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

Quantum Fourier Transform

Example 3 qubit QFT:

Shor’s Algorithm

Can be implemented by the Quantum Circuit:

Shor’s Algorithm

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

Shor’s Algorithm

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

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

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

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

The Hidden Abelian Subgroup Problem

The Simplest Example

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

The Hidden Abelian Subgroup Problem The Simplest Example Quantum circuit:

The Hidden Abelian Subgroup Problem

References Chris Lomont: Frederic Wang