 must be unitary. In the following, we shall consider five examples of 2 x 2 unitary operators or matrices and their effect on qubit transformation, namely: the Pauli matrices (called I, X, Y, and Z-gates) and the Hadamard matrix (also called a H-gate). Pauli matrices or I, X, Y, Z-gates The Pauli matrices (named after the physicist W. Pauli) or I, X, Y, Z-gates are defined (16.15) The different notations I = Y = Oy=02 and usually found in the related literature for historical reasons. It is easily established that the squares all the above matrices are equal to unity or X2 = Y2 = Z2 = (oi)2 = I2 I. The first matrix I is the identity matrix, which corresponds to the classical REPEAT gate and leaves the input qubit |q) = a|0) + B|1) invariant, according to |q')=Iq (16.16) The second matrix X (or X-Pauli) corresponds to the classical NOT gate. Indeed, we obtain |q'>=Xqq =(ii)(g) (16.17) 4 A transformation and associated matrix A is unitary if the following identity is satisfied A-Superscript(1) = (AT)* = A+. In this definition, A-Superscript(1) is the inverse matrix of A (such that A-1A=AA-1 = I), AT is the transposed matrix of A (with coefficients aij aji), and * denotes complex conjugation (aij a). The symbol + stands for Hermitian conjugation, which combines both transposition and complex conjugation (aij aii, i.e., A+ is the Hermitian conjugate of the matrix A. 16.2 Basic computations with 1-qubit quantum gates showing that the elementary qubits |0), |1) are switched into their counterparts |1), |0), which for a state superposition lq) = a|0) + B|1) amounts to swapping the amplitude probabilitie according to (a, B) -> (B,a). Thus, if the input lq) is in the pure state, for instance lq) = |0) (or a = 1,B=0), the output state is |q') = |1), and the reverse with |q>=||) which gives |q') = |0). In these two limiting cases, this operation, indeed, corresponds to that of a classical NOT gate. But generally, it is also correct to call X, 01, Ox a quantum NOT gate. The action of the Y-gate is both to swap the amplitude