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quantum-state-superposition

Superposition

Superposition is the quantum principle that a qubit can exist in a linear combination of basis states simultaneously. Unlike a classical bit forced into a definite 0 or 1, a quantum qubit occupies a weighted sum of states until measured.

A single-qubit superposition is written $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ where $\alpha$ and $\beta$ are complex amplitudes with normalization $|\alpha|^2 + |\beta|^2 = 1$. The quantities $|\alpha|^2$ and $|\beta|^2$ are probabilities: measuring the qubit yields 0 with probability $|\alpha|^2$ and 1 with probability $|\beta|^2$.

Equal Superposition

An equal superposition of all $n$-qubit basis states is the uniform superposition $|s\rangle = \frac{1}{\sqrt{2^n}} \sum_{x=0}^{2^n-1} |x\rangle$. This state has maximum entropy and appears in quantum algorithms like Grover's search (starting state) and quantum Fourier transform.

Superposition and Interference

Superposition enables quantum interference. If two paths through a quantum computation interfere constructively, amplitude adds; if destructively, amplitudes cancel. This is why quantum algorithms can amplify the probability of correct answers and suppress incorrect ones.

Creation and Manipulation

The Hadamard gate creates equal superposition from a computational basis state: $H|0\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle) = |+\rangle$. Rotation gates like $R_X(\theta)$ and $R_Y(\theta)$ create arbitrary superpositions by tuning the angle.

Measurement Collapses Superposition

Measurement forces superposition into a definite basis state. Before measurement, the qubit is in superposition; after measurement, it is in a definite state. This irreversible collapse is why quantum information cannot be cloned and why measurement-based quantum computing is delicate.

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