Supported Mathematical Notation
This page documents the LaTeX mathematical notation supported in our documentation and provides examples of quantum computing mathematical expressions.
Dirac Notation
Ket vectors (quantum states):
\[|\psi\rangle = \alpha|0\rangle + \beta|1\rangle\]
Bra vectors (complex conjugate transpose):
\[\langle\psi| = \alpha^*\langle 0| + \beta^*\langle 1|\]
Inner products (probability amplitudes):
\[\langle\phi|\psi\rangle = \sum_i \phi_i^* \psi_i\]
Outer products (projection operators):
\[|\psi\rangle\langle\phi| = \begin{pmatrix} \psi_0 \phi_0^* & \psi_0 \phi_1^* \\ \psi_1 \phi_0^* & \psi_1 \phi_1^* \end{pmatrix}\]
Common Quantum States
Computational basis states:
\[|0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad |1\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}\]
Superposition states:
\[|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle) = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix}\]
\[|-\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle) = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ -1 \end{pmatrix}\]
Multi-Qubit Notation
Tensor product:
\[|00\rangle = |0\rangle \otimes |0\rangle = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}\]
Bell states (maximally entangled):
\[|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)\]
\[|\Phi^-\rangle = \frac{1}{\sqrt{2}}(|00\rangle - |11\rangle)\]
\[|\Psi^+\rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle)\]
\[|\Psi^-\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |10\rangle)\]
Operators and Matrices
Pauli matrices:
\[\sigma_x = X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\]
\[\sigma_y = Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}\]
\[\sigma_z = Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\]
Hadamard gate:
\[H = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}\]
CNOT gate (controlled-X):
\[\text{CNOT} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix}\]
Probability and Measurement
Born rule (measurement probability):
\[P(|i\rangle) = |\langle i|\psi\rangle|^2\]
Normalization condition:
\[\langle\psi|\psi\rangle = \sum_i |\alpha_i|^2 = 1\]
Expectation value of observable \(A\):
\[\langle A \rangle = \langle\psi|A|\psi\rangle\]
Evolution and Dynamics
Unitary evolution:
\[|\psi(t)\rangle = U(t)|\psi(0)\rangle\]
Schrödinger equation:
\[i\hbar\frac{d}{dt}|\psi\rangle = H|\psi\rangle\]
Time evolution operator:
\[U(t) = e^{-iHt/\hbar}\]
Rendering Notes
- All equations are rendered using MathJax
- Complex expressions may take a moment to load