The Hidden Subgroup Problem

h𝐸𝑙𝑒𝑓𝑡 𝑒𝑟𝑖𝑜𝑠 h𝑀𝑜𝑠𝑐 𝑎𝑛𝑑𝑟𝑒𝑜𝑢

The Hidden Subgroup ProblemProblem 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 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:

Shor’s Algorithm

Purpose: Factor an Integer

Shor’s Algorithm

Purpose: Factor an Integer (e.g. )

1. Choose a random integer a (e.g. )2. Define a function :

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:

Shor’s Algorithm

1. =

Shor’s Algorithm

1. =

2. =

Shor’s Algorithm

1. =

2. =

3.

Shor’s Algorithm

1. =

2. =

3.

4.

First register collapses into a superposition of the preimages of

Shor’s Algorithm

Restrict the study in the domain with N a multiple of the period

4.

5.

Shor’s Algorithm

Restrict the study in the domain with N a multiple of the period

4.

5. 𝐹𝑁= 1

√ 𝑁 ∑𝑗=0

𝑁− 1

∑𝑖=0

𝑁− 1

𝑒− 2𝜋 𝚤

𝑁⋅ 𝑗𝑖

¿ 𝑗 ⟩ ¿

Shor’s Algorithm

Restrict the study in the domain with N a multiple of the period

4.

5. 𝐹𝑁= 1

√ 𝑁 ∑𝑗=0

𝑁− 1

∑𝑖=0

𝑁− 1

𝑒− 2𝜋 𝚤

𝑁⋅ 𝑗𝑖

¿ 𝑗 ⟩ ¿

¿𝜓 𝑓 ⟩= 1√𝑟 ∑

𝑗 :𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒𝑜𝑓 𝑚

𝑁 −1

𝑒− 2𝜋 𝚤

𝑁⋅ 𝑥0 𝑗

¿ 𝑗 ⟩

Shor’s Algorithm

¿𝜓 𝑓 ⟩= 1√𝑟 ∑

𝑗 :𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒𝑜𝑓 𝑚

𝑁 −1

𝑒− 2𝜋 𝚤

𝑁⋅ 𝑥0 𝑗

¿ 𝑗 ⟩

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

Shor’s Algorithm

¿𝜓 𝑓 ⟩= 1√𝑟 ∑

𝑗 :𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒𝑜𝑓 𝑚

𝑁 −1

𝑒− 2𝜋 𝚤

𝑁⋅ 𝑥0 𝑗

¿ 𝑗 ⟩

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 !

𝑎0=1→𝑎𝑟=1→ (𝑎𝑟 /2+1 ) (𝑎𝑟 /2−1 )=0𝑚𝑜𝑑 (𝑁0)

Shor’s Algorithm

¿𝜓 𝑓 ⟩= 1√𝑟 ∑

𝑗 :𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒𝑜𝑓 𝑚

𝑁 −1

𝑒− 2𝜋 𝚤

𝑁⋅ 𝑥0 𝑗

¿ 𝑗 ⟩

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 !

𝑎0=1→𝑎𝑟=1→ (𝑎𝑟 /2+1 ) (𝑎𝑟 /2−1 )=0𝑚𝑜𝑑 (𝑁0)

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 “black box” 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 don’t know M, d, H but we know G and we have a “machine” 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