nLab Pauli gate

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.

Related concepts

• quantum computation

• quantum gate, quantum circuit

• Hadamard gate

• XOR, CNOT

• Solovay-Kitaev theorem

References

For example:

• Michael A. Nielsen, Isaac L. Chuang, p. xxx in: Quantum computation and quantum information, Cambridge University Press (2000) [doi:10.1017/CBO9780511976667, pdf, pdf]