ined through its complex amplitudes (a, B) with respect to a given orthonormal base or pure states (|0), |1)), also referred to as computational basis. A quantum gate, call it A, has the effect of transforming an input qubit state 1q) into an output qubit state |q'), using predefined control qubits. Thus, the relation between the input and output states can be expressed in the equivalent matrix-vector relation: |q')=A|q). (16.13) In quantum vocabulary, the entity A that transforms 1q) into |q'> is referred to as an operator. Since the vector |q') must remain on the Bloch sphere surface (so that its squared amplitudes correspond to actual probabilities), it must be a unitary vector. For this reason, the transformation from |q) to |q') is called unitary, and A must be a unitary operator. We shall now further develop the notion of unitary transformation and the relation between unitary operator and matrix. For simplicity and to escape any further conceptual burden, SO far we have assumed that qubits are equivalent to vectors in a 2D space. This view is simplistic, but accurate. Consistently, we may assume that the transformation operator A may be defined through a 2 X 2 matrix, whose coefficients are defined by four complex numbers, Aij. From the definitions Eq. (16.3) and Eq. (16.13), we, thus, obtain the matrix-vector equation in the |0), |1) base: |q')=A|qy (16.14) (an +112B) |0) + (a21 a - a22B)|1) = a'/0) + B'|1). Such a relation between the input (a,B) and output (a', B') amplitudes does not account for the control qubits in the quantum gate, but for the time being this will suffice for our description. The result in Eq. (16.14) shows that the action of the quantum gate, or that of the quantum operator A, modifies the state amplitudes of the initial superposition, according to the transformations a B Since the output state amplitudes (a', B') must correspond to probabilities, it is required that Quantum bits and quantum gates = I have mentioned earlier, such a transformation and the associated operator/matrix,