1. The Hidden Subgroup Problem

2. Problem of great importance in Quantum Computation Most Q.A. that run exponentially faster than their classical counterparts fall into the framework of HSP Simon’s Algorithm, Shor’s Algorithm for factoring, Shor’s discrete logarithm algorithm equivalent to HSP

3. Quantum Fourier Transform

6. Example 3 qubit QFT:

7. Shor’s Algorithm

9. Can be implemented by the Quantum Circuit:

10. Shor’s Algorithm

17. Perform measurement: get a j (and thus a multiple of m) After k trials obtain k number multiples of m.

18. Shor’s Algorithm

20. Elements of Group Theory Group G: set of elements {g}, equipped with an internal composition law

21. Elements of Group Theory Group G: set of elements {g}, equipped with an internal composition law

22. Elements of Group Theory Group G: set of elements {g}, equipped with an internal composition law Subgroup: a non empty set which is a group on its own, under the same composition law

23. Elements of Group Theory Group G: set of elements {g}, equipped with an internal composition law Subgroup: a non empty set which is a group on its own, under the same composition law

24. The Hidden Abelian Subgroup Problem

26. The Simplest Example

27. The Hidden Abelian Subgroup Problem The Simplest Example We don’t know M, d, H but we know G and we have a “machine” performing the function f

28. The Hidden Abelian Subgroup Problem The Simplest Example Quantum circuit:

29. The Hidden Abelian Subgroup Problem

35. References Chris Lomont: http://arxiv.org/pdf/quant-ph/0411037v1.pdfhttp://arxiv.org/pdf/quant-ph/0411037v1.pdf Frederic Wang http://arxiv.org/ftp/arxiv/papers/1008/1008.0010.pdfhttp://arxiv.org/ftp/arxiv/papers/1008/1008.0010.pdf http://en.wikipedia.org/wiki/Quantum_Fourier_transform