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Quamtum Gate - X (Pauli-X)
Quantom Gate X is similar to NOT gate in classical computing. As NOT gate in classical computing convert bit 0 to bit 1 and convert bit 1 to bit 0, X gate in quantum computing convert the quatum bit |0> to |1> and convert the quatum bit |1> to |0>.
In symbol and mathematical form, it is presented as follows. It takes 1 bit as input and return 1 bit as output. If you look into the matrix form, you would notice that it takes 2x1 vector (representing 1 qubit) and swap the elements within the vector.
If I you plug in the state vector for |0> and |1> into X gate matrix equation, you can get the output as follows. You would see that the element in the statevector get swaped by the gate X matrix.
Quamtum Gate - Y (Pauli-Y)
Quamtum Gate - Z (Pauli-Z)