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Supplementary Information: Difference between revisions
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[[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> | [[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> | 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}}|+} | <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 Zm |+i. Hence, such Z-basis measurements invoke an extra Zm Pauli correction on all the neigbouring 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.<br/> | Thus, if measurement on 1 yields m, qubit 2 would be in the state Zm |+i. Hence, such Z-basis measurements invoke an extra Zm Pauli correction on all the neigbouring 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.<br/> | ||