# Quantum gate This are linear operators that are defined on [[What is qubit?|qubits]]. As quantum mechanics require that qubit norm is $1$, then the only operator allow are those that preserve the norm of a qubit. These are the Unitarian operators which satisfy $UU^\dagger=U^\dagger U=I$ where $U^\dagger$ is the transposed and conjugate operator of $U$. ## Examples [[X-gate (NOT gate)|X-gate]], [[Z-gate (Relative phase gate)|Z-gate]], [[Hadamard Gate (H-gate)|H-gate]] ## Theorem: Unitarian preserve inner product Proof: $\langle U\ket x,U\ket y\rangle=(U\ket x)^\dagger U\ket y =\ket x^\dagger U^\dagger U\ket y={\ket x}^\dagger I\ket y =\ket x^\dagger \ket y=\langle x,y\rangle$ ## Theorem: $U$ is Unitarian $\leftarrow\rightarrow$ $U$ preserves norm $U$ is Unitarian $\rightarrow$ $U$ preserves norm: $\| U\ket\psi\|^2=(U\ket\psi)^\dagger (U\ket\psi)= \ket\psi^\dagger U^\dagger U\ket\psi=\ket\psi^\dagger \ket\psi=\|\ket\psi\|^2$ $U$ preserves norm $\rightarrow$ $U$ is Unitarian: $\| U\ket\psi\|^2=\|\ket\psi\|^2 \Rightarrow \langle U\ket\psi, U\ket\psi\rangle=\langle\ket\psi,\ket\psi\rangle\Rightarrow \ket\psi^\dagger U^\dagger U\ket\psi=\ket\psi^\dagger \ket\psi$ . We have $\ket\psi^\dagger( U^\dagger U-I)\ket\psi=0$ . define $A= U^\dagger U-I$. set $\ket\psi=e_i$, $e_i^\dagger A e_i=0$ for standard basis $e_i$ means that the diagonal of $A$ contains only zeros. set $\ket\psi=e_i+e_j$ we get $A_{ij}+A_{ji}=0$ . as $A=A^\dagger$ we get $A_{ij}=A^*_{ji}$. combining this results the real part must be zero. set $\ket\psi=e_i+ie_j$, we get in similar way that the imaginary part of $A_{ij}$ is also zero. This means that $A=0$ and we showed that $U$ is Unitarian. ## Created 2026-07-12 15:31