Editing Glossary (section)

====Cluster States====
In case of multi-qubit quatum circuits, one needs a 2-dimensional graph state. Cluster State is a square lattice used as substrate for such computation. All the nodes are in <math>|+\rangle</math> entangled by <math>\mathrm{CZ}</math> indicated by the edges. It is known to be universal i.e. it can simulate any quatum gate.<br/>
<div id="5"> 
[[File:Cluster State.jpg|center|thumb|500px|Figure 5: Cluster State]]</div>
Each row would thus represent the teleporation of starting qubit in that row horizontally. On the other hand, vertical edges indicate different input qubits linked with multi-qubit gates (same as circuit model). For example, see Figure [[Supplementary Information#6|Figure 6]] to understand the conversion from circuit model to graph state model. As the computation relation follows <math>X=HZH</math>, thus, Figure [[Supplementary Information#6a|Figure 6a]] represents Circuit diagram for <math>\mathrm{CNOT}</math> gate in terms of <math>\mathrm{CZ}</math> gate and Single Qubit Gate <math>\mathrm{H}</math>.<br/>
<div id="6"><div id="6a"><div id="6b"><ul> 
<li style="display: inline-block;"> [[File:Circuit Diagram to implement C-NOT.jpg|frame|500px|(a)Circuit Diagram to implement C-NOT]] </li>
<li style="display: inline-block;"> [[File:Graph State Pattern for C-NOT.jpg|frame|500px|(b)Graph State Pattern for C-NOT]] </li>
<center><caption>Figure 6: Measurement Pattern from Circuit Model</caption></center>
</ul></div></div></div><br/>
[[Supplementary Information#6a|Figure 6b]] shows implementation of the first Hadamard gate on the second input state as measurement <math>\mathrm{M}_\mathrm{2}</math> on qubit <math>\mathrm{2}</math>. Then <math>\mathrm{CZ}</math> gate is implemented by the entangled qubits <math>\mathrm{3}</math> and <math>\mathrm{1}</math> in the graph state. Qubit <math>\mathrm{3}</math> is entangled to another qubit <math>\mathrm{4}</math> to record the output while measurement <math>\mathrm{M}_\mathrm{3}</math> on qubit <math>\mathrm{3}</math> implements the second Hadamard gate. Finally, the states to which qubits <math>\mathrm{(1)}</math> and <math>\mathrm{(4)}</math> are reduced to determine the output states of the two input qubits after <math>\mathrm{CNOT}</math> gate operation.</br>
It is evident that one needs to remove certain nodes from the cluster state in order to implement the above shown graph state. This can be done by Z-basis measurements. Such measurements would leave the remaining qubits in the cluster state with extra Pauli corrections. This can be explained as follows. Consider a two-dimesional graph state <math>\{\mathrm{1,2}\}</math>. If qubit <math>\mathrm{1}</math> is to be eliminated, we operate <math>\mathrm{CZ}</math> with <math>\mathrm{2}</math> as target and <math>\mathrm{1}</math> as control.<br/></br>
<math>{\mathrm{CZ}}_{\mathrm{12}}{|+\rangle}_{1}|+\rangle_2=|0\rangle_1|+\rangle_2+|1\rangle_1|-\rangle_2</math></br></br>
Thus, if measurement on <math>\mathrm{1}</math> yields <math>\mathrm{m}</math>, qubit <math>\mathrm{2}</math> would be in the state <math>\mathrm{Z}^\mathrm{m}|+\rangle</math>. Hence, such Z-basis measurements invoke an extra <math>\mathrm{Z}^\mathrm{m}</math> Pauli correction on all the neighbouring sites of 1 with 1 eliminated, in the resulting graph state. 
Thus, to summarise, we design a measurement pattern from gate teleportation circuit of the desired computation as shown above. The cluster state is converted into the required graph state by Z-basis measurement on extraneous sites. Measuring all the qubits in the required basis and we get the required computation in the form of classical outcome register from measurement of the last layer of qubits. If it is a quantum function, the last layer of qubits is the output quantum register.