s and to introduce a IT phase shift between the two states of the initial superposition, according to |q')=Yqq =(i o)(a) (16.18) = (i ) ? i(B0)-a1))=e(B0)+eira1)) where y = - -/2 is an immeasurable, or "unobservable" phase constant. Finally, the action of the Z-gate is only to introduce a IT phase shift between the two states of the initial superposition, according to |q'>=Zqq =(=-i)(a) (16.19) As can also be easily checked, the three Pauli matrices X, Y, Z exhibit the following properties: 0102=103 0203=io (16.20) 0301=102 One defines the commutator of two operators or matrices A, B as [A, B] = AB - BA. Two operators or matrices are said to commute if [A, B] = 0. Then we observe from the last equation in Eq. (16.20) that for i # j the Pauli matrices do not commute, i.e., = general case (16.21) where Sij is the Kronecker symbol.6 We, thus, have [01,02] = 20102 = 2io3, [02,03] = 20203 = 2101 and [03,01] = 20301 = 2io2. The commutator between any two Pauli 5 With complex numbers, a change of sign corresponds to a phase shift of or a multiplying factor of e+ir -1, sincee=)+isin)= = -1. Namely: dij = =1 forij,and8ij=0forij Quantum bits and quantum gates matrices can be generalized in the formula [0j,0j] = =2ieijkOk (16.22) where Eijk is the Levi-Civita symbol. Likewise, using the property (oi)2 = I, we can generalize Eq. (16.20) in the formula: (16.23) These properties of the Pauli matrices or I, X, Y, Z-gates are used extensively in quantum computing, as will be illustrated. In Appendix N, I show that the Pauli matrices constitute a universal base, making it possible to generate any unitary 2 x 2 matrices or unitary operators U, which represent the universal building blocks for 2-qubit quantum gates and circuits. Hadamard matrix gate or H-gate The Hadamard matrix gate, or H-gate, is defined as: (16.24) Using results in Eqs. (16.17) and (16.19), we can interpret the action of the Hadamard gate as follows: (16.25) [(B0)+a|1))+(a0)-B1))] =a|+)+8|-). It is seen from the above that the Hadamard gate transforms any input 