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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> | 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> | <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 1 yields m, qubit 2 would be in the state | 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. | |||
====Brickwork States==== | ====Brickwork States==== | ||