There exist phenomenon like the “interference pattern” that cannot be simply explained. One explanation is “probability with negative numbers”. So how can we simulate such a system?
And “if you cannot simulate a system efficiently, then the system can be seen as performing powerful computation”
About Quantum Computing
Idea:
setup the initial state of a quantum system → evolve wiring quantum mechanics → be useful for something
What is a bit?
In semiconductor, low voltage → 0, hight voltage → 1.
In quantum world, there are electrons spinning. Spinning “up” → 0, “down” → 1
For photons, there’s also horizontal and vertical
<aside> 💡 Quantum Mechanic First Law
If a system can be in one of two states |0>, |1>, then it can also be in a superposition of the two: $\alpha| 0> + \beta |1>$ where $\alpha$ and $\beta$ are two real numbers with $\alpha^2+\beta^2 = 1$
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A qubit is just a superposition of the 2 states. like 0.8|0> + 0.6|1>
<aside> 💡 Quantum Mechanic Second Law If you measure $|\Psi> = \alpha |0> + \beta|1>$ using a devicem then
the measurement device says “H” with probability $\alpha^2$, “V” with probability $\beta^2$. However, the output is certain
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So on the previous example, we feed in a superposition but we would only observe to a certain state
→ Measuring changes the state
→ No way to know $\alpha,\beta$ without changing them
<aside> 💡 Quantum Mechanic Third Law
a) Can build a device that rotates the state by an angle $\theta$
b) Can build a device that reflect the state about any line through origin
Corollary
For any two perpendicular vectors |u>, |v> on the unit circle, we can build a uv device that detect the probabilities in terms of u and v (change of basis or change of the state). So we can build a device for any |u>, |v> perpendiculars on the circle
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