The Hidden Subgroup Problem
hπΈππππ‘ πππππ hπππ π πππππππ’
The Hidden Subgroup ProblemProblem of great importance in Quantum Computationβ’ Most Q.A. that run exponentially faster than their classical
counterparts fall into the framework of HSPβ’ Simonβs Algorithm , Shorβs Algorithm for factoring , Shorβs discrete
logarithm algorithm equivalent to HSP
Quantum Fourier TransformDiscrete Fourier Transform , maps the sequence of complex numbers onto an N periodic sequence of complex numbers
Quantum Fourier TransformDiscrete Fourier Transform , maps the sequence of complex numbers onto an N periodic sequence of complex numbers
Quantum Fourier Transform , acts on a quantum state and transforms it in the
quantum state
Quantum Fourier TransformQFT as a unitary matrix:
Can implemented in a quantum circuit as a set of Hadamard and phase shift gates.
gates
Quantum Fourier TransformQFT as a unitary matrix:
Can implemented in a quantum circuit as a set of Hadamard and phase shift gates.
gates
Example 3 qubit QFT:
Shorβs Algorithm
Purpose: Factor an Integer
Shorβs Algorithm
Purpose: Factor an Integer (e.g. )
1. Choose a random integer a (e.g. )2. Define a function :
Shorβs 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:
Shorβs Algorithm
1. =
Shorβs Algorithm
1. =
2. =
Shorβs Algorithm
1. =
2. =
3.
Shorβs Algorithm
1. =
2. =
3.
4.
First register collapses into a superposition of the preimages of
Shorβs Algorithm
Restrict the study in the domain with N a multiple of the period
4.
5.
Shorβs Algorithm
Restrict the study in the domain with N a multiple of the period
4.
5. πΉπ= 1
β π βπ=0
πβ 1
βπ=0
πβ 1
πβ 2π π€
πβ ππ
ΒΏ π β© ΒΏ
Shorβs Algorithm
Restrict the study in the domain with N a multiple of the period
4.
5. πΉπ= 1
β π βπ=0
πβ 1
βπ=0
πβ 1
πβ 2π π€
πβ ππ
ΒΏ π β© ΒΏ
ΒΏπ π β©= 1βπ β
π :ππ’ππ‘ππππππ π
π β1
πβ 2π π€
πβ π₯0 π
ΒΏ π β©
Shorβs Algorithm
ΒΏπ π β©= 1βπ β
π :ππ’ππ‘ππππππ π
π β1
πβ 2π π€
πβ π₯0 π
ΒΏ π β©
Perform measurement: get a j (and thus a multiple of m)After k trials obtain k number multiples of m.
Shorβs Algorithm
ΒΏπ π β©= 1βπ β
π :ππ’ππ‘ππππππ π
π β1
πβ 2π π€
πβ π₯0 π
ΒΏ π β©
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 !
π0=1βππ=1β (ππ /2+1 ) (ππ /2β1 )=0πππ (π0)
Shorβs Algorithm
ΒΏπ π β©= 1βπ β
π :ππ’ππ‘ππππππ π
π β1
πβ 2π π€
πβ π₯0 π
ΒΏ π β©
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 !
π0=1βππ=1β (ππ /2+1 ) (ππ /2β1 )=0πππ (π0)
One of the factors may has a common factor with
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
Identity element e:
Inverse element
Elements of Group Theory
Group G: set of elements {g} , equipped with an internal composition law
Identity element e:
Inverse element
If : Abelian GroupSubgroup: 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
Identity element e:
Inverse element
If : Abelian GroupSubgroup: 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
The Hidden Abelian Subgroup ProblemLet 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.
The Hidden Abelian Subgroup ProblemLet 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 βblack boxβ that performs
The Hidden Abelian Subgroup ProblemThe Simplest Example
Let e.g. separates cosets
and
The Hidden Abelian Subgroup ProblemThe Simplest Example
Let e.g. separates cosets
and
We donβt know M, d, H but we know G and we have a βmachineβ performing the function f
The Hidden Abelian Subgroup ProblemThe Simplest Example
Map:
Quantum circuit:
The Hidden Abelian Subgroup Problem
1. =
The Hidden Abelian Subgroup Problem
=
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.
The Hidden Abelian Subgroup Problem
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.
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.
References
Chris Lomont: http://arxiv.org/pdf/quant-ph/0411037v1.pdfFrederic Wang http://arxiv.org/ftp/arxiv/papers/1008/1008.0010.pdf
http://en.wikipedia.org/wiki/Quantum_Fourier_transform
Top Related