Standard Notation

Question

Should we agree on some standard notations? This should make it much easier to understand new posts once a notation is established.

In particular, I think it would be helpful to agree on symbols for common gates beforehand. This applies to formulas and circuit diagrams.

I suggest we start a list of notation here, then we can refer to this thread to conveniently define the symbols we use and to encourage new users to follow our convention. Use one answer for a single symbol or a group of symbols that logically belong together. Give the symbol and the description in a header, and give context, explanation, references below it.

Different notations for the same entity might be acceptable, and the votes will indicate the most popular variant. We can also discuss it in comments, of course.

Answer 1

H: Hadamard

Definition/notation: $n$-qubit ($2^n$-dimensional) Hadamard denoted by $H^{\otimes n}$, with $H^{\otimes 0} = 1$ and the single qubit (2 dimensional) Hadamard $H = H^{\otimes 1}$

$$H = \frac{1}{\sqrt{2}}\begin{pmatrix}1 & 1 \\ 1 & -1\end{pmatrix},$$ $$H^{\otimes n} = H\otimes H^{\otimes \left(n-1\right)}$$

Mathjax: $H$, $H^{\otimes n}$

See also: Hadamard

Answer 2

$\operatorname{tr}$: trace

Definition: $$\operatorname{tr}A= \sum_k \langle k | A |k\rangle = \sum_k A_{kk}$$
MathJax: \operatorname{tr}
See also: trace

Answer 3

$X$, $Y$, $Z$: Pauli matrices

Definition: $$ \begin{array}{ccccc} X=\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array}\right), &&Y=\left(\begin{array}{cc} 0 & -i \\ i & 0 \\ \end{array}\right), &&Z=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \\ \end{array} \right) \end{array} $$ MathJax: $X$, $Y$, $Z$
See also: Pauli matrices

Answer 4

$^{\mathsf{T}}$, $^\dagger$: transpose, conjugate transpose

Definition: The transpose of a matrix $M$, denoted by $M^{\mathsf{T}}$, has elements $M^{\mathsf{T}}_{ij} = M_{ji}$. The conjugate transpose additionally performs a complex conjugation $^*$: $M^{\dagger}_{ij} = M_{ji}^*$.

MathJax: $^{\mathsf{T}}$, $^\dagger$

See also: transpose, conjugate transpose

Answer 5

$\rho$: Density operator

Definition: $\rho$ is a positive semidefinite, normalized operator ($\rho\geq 0$, $\mathrm{tr } \rho = 1$)
MathJax: $\rho$
See also: Density matrix

Answer 6

$\mathbb{I}$: Identity operator

Definition: $$\mathbb{I}=\left( \begin{array}{cccc} 1 & 0 & \dots & 0 \\ 0 & 1 & & 0 \\ \vdots & & \ddots & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} \right)$$ MathJax: \mathbb{I}

See also: Identity Matrix

Answer 7

$C(U)_{ij}$: controlled-Unitary

Definition: controlled Unitary, where $i$ is the control and $j$ is the target.
Mathjax: C(U)_{ij}
circuit diagram (created with overleaf):

Special cases:
$U=X$ (CNOT gate)

$U=Z$ (controlled-Phase gate)

See also: quantum gates