Units#

Review & Summary#

Qubit#

A two-state quantum system; the quantum generalization of a classical bit.

Ket notation \(|\psi\rangle\)#

Quantum state vector in Hilbert space. In a chosen basis: \(|\psi\rangle = \alpha |0\rangle + \beta |1\rangle\) with \(|\alpha|^2 + |\beta|^2 = 1\).

Bloch sphere#

Geometric representation of qubit pure states on a unit sphere: \(|\psi\rangle = \cos(\theta/2)|0\rangle + e^{i\phi}\sin(\theta/2)|1\rangle\).

Pauli matrices#

\(\sigma^x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\), \(\sigma^y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}\), \(\sigma^z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\). Eigenvalues \(\pm 1\).

Born rule#

Probability of outcome \(m\): \(p(m|O,\psi) = \langle\psi|P_{O=m}|\psi\rangle = |\langle m|\psi\rangle|^2\).

Uncertainty principle#

\(\sigma_A \sigma_B \geq \tfrac{1}{2}|\langle [A,B] \rangle|\).

Schrödinger equation#

\(i\hbar \partial_t |\psi\rangle = H|\psi\rangle\). Time evolution: \(U(t) = e^{-iHt/\hbar}\).