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→A General Introduction to Quantum Information
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==A General Introduction to Quantum Information== | ==A General Introduction to Quantum Information== | ||
Quantum computation is marked by a set of unitary matrices (quantum gates) acting on qubit states followed by measurement. The most used representation is the circuit model of computation, comprising straight lines and boxes. The horizontal lines represent qubits and boxes represent single qubit unitary gates. A two qubit unitary gate links one qubit from another via vertical lines. Some useful notations are given below.<br/> | Quantum computation is marked by a set of unitary matrices (quantum gates) acting on qubit states followed by measurement. The most used representation is the circuit model of computation, comprising straight lines and boxes. The horizontal lines represent qubits and boxes represent single qubit unitary gates. A two qubit unitary gate links one qubit from another via vertical lines. Some useful notations are given below.<br/> | ||
<math>|+\rangle=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle),\quad |-\rangle=\frac{1}{\sqrt{2}}(|0\rangle-|1\rangle)</math></br> | ===Quantum States=== | ||
*<math>|+\rangle=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle),\quad |-\rangle=\frac{1}{\sqrt{2}}(|0\rangle-|1\rangle)</math></br> | |||
*Bell/ EPR pairs: | |||
*GHZ States: | |||
*W States: | |||
===Unitary Operations=== | |||
*X (NOT gate): <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> | *X (NOT gate): <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> | ||
*Z (Phase gate): <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> | *Z (Phase gate): <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> | ||
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*H (Hadamard gate): <math>H|0\rangle \,\to\,\ |+\rangle </math> or <math>H|1\rangle \,\to\,\ |-\rangle </math> | *H (Hadamard gate): <math>H|0\rangle \,\to\,\ |+\rangle </math> or <math>H|1\rangle \,\to\,\ |-\rangle </math> | ||
=== | <math>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> | |||
=== Hierarchy of Quantum Gates === | === Hierarchy of Quantum Gates === | ||
There are different class of quantum gates as follows, | There are different class of quantum gates as follows, | ||