Quantum Dots • Josephson Junctions • Nuclear Spins in Silicon / Molecules • Linear Optics (KLM proposal) E. Knill, R. Laflamme and G. J. Milburn, "A scheme forefficientquantum computation with linear optics",Nature 409, 46 (2001).

3. Three Key Principles in KLM Proposal • Conditional Non-linear Sign (NS) gates for two photon states (today) • Teleportation to achieve efficiency • Error Correction to achieve scalability

4. Conditional Non-linear Sign (NS) gates for two photon states • Part I – Optical Components • Part II – Quantum Logic Gates • Part III – Solving Quantum Algorithms

5. Part I - Optical Components

6. Dual Rail Basis b’ a’ b a Qubit: As opposed to:

7. B Bt b b b’ b’ a a a’ a’ Beamsplitter

8. Phase Shifter b b’ p a a’

9. B p Hadamard gate b b’ a a’ H > > | yout | yin

10. b b’ K a a’ Kerr gate Quantum Classical

11. Part II – Quantum Logic Gates

12. > > > > | A | B | A’ | B’ CNOT – Version I (non-linear components) d’ d 0 0 c’ c = | 1 L > = | 1 L > K 1 1 B B b’ b 0 1 a’ a = | 0 L > = | 1 L > p p 1 0 H H UCN

13. CNOT – Version I (non-linear components) A B A’ B’ 0 0 1 1 0 1 0 1 0 0 1 1 0 1 1 0

14. Non-linear Sign change gate ( NS gate ) | yc > = c2 | 2 > c1 | 1 > c0 | 0 > + + Btcb c’ c Bba Bba b’ b | 1 > q2 a’ a | 0 > q1 q1 | y 1 | y 2 | y 3 | y 4 > > > > c c’ NS b b’ | 1 > | 1 > Detection Condition (D.C) a a’ | 0 > | 0 >

15. Non-linear Sign change gate ( NS gate )

16. Non-linear Sign change gate ( NS gate )

17. > > > > | A’ | B | A | B’ CNOT – Version II (linear components based on KLM NS gate proposal) b’ b 0 0 > | 1 | 1 > NS D. C | 0 > | 0 > = | 1 L > = | 1 L > a a’ Bad Bad 1 1 Bba Bba NS d’ d | 1 > > 0 | 1 1 D. C | 0 > | 0 > c’ c = | 0 L > = | 1 L > 1 0 | y x | y y > >

18. CNOT – Version II (linear components) A B A’ B’ 0 0 1 1 0 1 0 1 0 0 1 1 0 1 1 0

19. Theoretical Experimental J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph and D. Branning, “Demonstration of an all-optical quantum computer controlled-NOT gate",Nature 426, 264 (2003). “The next step will be to incorporate the gate reported here in simple optical circuits to demonstrate simple algorithms and error correcting schemes.”

20. Part III – Solving Quantum Algorithms

21. Deutsch’s Problem Classical: Requires at least 2 attempts to find out if f(x) is balanced or constant Quantum:

22. Quantum Optical Circuit for Deutsch’s Problem B B d’ d 1 | 0 L > | x > | x > c’ c p p 0 H H Uf B b’ b 0 | 1 L > | y + f(x) > | y > a’ a p 1 H Depends on f(x)

23. Quantum Optical Circuit for generalised Deutsch/Jozsa Algorithm Classical: Requires around this many attempts to find out if f(x) is balanced or constant Quantum: Efficiency problems from Detection Conditions for NS gates in Quantum CNOT gates +/- 1 if f(x) is const 0 if f(x) is balanced Amplitude for all zero states: is

24. Next Time • Teleportation to achieve efficiency • Error Correction to achieve scalability • Decoherence-free subspaces References: E. Knill, R. Laflamme and G. J. Milburn, "A scheme forefficientquantum computation with linear optics",Nature 409, 46 (2001). J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph and D. Branning, “Demonstration of an all-optical quantum computer controlled-NOT gate",Nature 426, 264 (2003). D. Deutsch and R. Jozsa, "Rapid solution of problems by quantum computation", Proceedings of the Royal Society of London A 439 (1992) 553-558. M. A. Nielsen and I. L. Chuang, “Quantum Computation and Quantum Information”, Cambridge University Press, Cambridge (2000).