the hidden subgroup problem. problem of great importance in quantum computation most q.a. that run...
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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