Superposition

Superposition allows a quantum system to exist in multiple states at once. ### Mathematical Formulation A single-qubit state $|psi\rangle$ is written as: $$|\psi\rangle = \alpha |0\rangle + \beta |1\rangle$$ where $\alpha, \beta \in \mathbb{C}$ are complex probability amplitudes satisfying: $$|\alpha|^2 + |\beta|^2 = 1$$ ### The Hadamard Gate The Hadamard gate ($H$) transforms computational basis states into equal superpositions: $$H |0\rangle = \frac{|0\rangle + |1\rangle}{\sqrt{2}} = |+\rangle$$