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The Hidden Subgroup Problem
𝐸𝑙𝑒𝑓𝑡ℎ𝑒𝑟𝑖𝑜𝑠 𝑀𝑜𝑠𝑐ℎ𝑎𝑛𝑑𝑟𝑒𝑜𝑢
The Hidden Subgroup Problem
Problem of great importance in Quantum Computation
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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
Discrete Fourier Transform , maps the sequence of complex numbers 𝑥0 , 𝑥1 , … , 𝑥𝑁−1
onto an N periodic sequence of complex numbers 𝑋0 , 𝑋1 , … , 𝑋𝑁−1
𝑁−1
𝑋𝑘 =
𝑥𝑛 ⋅
𝑛=0
2𝜋𝚤
−
𝑒 𝑁 ⋅𝑘𝑛
𝑁−1
𝑥𝑛 ⋅ 𝜔−𝑘𝑛
=
𝑛=0
Quantum Fourier Transform
Discrete Fourier Transform , maps the sequence of complex numbers 𝑥0 , 𝑥1 , … , 𝑥𝑁−1
onto an N periodic sequence of complex numbers 𝑋0 , 𝑋1 , … , 𝑋𝑁−1
𝑁−1
𝑋𝑘 =
𝑥𝑛 ⋅
2𝜋𝚤
−
𝑒 𝑁 ⋅𝑘𝑛
𝑁−1
𝑥𝑛 ⋅ 𝜔−𝑘𝑛
=
𝑛=0
𝑛=0
Quantum Fourier Transform , acts on a quantum state
it in the quantum state
|𝑘 =
𝑁−1
𝑘=0 𝑥𝑘 ⋅
𝑁−1
1
√𝑁 𝑛=0
𝑁−1
𝑛=0 𝑥𝑛
⋅ |𝑛 and transforms
|𝑘
|𝑛 ⋅
2𝜋𝚤
−
𝑒 𝑁 ⋅𝑘𝑛
𝑁−1
|𝑛 ⋅ 𝜔−𝑘𝑛
=
𝑛=0
Quantum Fourier Transform
QFT as a unitary matrix:
𝐹𝑁 =
1
𝑁−1 𝑁−1
√𝑁 𝑛=0 ℓ=0
2𝜋𝚤
−
𝑒 𝑁 ⋅𝑘ℓ |ℓ
Can implemented in a quantum circuit as a set
of Hadamard and phase shift gates.
𝑛(𝑛+1)
2
gates
𝑘|
Quantum Fourier Transform
QFT as a unitary matrix:
𝐹𝑁 =
1
𝑁−1 𝑁−1
√𝑁 𝑛=0 ℓ=0
2𝜋𝚤
−
𝑒 𝑁 ⋅𝑘ℓ |ℓ
Can implemented in a quantum circuit as a set
of Hadamard and phase shift gates.
𝑛(𝑛+1)
2
gates
Example 3 qubit QFT:
𝑘|
Shor’s Algorithm
Purpose: Factor an Integer 𝑁0
Shor’s Algorithm
Purpose: Factor an Integer 𝑁0 (e.g.
𝑁0 = 21)
1. Choose a random integer a (e.g. 𝑎 = 11)
2. Define a function 𝑓 𝑥 = 𝑎 𝑥 𝑚𝑜𝑑𝑢𝑙𝑜 𝑁0 : 1,11,16,8,4,2,1,11,16
Shor’s Algorithm
Purpose: Factor an Integer 𝑁0 (e.g.
𝑁0 = 21)
1. Choose a random integer a (e.g. 𝑎 = 11)
2. Define a function 𝑓 𝑥 = 𝑎 𝑥 𝑚𝑜𝑑𝑢𝑙𝑜 𝑁0 : 1,11,16,8,4,2,1,11,16
Can be implemented by the Quantum Circuit:
Shor’s Algorithm
1. |𝜓0 =|0
⊗𝑛
|0
⊗𝑛
Shor’s Algorithm
1. |𝜓0 =|0
⊗𝑛
|0
⊗𝑛
2. |𝜓1 = 𝐻
⊗𝑛
|0
⊗𝑛
|0
⊗𝑛
=
1
√𝑁
𝑥∈2𝑛
|𝑥 |0
⊗𝑛
Shor’s Algorithm
1. |𝜓0 =|0
2. |𝜓1 = 𝐻
3. |𝜓2 =
⊗𝑛
⊗𝑛
|0
|0
1
𝑈𝑓
√𝑁
⊗𝑛
⊗𝑛
|0
𝑥∈2𝑛 |𝑥
⊗𝑛
|0
=
1
√𝑁
⊗𝑛
=
𝑥∈2𝑛
1
√𝑁
|𝑥 |0
𝑥∈2𝑛 |𝑥
⊗𝑛
|𝑓(𝑥)
Shor’s Algorithm
1. |𝜓0 =|0
2. |𝜓1 = 𝐻
3. |𝜓2 =
4. |𝜓3 =
⊗𝑛
⊗𝑛
|0
|0
⊗𝑛
⊗𝑛
|0
1
𝑈𝑓
𝑛 |𝑥
𝑥∈2
√𝑁
1
𝑚−1
|𝑥0 +
𝑘=0
√𝑚
⊗𝑛
|0
=
1
√𝑁
⊗𝑛
=
𝑥∈2𝑛
1
√𝑁
|𝑥 |0
𝑥∈2𝑛 |𝑥
⊗𝑛
|𝑓(𝑥)
𝑘 ⋅ 𝑟 |𝑓(𝑥0 )
First register collapses into a superposition of the preimages of 𝑓(𝑥)
Shor’s Algorithm
4. |𝜓3 =
1
√𝑚
𝑚−1
𝑘=0 |𝑥0
+𝑘⋅𝑟
Restrict the study in the domain 𝑍𝑁 ≤ 𝑍 with N a multiple of the period 𝑁 = 𝑚 ⋅ 𝑟
5. |𝜓4 =
1
𝑈𝑄𝐹𝑇
√𝑚
𝑚−1
𝑘=0 |𝑥0
+𝑘⋅𝑟
Shor’s Algorithm
4. |𝜓3 =
1
√𝑚
𝑚−1
𝑘=0 |𝑥0
+𝑘⋅𝑟
Restrict the study in the domain 𝑍𝑁 ≤ 𝑍 with N a multiple of the period 𝑁 = 𝑚 ⋅ 𝑟
5. |𝜓4 =
1
𝑈𝑄𝐹𝑇
√𝑚
𝑚−1
𝑘=0 |𝑥0
+𝑘⋅𝑟
𝐹𝑁 =
1
𝑁−1 𝑁−1
√𝑁 𝑗=0
𝑖=0
2𝜋𝚤
− 𝑁 ⋅𝑗𝑖
𝑒
|𝑗 𝑖|
Shor’s Algorithm
4. |𝜓3 =
1
√𝑚
𝑚−1
𝑘=0 |𝑥0
+𝑘⋅𝑟
Restrict the study in the domain 𝑍𝑁 ≤ 𝑍 with N a multiple of the period 𝑁 = 𝑚 ⋅ 𝑟
5. |𝜓4 =
1
𝑈𝑄𝐹𝑇
√𝑚
|𝜓𝑓 =
1
𝑚−1
𝑘=0 |𝑥0
𝑁−1
√𝑟 𝑗:𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 𝑚
+𝑘⋅𝑟
𝐹𝑁 =
2𝜋𝚤
− 𝑁 ⋅𝑥0 𝑗
𝑒
|𝑗
1
𝑁−1 𝑁−1
√𝑁 𝑗=0
𝑖=0
2𝜋𝚤
− 𝑁 ⋅𝑗𝑖
𝑒
|𝑗 𝑖|
Shor’s Algorithm
|𝜓𝑓 =
1
𝑁−1
√𝑟 𝑗:𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 𝑚
2𝜋𝚤
− 𝑁 ⋅𝑥0 𝑗
𝑒
|𝑗
Perform measurement: get a j (and thus a multiple of m)
After k trials obtain k number multiples of m.
Shor’s Algorithm
|𝜓𝑓 =
1
𝑁−1
√𝑟 𝑗:𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 𝑚
2𝜋𝚤
− 𝑁 ⋅𝑥0 𝑗
𝑒
|𝑗
Perform measurement: get a j (and thus a multiple of m)
After k trials obtain k number multiples of m.
𝑁
𝑚 = 𝐺𝐶𝐷(𝑗1 , 𝑗2 , … , 𝑗𝑘 ) . It is 𝑟 = . Period is found !
𝑚
𝑎0 = 1 → 𝑎𝑟 = 1 → 𝑎𝑟/2 + 1 𝑎𝑟/2 − 1 = 0
𝑚𝑜𝑑(𝑁0 )
Shor’s Algorithm
|𝜓𝑓 =
1
𝑁−1
√𝑟 𝑗:𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 𝑚
2𝜋𝚤
− 𝑁 ⋅𝑥0 𝑗
𝑒
|𝑗
Perform measurement: get a j (and thus a multiple of m)
After k trials obtain k number multiples of m.
𝑁
𝑚 = 𝐺𝐶𝐷(𝑗1 , 𝑗2 , … , 𝑗𝑘 ) . It is 𝑟 = . Period is found !
𝑚
𝑎0 = 1 → 𝑎𝑟 = 1 → 𝑎𝑟/2 + 1 𝑎𝑟/2 − 1 = 0
𝑚𝑜𝑑(𝑁0 )
One of the factors may has a common factor with 𝑁0
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
∀𝑎, 𝑏 ∈ 𝐺 → 𝑎 ∘ 𝑏 = 𝑐 ∈ 𝐺
𝑎∘𝑏 ∘𝑐 =𝑎∘ 𝑏∘𝑐
Identity element e: 𝑒 ∘ 𝑎 = 𝑎 ∘ 𝑒 = 𝑎
Inverse element ∀𝑎 ∈ 𝐺 ∃ 𝑎−1 : 𝑎−1 ∘ 𝑎 = 𝑎 ∘ 𝑎−1 = 𝑒
Elements of Group Theory
Group G: set of elements {g} , equipped with an internal composition law
∀𝑎, 𝑏 ∈ 𝐺 → 𝑎 ∘ 𝑏 = 𝑐 ∈ 𝐺
𝑎∘𝑏 ∘𝑐 =𝑎∘ 𝑏∘𝑐
Identity element e: 𝑒 ∘ 𝑎 = 𝑎 ∘ 𝑒 = 𝑎
Inverse element ∀𝑎 ∈ 𝐺 ∃ 𝑎−1 : 𝑎−1 ∘ 𝑎 = 𝑎 ∘ 𝑎−1 = 𝑒
If 𝑎 ∘ 𝑏 = 𝑏 ∘ 𝑎 : Abelian Group
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
∀𝑎, 𝑏 ∈ 𝐺 → 𝑎 ∘ 𝑏 = 𝑐 ∈ 𝐺
𝑎∘𝑏 ∘𝑐 =𝑎∘ 𝑏∘𝑐
Identity element e: 𝑒 ∘ 𝑎 = 𝑎 ∘ 𝑒 = 𝑎
Inverse element ∀𝑎 ∈ 𝐺 ∃ 𝑎−1 : 𝑎−1 ∘ 𝑎 = 𝑎 ∘ 𝑎−1 = 𝑒
If 𝑎 ∘ 𝑏 = 𝑏 ∘ 𝑎 : Abelian Group
Subgroup: 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
𝑔𝐻 = 𝑔ℎ | ℎ ∈ 𝐻
Two cosets of H can either totally match or be totally different
The Hidden Abelian Subgroup Problem
Let G be a group , H a subgroup and X a set.
Let 𝑔1 𝐻 = 𝑔2 𝐻 . A function 𝑓 𝑔 separates the cosets of H iff 𝑓 𝑔1 = 𝑓 𝑔2 .
The function 𝑓 separates the cosets.
The Hidden Abelian Subgroup Problem
Let G be a group , H a subgroup and X a set.
Let 𝑔1 𝐻 = 𝑔2 𝐻 . A function 𝑓 𝑔 separates the cosets of H iff 𝑓 𝑔1 = 𝑓 𝑔2 .
The function 𝑓 separates the cosets.
HSP: determine the subgroup H using information gained by the evaluation of 𝑓.
Assume that elements of G are encoded to basis states of a Quantum Computer.
Assume that exists a “black box” that performs 𝑈𝑓
The Hidden Abelian Subgroup Problem
The Simplest Example
Let 𝐺 = 𝑍𝑁 , + e.g. 𝑍3 = 0,1,2
𝑓: 𝐺 → 𝑋 separates cosets
𝐻 = 𝑑 and 𝐻 = 𝑀
The Hidden Abelian Subgroup Problem
The Simplest Example
Let 𝐺 = 𝑍𝑁 , + e.g. 𝑍3 = 0,1,2
𝑓: 𝐺 → 𝑋 separates cosets
𝐻 = 𝑑 and 𝐻 = 𝑀
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
Map: 𝐺: |0 , |1 , |2 , … , |𝑁 − 1
𝐻: |0 , |𝑑 , |2𝑑 , … , | 𝑀 − 1 𝑑
Quantum circuit:
The Hidden Abelian Subgroup Problem
1. |𝜓0 =|0 |0
The Hidden Abelian Subgroup Problem
1. |𝜓0
=|0 |0
2. |𝜓1 = UQFT |0 |0 =
1
√𝑁
𝑁−1
|𝑗 |0
𝑗=0
The Hidden Abelian Subgroup Problem
1. |𝜓0
=|0 |0
2. |𝜓1 = UQFT |0 |0 =
1
√𝑁
𝑁−1
|𝑗 |0
𝑗=0
3. |𝜓2 = 𝑈𝑓
1
√𝑁
𝑁−1
|𝑗 |0 =
𝑗=0
1
√𝑁
𝑁−1
|𝑗 |𝑓(𝑗)
𝑗=0
Measure the second register. The function acquires a certain value 𝑓(𝑗0 ). 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
|𝜓3 =
1
√𝑀
|𝑗0 + ℎ |𝑓(𝑗0 ) =
ℎ∈𝐻
|𝜓4 =
1
𝑛
√𝑑 𝑖=0
1
√𝑀
2𝜋𝚤𝑗0
𝑒 𝑁 ⋅𝑡𝑀
𝑀−1
|𝑗0 + 𝑠𝑑 |𝑓(𝑗0 )
𝑠=0
|𝑡𝑀
The Hidden Abelian Subgroup Problem
|𝜓3 =
1
√𝑀
|𝑗0 + ℎ |𝑓(𝑗0 ) =
ℎ∈𝐻
|𝜓4 =
1
𝑛
√𝑑 𝑖=0
1
√𝑀
2𝜋𝚤𝑗0
𝑒 𝑁 ⋅𝑡𝑀
𝑀−1
|𝑗0 + 𝑠𝑑 |𝑓(𝑗0 )
𝑠=0
|𝑡𝑀
A measurement will yield a value for M. Repeat and use Euclidean algorithm for the GCD to find
M. Since 𝑁 = 𝑀 ⋅ 𝑑 the generating set can be determined and thus H.
The Hidden Abelian Subgroup Problem
|𝜓3 =
1
√𝑀
|𝑗0 + ℎ |𝑓(𝑗0 ) =
ℎ∈𝐻
|𝜓4 =
1
𝑛
√𝑑 𝑖=0
1
√𝑀
2𝜋𝚤𝑗0
𝑒 𝑁 ⋅𝑡𝑀
𝑀−1
|𝑗0 + 𝑠𝑑 |𝑓(𝑗0 )
𝑠=0
|𝑡𝑀
A measurement will yield a value for M. Repeat and use Euclidean algorithm for the GCD to find
M. Since 𝑁 = 𝑀 ⋅ 𝑑 the generating set can be determined and thus H.
References
Chris Lomont: http://arxiv.org/pdf/quant-ph/0411037v1.pdf
Frederic Wang http://arxiv.org/ftp/arxiv/papers/1008/1008.0010.pdf
http://en.wikipedia.org/wiki/Quantum_Fourier_transform