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"While I am certainly not asking you to close your eyes to the experiences of earlier generations, I want to advise you not to conform too soon and to resist the pressure of practical necessity. Free imagination is the inestimable prerogative of youth and it must be cherished and guarded as a treasure."
AP Photo/Scanpix.
Lisens: Begrenset gjenbruk
Felix Bloch
1905-1983
Bloch sphere and quantum bit
布洛赫球及量子位元
Electron: a particle & a wave(state)
9.1093837015(28)×10−31 kg
⎢𝚿⟩
Single electron
Superposition
⎢𝚿⟩ =𝛼⎢𝚿1⟩ + 𝛽⎢𝚿2⟩ + ...
×
Spin ½ system
Spin ½ system in quantum state
|↑⟩
|↓⟩
|0⟩
|1⟩
Bloch Sphere
Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit)
Quantum states
Point on surface:
Pure quantum state
Point insided:
Mixed quantum state
|↑⟩
|↓⟩
Spin ½ & Qbit
|↑⟩
|↓⟩
|0⟩
|1⟩
Spin 1/2
qubit
Two eigenstates are orthogonal
|↑⟩≠ ∑ any other |?⟩ (|↓⟩)
⟨↓|↑⟩ = 0
1 |
0 |
|↑⟩:
|↓⟩:
0 |
1 |
⟨↓|↑⟩ =
0 | 1 |
1 |
0 |
Superpostion of two eigenstates
|𝚿⟩= 𝛂|↑⟩+ 𝝱|↓⟩
Probability:
⟨𝚿|𝚿⟩ = 𝛂2⟨↑|↑⟩+𝛂𝝱*⟨↓|↑⟩+𝝱𝛂*⟨↑|↓⟩+𝝱2⟨↓|↓⟩
Probability = 1
𝛂2+𝝱2 = 1
State in terms of 𝜃 & 𝜑
𝛂2+𝝱2 = 1
𝛂 = cos𝜃ei𝛿
𝝱 = sin𝜃ei𝜂
|𝚿⟩ = cos𝛄ei𝛿|↑⟩+ sin𝛄ei𝜂|↓⟩
= ei𝛿(cos𝛄|↑⟩+ sin𝛄ei(𝜂-𝛿)|↓⟩)
𝛿-𝜂 ≡ 𝜑
2-level quantum state
|𝚿⟩=cos𝛄|↑⟩+ sin𝛄ei(𝜂-𝛿)|↓⟩
State in terms of 𝜃 & 𝜑
= cos(𝜃/2)|↑⟩+ sin(𝜃/2)ei𝜑|↓⟩
North Pole: 𝛄=0 ( |↑⟩ )
|𝚿⟩=cos𝛄|↑⟩+ sin𝛄ei(𝜂-𝛿)|↓⟩
|↑⟩
|↓⟩
South Pole: 𝛄=𝛑/2 (|↓⟩)
|𝚿(𝜃,𝜑)⟩
⎢𝚿(𝜃,𝜑)⟩ : Eigenstate
|𝚿(𝜃,𝜑)⟩ = ℏ/2 |𝚿(𝜃,𝜑)⟩
Bit vs Qubit
0 &1
Before:
𝛂|0⟩+ 𝝱|1⟩
After:
|0⟩ or |1⟩
Qubit states
1 |
0 |
0 |
1 |
1 |
1 |
1 |
-1 |
|0⟩
|1⟩
(|0⟩+|1⟩)
(|0⟩-|1⟩)
|+⟩
|-⟩
Single Qubit Operation
NOT
GATE
0 | 1 |
1 | 0 |
NOT |0⟩ = |1⟩
NOT |1⟩ = |0⟩
X
Single Qubit Operation
H
GATE
H |0⟩ = |+⟩
H |1⟩ = | -⟩
Y rotation
0 | 1 |
1 | 0 |
NOT
GATE
H
GATE