Superposition (Quantum Computing)

quantum-mechanics quantum-computing

Definition

Superposition (Quantum Computing)

A superposition is a quantum state that is a linear combination of basis states.

For $n$ qubits,

$$|\psi ⟩=\sum _{i=0}^{2^n-1}{\alpha }_i|i⟩,{\alpha }_i\in \mathbb{C},\sum _i|{\alpha }_i{|}^2=1.$$

The coefficients ${\alpha }_i$ are probability amplitudes; the normalisation condition ensures $|\psi ⟩$ is a unit vector in Hilbert space. This generalises the classical bit, which can only be $|0⟩$ or $|1⟩$.

Measurement

Upon measurement in the computational basis, outcome $|i⟩$ occurs with probability $|{\alpha }_i{|}^2$ and the state collapses to $|i⟩$.