# Pauli

 Other toolboxes required Pauli Produces a Pauli operator none GellMannGenGellMannGenPauliPauliChannel Special states, vectors, and operators

Pauli is a function that produces the 2-by-2 Pauli X, Y, Z, or identity operator, as defined here:

$X = \begin{bmatrix}0 & 1\\ 1 & 0\end{bmatrix}, \ \ Y = \begin{bmatrix}0 & -i\\ i & 0\end{bmatrix}, \ \ Z = \begin{bmatrix}1 & 0\\ 0 & -1\end{bmatrix}, \ \ I = \begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix}.$

This function can also produce multi-qubit Pauli operators.

## Syntax

• P = Pauli(IND)
• P = Pauli(IND,SP)

## Argument descriptions

• IND: An index indicating which Pauli operator you would like to be generated. Values of 1, 2, 3, and 0 correspond to the Pauli X, Y, Z, and identity operators, respectively. Values of 'I', 'X', 'Y', and 'Z' are also accepted, and indicate the Pauli identity, X, Y, and Z operators, respectively. If IND is a vector then this function returns a multi-qubit Pauli operator whose action on the K-th qubit is the same as Pauli(IND(K)).
• SP (optional, default 1): A flag (either 1 or 0) indicating that the Pauli operator produced should or should not be sparse.

## Examples

### Single-qubit examples

>> full(Pauli('X'))

ans =

0     1
1     0

>> full(Pauli(1))

ans =

0     1
1     0

>> full(Pauli(0))

ans =

1     0
0     1

>> full(Pauli('Y'))

ans =

0.0000 + 0.0000i   0.0000 - 1.0000i
0.0000 + 1.0000i   0.0000 + 0.0000i

>> Pauli('Z',1) % sparse Pauli Z operator

ans =

(1,1)        1
(2,2)       -1

### Multi-qubit examples

>> full(Pauli('XZI')) % the three-qubit Pauli operator X \otimes Z \otimes I

ans =

0     0     0     0     1     0     0     0
0     0     0     0     0     1     0     0
0     0     0     0     0     0    -1     0
0     0     0     0     0     0     0    -1
1     0     0     0     0     0     0     0
0     1     0     0     0     0     0     0
0     0    -1     0     0     0     0     0
0     0     0    -1     0     0     0     0

>> Pauli([0,1,2]) % the three-qubit Pauli operator I \otimes X \otimes Y

ans =

(4,1)           0 + 1.0000i
(3,2)           0 - 1.0000i
(2,3)           0 + 1.0000i
(1,4)           0 - 1.0000i
(8,5)           0 + 1.0000i
(7,6)           0 - 1.0000i
(6,7)           0 + 1.0000i
(5,8)           0 - 1.0000i