Editing Quantum logic gate (section)

=== Phase shift gates ===
The phase shift is a family of single-qubit gates that map the basis states <math>|0\rangle \mapsto |0\rangle</math> and <math>|1\rangle \mapsto e^{i\varphi}|1\rangle</math>. The probability of measuring a <math>|0\rangle</math> or <math>|1\rangle</math> is unchanged after applying this gate, however it modifies the phase of the quantum state. This is equivalent to tracing a horizontal circle (a line of constant latitude), or a rotation about the z-axis on the [[Bloch sphere]] by <math>\varphi</math> radians. The phase shift gate is represented by the matrix:

:<math>P(\varphi) = \begin{bmatrix} 1 & 0 \\ 0 & e^{i \varphi} \end{bmatrix}</math>

where <math>\varphi</math> is the ''phase shift'' with the [[Periodic function|period]] {{math|2&pi;}}. Some common examples are the ''T'' gate where <math display="inline">\varphi = \frac{\pi}{4}</math> (historically known as the <math>\pi /8</math> gate), the phase gate (also known as the S gate, written as ''S'', though ''S'' is sometimes used for SWAP gates) where <math display="inline">\varphi= \frac{\pi}{2}</math> and the [[#Z|Pauli-''Z'' gate]] where <math>\varphi = \pi.</math>

The phase shift gates are related to each other as follows:

:<math> Z = \begin{bmatrix} 1 & 0 \\ 0 & e^{i \pi} \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} = P\left(\pi\right)</math>

:<math> S = \begin{bmatrix} 1 & 0 \\ 0 & e^{i \frac{\pi}{2}} \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & i \end{bmatrix} = P\left(\frac{\pi}{2}\right)=\sqrt{Z}</math>

:<math> T = \begin{bmatrix} 1 & 0 \\ 0 & e^{i \frac{\pi}{4}} \end{bmatrix} =P\left(\frac{\pi}{4}\right) = \sqrt{S} = \sqrt[4]{Z}</math>

Note that the phase gate <math>P(\varphi)</math> is not [[Hermitian matrix|Hermitian]] (except for all <math>\varphi = n\pi, n \in \mathbb{Z}</math>). These gates are different from their Hermitian conjugates: <math>P^\dagger(\varphi)=P(-\varphi)</math>. The two [[Hermitian adjoint|adjoint]] (or [[conjugate transpose]]) gates <math>S^\dagger</math> and <math>T^\dagger</math> are sometimes included in instruction sets.<ref>{{cite web|url=https://qiskit.org/documentation/stubs/qiskit.circuit.library.TdgGate.html#qiskit.circuit.library.TdgGate|title=TdgGate}} [[Qiskit]] online documentation.</ref><ref>{{cite web|url=https://www.quantum-inspire.com/kbase/t-dagger-gate/|title=T dagger Gate}} cQASM online documentation.</ref>

{{anchor|Hadamard|Hadamard gate|H gate}}