# nLab Pauli gate

Contents

### Context

#### Computation

intuitionistic mathematics

# Contents

## Idea

In qbit-based quantum computation, by the Pauli gates one means the linear basis of quantum gates on single qbits, hence on the 2-dimensional Hilbert spaces $QBit \simeq \mathbb{C}^2$, which, in terms of the canonical quantum measurement-basis $\mathbb{C}^2 \simeq Span\big( \{ \vert 0 \rangle ,\, \vert 1 \rangle\} \big)$, are given by the Pauli matrices.

Explicitly this means that (in the conentional normalization) the:

1. Pauli-X gate (or quantum NOT gate) is given by the matrix

$X \;\;\coloneqq\;\; \left[ \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right]$
2. Pauli-Y gate is given by the matrix

$X \;\;\coloneqq\;\; \left[ \begin{array}{cc} 0 & - \mathrm{i} \\ \mathrm{i} & 0 \end{array} \right]$
3. Pauli-Z gate is given by the matrix

$X \;\;\coloneqq\;\; \left[ \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right]$

## Properties

### Relation to Hadamard gates and the ZX-calculus

The Hadamard gate transforms the eigenstates $\vert 0 \rangle$, $\vert 1 \rangle$ of the Pauli Z-gate into those $\propto \vert 0 \rangle \pm \vert 1 \rangle$ of the Pauli-X gate, a relation that is elaborated on by the correspondingly named ZX-calculus.

For example: