hadamard.md

Hadamard Gate

The hadamard's gate (H-gate) primary function is to place a qubit into a superposition. It places the qubit into an equal 50/50 probability between |0⟩ and |1⟩.

When represented mathematically to a basis state |0⟩ you get

H |0⟩ = 1/√2 (|0⟩ + |1⟩).

To expand on why 1/√2 is used, lets look at the basis states as column vectors.

|0⟩ = | 1 |
      | 0 |

|1⟩ = | 0 |
      | 1 |

This is the Hadamard gate represented as a matrix

H = 1/√2 * | 1  1 |
           | 1 -1 |

Now we do matrix multiplication, in this example with |0⟩

H |0⟩ = (1/√2) * | 1  1 | * | 1 | 
                 | 1 -1 |   | 0 |            

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

      = (1/√2) * | 1 |
                 | 1 |

The Hadamard gate is reverisble, meaning if you apply it twice to the same qubit you end up in the same state.

Born Rule

The Born rule is the reason for 1/√2. In quantum mechanics we need a system to interpret a wave function of a quantum system into a definite result. The Born rule states that the probability of obtaining a particular measurement is equal to the square of the absolute value of the probability amplitude.

Probability of outcome = amplitude²

Hence if the amplitude is 1/√2

(1/√2)² = 1/2 = 50%

And this allows us to arrive at the 50% probability of collapsing into either |0⟩ or |1⟩.