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→Unitary Operations
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===Unitary Operations=== | ===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{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>\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. | 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{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>X= | <math><math>R(\theta)</math>= | ||
\left[ {\begin{array}{cc} | |||
1 & 0 \\ | |||
0 & e^{i\theta} \\ | |||
\end{array} }\right],\quad | |||
X= | |||
\left[ {\begin{array}{cc} | \left[ {\begin{array}{cc} | ||
0 & 1 \\ | 0 & 1 \\ | ||
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*<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-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> | <math> | ||
\text{Controlled-NOT( | \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- | <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> | The commutation relations for the above gates are as follows:</br> | ||