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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/>
   
   
*X (NOT gate): X |0i → |1i, X |1i → |0i, X |+i → |+i, X |−i → −|−i
<math>\ket{+}=\frac{1}{\sqrt{2}}(\ket{0}+\ket{1}),</math></br>
*Z (Phase gate): Z |+i → |−i, Z |−i → |+i, Z |0i → |0i, Z |1i → −|1i
<math>\ket{-}=\frac{1}{\sqrt{2}}(\ket{0}-\ket{1})</math>
Thus, |0i, |1i are eigenstates of Z gate and |+i, |−i are eigenstates of X gate.
Thus, |0i, |1i are eigenstates of Z gate and |+i, |−i are eigenstates of X gate.
*H (Hadamard gate): H |0i → |+i or H |1i → |−i
</br>X (NOT gate):  <math>X\ket{0}\,\to\,\ \ket{1},\quad X\ket{1}\,\to\,\ \ket{0},\quad X\ket{+}\,\to\,\ \ket{+}, X\ket{-}\,\to\,\ -\ket{-}</math>
   
</br>Z (Phase gate): <math>Z\ket{+}\,\to\,\ \ket{-},\quad Z\ket{-}\,\to\,\ \ket{+},\quad Z\ket{0}\,\to\,\ \ket{0},\quad Z\ket{1}\,\to\,\ -\ket{1}</math></br>
*Controlled-U(CU): 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 |+i 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.
Thus, <math>\ket{0}</math>, <math>\ket{1}</math> are eigenstates of Z gate and <math>\ket{+}</math>, <math>\ket{-}</math> are eigenstates of X gate.
*Controlled-NOT(CX):  
</br> (Hadamard gate): <math>H\ket{0}\,\to\,\ \ket{+}</math> or <math>H\ket{1}\,\to\,\ \ket{-}</math>
*Controlled-Phase(CZ): <br/>
The commutation relations for the above gates are as follows:<br/>
* XH = HZ
*      XZ = −ZX
*(X ⊗ I)CZ = CZ(X ⊗ Z), (Z ⊗ I)CZ = CZ(Z ⊗ I) 1 <br/>
 
===Unitary Operation===
===Unitary Operation===
=== Hierarchy of Quantum Gates ===
=== Hierarchy of Quantum Gates ===
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