Quantum Computing Basics: Qubits, Superposition, and Measurement

Single Qubit

In classical computing, the basic unit of information is a bit. A bit can only be in one of two states, either zero or one, and therefore it can be physically implemented by a two-state device.

A qubit is the basic unit of information in quantum computing. Unlike a bit, a qubit is not restricted to only $0$ or $1$ before measurement.

The two computational basis states are written as $|0\rangle$ and $|1\rangle$.

The two computational basis states are

$$ |0\rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \qquad |1\rangle = \begin{bmatrix} 0 \\ 1 \end{bmatrix}. $$

A general single-qubit state is written as

$$ |\psi\rangle = \alpha |0\rangle + \beta |1\rangle. $$

Here, $\alpha$ and $\beta$ are complex numbers called probability amplitudes.

Using column-vector notation, the same qubit can be written as

$$ |\psi\rangle = \begin{bmatrix} \alpha \\ \beta \end{bmatrix}. $$

For the qubit to be physically valid, the amplitudes must satisfy the normalization condition

$$ |\alpha|^2 + |\beta|^2 = 1. $$

The probability of measuring $0$ is $|\alpha|^2$, and the probability of measuring $1$ is $|\beta|^2$.

Therefore,

$$ P(0) = |\alpha|^2, \qquad P(1) = |\beta|^2. $$