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Quantum Algorithms Towards quantum codebreaking

Quantum Algorithms Towards quantum codebreaking. Artur Ekert. More general oracles. Quantum oracles do not have to be of this form. e.g. generalized controlled-U operation. n qubits. m qubits. Phase estimation problem. n qubits. m qubits. Phase estimation algorithm.

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Quantum Algorithms Towards quantum codebreaking

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  1. Quantum AlgorithmsTowards quantum codebreaking Artur Ekert

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

  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 STEP 1: H n qubits m qubits Recall Quantum Fourier Transform:

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

  7. Phase estimation algorithm Probability 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 0000 0001 1110 1111

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

  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 Estimate of p1 with prob. ||2 H Fny Estimate of p2 with prob. ||2

  14. Solving order-finding via phase estimation

  15. Solving order-finding via phase estimation

  16. Shor’s Factoring Algorithm H F2ny 2n qubits n qubits 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 0 1 1 1 0 1 0 1 energy annealing configurations 011101…01

  19. Adiabatic Annealing Final complicated Hamiltonian Initial simple Hamiltonian

  20. Coherent quantum phenomena in nature ?

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

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