Glossary

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.

Quantum States

• $|+\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle +|1\rangle ),\quad |-\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle -|1\rangle )$
  
• Bell/ EPR pairs:
• GHZ States:
• W States:

Unitary Operations

• X (NOT gate): $X|0\rangle \,\to \,\ |1\rangle ,\quad X|1\rangle \,\to \,\ |0\rangle ,\quad X|+\rangle \,\to \,\ |+\rangle ,\quad *X|-\rangle \,\to \,\ -|-\rangle$
• Z (Phase gate): $Z|+\rangle \,\to \,\ |-\rangle ,\quad Z|-\rangle \,\to \,\ |+\rangle ,\quad Z|0\rangle \,\to \,\ |0\rangle ,\quad Z|1\rangle \,\to \,\ -|1\rangle$
  

Thus, $|0\rangle$, $|1\rangle$ are eigenstates of Z gate and $|+\rangle$, $|-\rangle$ are eigenstates of X gate.

• H (Hadamard gate): $H|0\rangle \,\to \,\ |+\rangle$ or $H|1\rangle \,\to \,\ |-\rangle$

$X=\left[{\begin{array}{cc}0&1\\1&0\\\end{array}}\right],\quad Z=\left[{\begin{array}{cc}1&0\\0&-1\\\end{array}}\right],\quad H={\frac {1}{\sqrt {2}}}\left[{\begin{array}{cc}1&1\\1&-1\\\end{array}}\right]\quad$

• 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 $|+\rangle$ 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.

${\text{Controlled-NOT(CX or CNOT): }}CX_{ij}|+\rangle _{i}|0\rangle _{j}\,\to \,\ {\frac {1}{\sqrt {2}}}(|0_{i}0_{j}\rangle +|1_{i}1_{j}\rangle )$
${\text{Controlled-Phase(CZ): }}CZ_{ij}|+\rangle _{i}|+\rangle _{j}\,\to \,\ {\frac {1}{\sqrt {2}}}(|0_{i}+_{j}\rangle +|1_{i}-_{j}\rangle )$

The commutation relations for the above gates are as follows:
$XH=HZ,\quad XZ=-ZX$
$(X\otimes I)CZ=CZ(X\otimes Z),\quad (Z\otimes I)CZ=CZ(Z\otimes I)$

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 $U^{2}=I$
• 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
• 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 $UT=P^{a}U'T$, where P is the phase gate and U, U' are any two Pauli gates depending on C. Parameter $a\epsilon {0,1}$ is obtained from U, such that $P^{0}=I$, $P^{1}=P$.
  

To summarize, if $C^{1}=$P, $C^{2}=$C, $C^{3}=$T, then $C^{k}=\{U:UQU=C^{k-1}|Q\epsilon C^{1}\}$

Magic States

Eigen states of T gates

Universal Resource

A set of \ket{+_\theta} states on which applying Clifford operations is enough for universal quantum computation.