Normalization

Valid Quantum States

Not every pair of numbers is a valid qubit. The normalization condition requires:

|alpha|^2 + |eta|^2 = 1

This ensures the probabilities of all outcomes sum to 1. The quantity sqrt{alpha^2 + eta^2} is called the norm of the state.

import math

def norm(state):
    return math.sqrt(state[0]**2 + state[1]**2)

def is_normalized(state):
    return abs(norm(state) - 1.0) < 1e-9

print(is_normalized([1.0, 0.0]))   # True
print(is_normalized([0.6, 0.8]))   # True  (0.36 + 0.64 = 1)
print(is_normalized([3.0, 4.0]))   # False (9 + 16 = 25, norm = 5)

To fix an unnormalized state, divide each amplitude by the norm:

def normalize(state):
    n = norm(state)
    return [state[0] / n, state[1] / n]

print(normalize([3.0, 4.0]))  # [0.6, 0.8]

Your Task

Implement norm(state), is_normalized(state), and normalize(state).