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===Unitary Operations=== *<math>\text{X (NOT gate)}</math>: <math>X|0\rangle\,\to\,\ |1\rangle,\quad X|1\rangle\,\to\,\ |0\rangle,\quad X|+\rangle\,\to\,\ |+\rangle,\quad *X|-\rangle\,\to\,\ -|-\rangle</math> *<math>\text{Z}</math>: <math>Z|+\rangle \,\to\,\ |-\rangle,\quad Z|-\rangle \,\to\,\ |+\rangle,\quad Z|0\rangle \,\to\,\ |0\rangle,\quad Z|1\rangle \,\to\,\ -|1\rangle </math></br> Thus, <math>|0\rangle </math>, <math>|1\rangle </math> are eigenstates of Z gate and <math>|+\rangle</math>, <math>|-\rangle </math> are eigenstates of X gate. *<math>\text{H (Hadamard gate)}</math>: <math>H|0\rangle \,\to\,\ |+\rangle </math> or <math>H|1\rangle \,\to\,\ |-\rangle </math> *<math>\text{P (Phase Gate)}</math>: Gates in this class operate on a single qubit. They are represented by 2 x 2 matrices of the form <math>R(\theta)</math>, as shown below. Here <math>\theta</math> is the phase shift. <math>R(\theta)= \left[ {\begin{array}{cc} 1 & 0 \\ 0 & e^{i\theta} \\ \end{array} }\right],\quad X= \left[ {\begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array} }\right],\quad Z= \left[ {\begin{array}{cc} 1 & 0 \\ 0 & -1 \\ \end{array} }\right],\quad H=\frac{1}{\sqrt{2}} \left[ {\begin{array}{cc} 1 & 1 \\ 1 & -1 \\ \end{array} }\right]\quad </math> *<math>\text{Controlled-U(CU)}</math>: uses two inputs, control qubit and target qubit. It operates U on the second(target) qubit only when the first (source) qubit is 1. C-U gates are used to produce entangled states, when the target qubit is <math>|+\rangle</math> and control qubit is not an eigenstate of U. In the given equation 'i' denotes the source qubit and 'j', the target qubit. Following are two important C-U gates. <math> \text{Controlled-NOT(C-X or CNOT): }CX_{ij}|+\rangle_i|0\rangle_j\,\to\,\ \frac{1}{\sqrt{2}} (|0_i0_j\rangle+|1_i1_j\rangle)</math></br> <math>\text{Controlled-Z (C-Z):}CZ_{ij}|+\rangle_i|+\rangle_j\,\to\,\ \frac{1}{\sqrt{2}} (|0_i+_j\rangle+|1_i-_j\rangle)</math> The commutation relations for the above gates are as follows:</br> <math> XH=HZ,\quad XZ=-ZX</math></br> <math>(X\otimes I)CZ=CZ(X\otimes Z),\quad(Z\otimes I)CZ=CZ(Z\otimes I)</math>
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