The Hidden Subgroup Problem. Problem of great importance in Quantum Computation Most Q.A. that run...
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![Page 1: The Hidden Subgroup Problem. Problem of great importance in Quantum Computation Most Q.A. that run exponentially faster than their classical counterparts.](https://reader037.fdocuments.in/reader037/viewer/2022110322/56649d085503460f949da2b2/html5/thumbnails/1.jpg)
The Hidden Subgroup Problem
h𝐸𝑙𝑒𝑓𝑡 𝑒𝑟𝑖𝑜𝑠 h𝑀𝑜𝑠𝑐 𝑎𝑛𝑑𝑟𝑒𝑜𝑢
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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
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Quantum Fourier TransformDiscrete Fourier Transform , maps the sequence of complex numbers onto an N periodic sequence of complex numbers
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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
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Quantum Fourier TransformQFT as a unitary matrix:
Can implemented in a quantum circuit as a set of Hadamard and phase shift gates.
gates
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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:
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Shor’s Algorithm
Purpose: Factor an Integer
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Shor’s Algorithm
Purpose: Factor an Integer (e.g. )
1. Choose a random integer a (e.g. )2. Define a function :
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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:
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Shor’s Algorithm
1. =
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Shor’s Algorithm
1. =
2. =
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Shor’s Algorithm
1. =
2. =
3.
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Shor’s Algorithm
1. =
2. =
3.
4.
First register collapses into a superposition of the preimages of
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Shor’s Algorithm
Restrict the study in the domain with N a multiple of the period
4.
5.
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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 ⟩ ¿
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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 ⟩
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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.
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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)
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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
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Elements of Group Theory
Group G: set of elements {g} , equipped with an internal composition law
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Elements of Group Theory
Group G: set of elements {g} , equipped with an internal composition law
Identity element e:
Inverse element
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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
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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
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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.
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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
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The Hidden Abelian Subgroup ProblemThe Simplest Example
Let e.g. separates cosets
and
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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
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The Hidden Abelian Subgroup ProblemThe Simplest Example
Map:
Quantum circuit:
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The Hidden Abelian Subgroup Problem
1. =
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The Hidden Abelian Subgroup Problem
=
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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.
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The Hidden Abelian Subgroup Problem
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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.
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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.
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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