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<math>\text{Controlled-Phase(CZ): }CZ_{ij}|+\rangle_i|+\rangle_j\,\to\,\ \frac{1}{\sqrt{2}} (|0_i+_j\rangle+|1_i-_j\rangle)</math> | <math>\text{Controlled-Phase(CZ): }CZ_{ij}|+\rangle_i|+\rangle_j\,\to\,\ \frac{1}{\sqrt{2}} (|0_i+_j\rangle+|1_i-_j\rangle)</math> | ||
The commutation relations for the above gates are as follows: | The commutation relations for the above gates are as follows:</br> | ||
<math> | <math> | ||
XH=HZ,\quad XZ=-ZX</math></br> | XH=HZ,\quad XZ=-ZX</math></br> | ||
<math>(X\otimes I)CZ=CZ(X\otimes Z),\quad(Z\otimes I)CZ=CZ(Z\otimes I)</math> | <math>(X\otimes I)CZ=CZ(X\otimes Z),\quad(Z\otimes I)CZ=CZ(Z\otimes I)</math> | ||
=== Hierarchy of Quantum Gates === | === Hierarchy of Quantum Gates === | ||
There are different class of quantum gates as follows, | There are different class of quantum gates as follows, | ||