Part 3/7:
- Matrices: Quantum operations are often expressed in terms of matrices, such as unitary matrices, which have special properties that preserve vector lengths when transforming states.
Qubits and Quantum States
A qubit's state can be mathematically represented by a column vector. For example, the states ( |0⟩ ) and ( |1⟩ ) correspond to the column vectors ( \begin{bmatrix} 1 \ 0 \end{bmatrix} ) and ( \begin{bmatrix} 0 \ 1 \end{bmatrix} ), respectively. Importantly, qubits can exist in superpositions like ( \alpha|0⟩ + \beta|1⟩ ), where ( |α|^2 + |β|^2 = 1 ).