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The Hidden Subgroup Problem

The Hidden Subgroup Problem. The Hidden Subgroup Problem. Problem of great importance in Quantum Computation Most Q.A. that run exponentially faster than their classical counterparts fall into the framework of HSP

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The Hidden Subgroup Problem

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  1. The Hidden Subgroup Problem

  2. The Hidden Subgroup Problem • 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. Discrete Fourier Transform , maps the sequence of complex numbers onto an N periodic sequence of complex numbers Quantum Fourier Transform

  4. Discrete Fourier Transform , maps the sequence of complex numbers onto an N periodic sequence of complex numbers Quantum Fourier Transform Quantum Fourier Transform , acts on a quantum state and transforms it in the quantum state

  5. QFT as a unitary matrix: Quantum Fourier Transform Can implemented in a quantum circuit as a set of Hadamard and phase shift gates. gates

  6. QFT as a unitary matrix: Quantum Fourier Transform Can implemented in a quantum circuit as a set of Hadamard and phase shift gates. gates Example 3 qubit QFT:

  7. Shor’s Algorithm Purpose: Factor an Integer

  8. Shor’s Algorithm Purpose: Factor an Integer (e.g. ) 1. Choose a random integer a (e.g. ) 2. Define a function :

  9. Shor’s Algorithm Purpose: Factor an Integer (e.g. ) 1. Choose a random integer a (e.g. ) 2. Define a function : Can be implemented by the Quantum Circuit:

  10. Shor’s Algorithm 1. =

  11. Shor’s Algorithm 1. = 2. =

  12. Shor’s Algorithm 1. = 2. = 3.

  13. Shor’s Algorithm 1. = 2. = 3. 4. First register collapses into a superposition of the preimages of

  14. Shor’s Algorithm 4. Restrict the study in the domain with N a multiple of the period 5.

  15. Shor’s Algorithm 4. Restrict the study in the domain with N a multiple of the period 5.

  16. Shor’s Algorithm 4. Restrict the study in the domain with N a multiple of the period 5.

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

  18. Shor’s Algorithm Perform measurement: get a j (and thus a multiple of m) After k trials obtain k number multiples of m. . It is . Period is found !

  19. Shor’s Algorithm Perform measurement: get a j (and thus a multiple of m) After k trials obtain k number multiples of m. . It is . Period is found ! One of the factors may has a common factor with

  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 Identity element e: Inverse element

  22. Elements of Group Theory Group G: set of elements {g} , equipped with an internal composition law Identity element e: Inverse element If : Abelian Group 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 Identity element e: Inverse element If : Abelian Group Subgroup: a non empty set which is a group on its own, under the same composition law Cosets: H a subgroup of G. Choose an element g. The (left) coset of H in terms of g is Two cosets of H can either totally match or be totally different

  24. The Hidden Abelian Subgroup Problem Let G be a group , H a subgroup and X a set. Let . A function separates the cosets of H iff . The function separates the cosets.

  25. The Hidden Abelian Subgroup Problem Let G be a group , H a subgroup and X a set. Let . A function separates the cosets of H iff . The function separates the cosets. HSP: determine the subgroup H using information gained by the evaluation of . Assume that elements of G are encoded to basis states of a Quantum Computer. Assume that exists a “black box” that performs

  26. The Hidden Abelian Subgroup Problem The Simplest Example Let e.g. separates cosets and

  27. The Hidden Abelian Subgroup Problem The Simplest Example Let e.g. separates cosets and 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 Map: Quantum circuit:

  29. The Hidden Abelian Subgroup Problem 1. =

  30. The Hidden Abelian Subgroup Problem =

  31. The Hidden Abelian Subgroup Problem = Measure the second register. The function acquires a certain value . The first register has to collapse to those j that belong to the coset of H. Entanglement : computational speed up.

  32. The Hidden Abelian Subgroup Problem

  33. The Hidden Abelian Subgroup Problem A measurement will yield a value for M. Repeat and use Euclidean algorithm for the GCD to find M. Since the generating set can be determined and thus H.

  34. The Hidden Abelian Subgroup Problem A measurement will yield a value for M. Repeat and use Euclidean algorithm for the GCD to find M. Since the generating set can be determined and thus H.

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

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