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→‎Hierarchy of Quantum Gates

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 === Hierarchy of Quantum Gates ===
 There are different class of quantum gates as follows,
-*'''Pauli Gates(U):''' Single Qubit Gates I (Identity), X, Y, Z. All the gates in this set follow U2 = I
+*Pauli Gates(U): Single Qubit Gates I (Identity), X, Y, Z. All the gates in this set follow <math>U^2=I</math>
-*'''Clifford Gates(C):''' Pauli Gates, Phase Gate, C-NOT. This set of gates can be simulated on classical computer. All the gates in this set follow CU=U’C, where U and U’ are two different Pauli gates depending on C
+*Clifford Gates(C): Pauli Gates, Phase Gate, C-NOT. This set of gates can be simulated on classical computer. All the gates in this set follow CU=U'C, where U and U' are two different Pauli gates depending on C
-*'''Toffoli Gate:''' A three qubit gate that does not belong to Clifford Group
+*Universal Set of gates: This set consists of all Clifford gates and one Non-Clifford gate (T gate). If a model can realise Universal Set of gates, it can imlement any quantum computation efficiently. T gates follow <math>UT=P^aU'T</math>, where P is the phase gate and U, U' are any two Pauli gates depending on C. Parameter <math>a\epsilon{0,1}</math> is obtained from U, such that <math>P^0=I</math>, <math>P^1=P</math>.</br>
-*'''T Gates:''' \sqrt{Z} Although a member of Clifford Gate, its eigen states can be used as acillas to make quantum gate sthat are not!
+To summarize, if <math>C^1=</math>P, <math>C^2=</math>C, <math>C^3=</math>T, then <math>C^{k}=\{U:UQU=C^{k-1}|Q\epsilon C^1\}</math>
-*'''Universal Set of gates:''' This set consists of all Clifford gates and one Non-Clifford gate (or T gate). One can also say one Toffoli gate and Hadamard gate constitute the set of Universal Gates. If a model can realise Universal Set of gates, it can imlpement any quantum computation efficiently. T gates follow UT = PaU0T, where P is the phase gate and U, U’ are any two Pauli gates depending on C. Parameter 1 is obtained from U, such that P0 = I, P1 = P.
-To summarize, the hierarchy of quantum can be defined as such.<br/>
-If C(1)=P, C(2)=C, C(3)=T, then
-C(n)={U:UQU\dagger=C(n-1),Q\epsilon C(1)}
 ===Magic States===