nLab Pauli gate




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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 2QBit \simeq \mathbb{C}^2, which, in terms of the canonical quantum measurement-basis 2Span({|0,|1})\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[0 1 1 0] X \;\;\coloneqq\;\; \left[ \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right]
  2. Pauli-Y gate is given by the matrix

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


Relation to Hadamard gates and the ZX-calculus

The Hadamard gate transforms the eigenstates |0\vert 0 \rangle, |1\vert 1 \rangle of the Pauli Z-gate into those |0±|1\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:

Created on March 7, 2023 at 15:10:18. See the history of this page for a list of all contributions to it.