1. Quantum Algorithms Towards quantum codebreaking Artur Ekert

2. More general oracles Quantum oracles do not have to be of this form n qubits m qubits e.g. generalized controlled-U operation

3. Phase estimation problem n qubits m qubits

4. Phase estimation algorithm Suppose p is an n-bit number: Recall Quantum Fourier Transform:

5. Phase estimation algorithm n qubits m qubits H STEP 1: Recall Quantum Fourier Transform:

6. Phase estimation algorithm n qubits m qubits H STEP 2: Apply the reverse of the Quantum Fourier Transform FnyFny But what if p’ has more than n bits in its binary representation ?

7. Phase estimation algorithm 00000001 001000110100010101100111100010011010101111001101 11101111 Probability

8. Phase estimation - solution n qubits m qubits H FnyFny

9. Order-finding problem PRELIMINARY DEFINITIONS: This is a group under multiplication mod N For example

10. Order-finding problem PRELIMINARY DEFINITIONS: For example (period 6)

11. Order-finding problem Order finding and factoring have the same complexity. Any efficient algorithm for one is convertible into an efficient algorithm for the other.

12. Solving order-finding via phase estimation n qubits m qubits Suppose we are given an oracle that multiplies y by the powers of a

13. Solving order-finding via phase estimation H FnyFny Estimate of p 1 with prob. |  | 2 Estimate of p 2 with prob. |  | 2

14. Solving order-finding via phase estimation

16. Shor’s Factoring Algorithm 2n qubits n qubits H F 2n y Quantum factorization of an n bit integer N

17. Wacky ideas for the future Particle statistics in interferometers, additional selection rules ? Beyond sequential models – quantum annealing? Holonomic, geometric, and topological quantum computation? Discover (rather than invent) quantum computation in Nature?

18. Beyond sequential models … Interacting spins configurations energy 00011111 011101…01 annealing

19. Adiabatic Annealing Initial simple Hamiltonian Final complicated Hamiltonian

20. Coherent quantum phenomena in nature ?

21. Further Reading http://cam.qubit.org Centre for Quantum Computation University of Cambridge, DAMTP