Quantum computing

Some math

Magnitude of a complex number

|a + ib| = √(a² + b²)

Eg:

|2 + 4i| = √(4 + 16) = 5

Quantum mechanics

Double-slit experiment

    │       │             │
    │       │             │
    │       │ slit 1      │ 
    │                     │
    │       │             │
    │       │             │
src                       │
    │       │             │
    │       │             │
    │                     │
    │       │ slit 2      │
    │       │             │
    │       │             │
 Screen1  Screen2     Detector

Superposition principle

If a system can be in any of a set of k states, it can also be in any linear superposition of the k states.

α₀|0⟩ + α₁|1⟩ + ...  + αₖ₋₁|k-1⟩

where

αᵢ∈ ℂ     // probability amplitude

and

k-1
 ∑  |αᵢ|² = 1   // DOUBT: What's the need for absolute here? It's already squared, right?
i=0

ie, the system is in state i with probability αᵢ.

Examples

Example 1:

Number of states = k = 3

(1/√3) |0⟩ + (i/√3) |1⟩ + (1/√3) |2⟩

 2
 ∑  |αᵢ|² = (1/3) + (1/3) + (1/3) = 1     ✓
i=0

Example 2:

Number of states = k = 3

(1/2) |0⟩ - (1/2) |1⟩ + ((1/2) + (i/2)) |2⟩

 2
 ∑  |αᵢ|² = (1/4) + (1/4) + ((1/4 + 1/4)) = 1     ✓
i=0

Measurement axioms

Geometrical representation of a quantum system

As vectors (or as matrices).

|ψ⟩ = α₀|0⟩ + α₁|1⟩ + ...  + αₖ₋₁|k-1⟩

Could be written as

⎡  α₀  ⎤
⎢  α₁  ⎥
⎢  ..  ⎥
⎢  ..  ⎥
⎣ αₖ₋₁ ⎦

which will be a unit vector (because magnitude would be 1).

(The usual vector notation with little arrows on top is known as the Dirac notation.)

Qubits

Systems with just 2 levels.

ie, a k-level system with k=2.

Like a Hydrogen atom with 2 energy states.

+---------------+ |1⟩
|               |
|   +---+       |
|   |   |       |
|   | █ |       |
|   |   |       |
|   +---+ |0⟩   |
|               |
+---------------+

A representation of an atom with 2 energy states.

The █ is the atom's nucleus.

An example system:

(1/√2) |0⟩ + ((1/2) + (i/2)) |1⟩

Qubit is like

|ψ⟩ = α₀ |0⟩ + α₁ |1⟩


        |1⟩

        ^                
        |                
     α₁ |┄┄┄┄┄┄┄. |ψ⟩     
        |      /┆        
        |     / ┆        
        |    /  ┆        
        |   /   ┆        
        |  /    ┆        
        | /     ┆        
        |/θ     ┆        
        +------------> |0⟩
               α₀

Probabilities:

Measurement in arbitrary basis

        |1⟩        |u'⟩      
         │        /      
         │       / 
         │      /  
         │     /    
         │    /      
         │   /      .... |ψ⟩ = α₀ |0⟩ + α₁ |1⟩
         │  /    ....  
         │ /  ....      
         │/....  θ       
         +────────────> |0⟩
          \
           \
            \           θ'
             \
              \
               \
                \
                 \ |u⟩

(the dotted line is meant to be a straight line ☻ )

 - θ = ∠ between |ψ⟩ and |0⟩
 - θ' = ∠ between |ψ⟩ and |u⟩
u⟩ and u'⟩ are the arbitrary basis.
ψ⟩ can also be written as: 
|ψ⟩ = cos²θ' |u⟩ + sin²θ'

Hadamard basis

A special basis.

Consists of |+⟩ and |-⟩.

|+⟩ = (1/√2) |0⟩ + (1/√2) |1⟩
|-⟩ = (1/√2) |0⟩ - (1/√2) |1⟩

General

Doubts