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===Graph states=== | ===Graph states=== | ||
The above operation can also be viewed as a graph state with two nodes and one edge. The qubit 1 is measured in a rotated basis <math>\mathrm{HZ}(\theta)</math>, thus leaving qubit 2 in desired state and Pauli Correction <math>\mathrm{X}^|\mathrm{s1}\rangle \mathrm{HZ}(\theta_1)|\psi\rangle</math>, where <math>\mathrm{s1}</math> is the measurement outcome of qubit 1. See Figure \ref{3}.[[Appendix#3|Figure 3]]<br/> | |||
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[[File:Graph States for Single Qubit States.jpg|center|thumb|1000px|Figure 3: Graph State for Single Qubit Gates]]</div> | [[File:Graph States for Single Qubit States.jpg|center|thumb|1000px|Figure 3: Graph State for Single Qubit Gates]]</div> | ||
Now, suppose we need to operate the state with two unitary gates <math>\mathrm{Z}(\theta_1)</math> and <math>\mathrm{Z}(\theta_2)</math>. This can be done by taking the output state of <math>\mathrm{Z}(\theta_1)</math> gate as the input state of <math>\mathrm{Z}(\theta_2)</math> gate and then repeating gate teleportation for this setup, as described above. Thus, following the same pattern for graph states we have now three nodes (two measurement qubits for two operators and one output qubit) with two edges, entangled as one dimensional chain (See [[Supplementary Information#4|Figure 4]]).<br/> | Now, suppose we need to operate the state with two unitary gates <math>\mathrm{Z}(\theta_1)</math> and <math>\mathrm{Z}(\theta_2)</math>. This can be done by taking the output state of <math>\mathrm{Z}(\theta_1)</math> gate as the input state of <math>\mathrm{Z}(\theta_2)</math> gate and then repeating gate teleportation for this setup, as described above. Thus, following the same pattern for graph states we have now three nodes (two measurement qubits for two operators and one output qubit) with two edges, entangled as one dimensional chain (See [[Supplementary Information#4|Figure 4]]).<br/> | ||
<div id="4"> | <div id="4"> | ||
[[File:Gate Teleportation for Multiple Qubit Gates.jpg|center|thumb|500px|Figure 4: Gate Teleporation for Multiple Single Qubit Gates]]</div> | |||
The measurement on qubit <math>\mathrm{1}</math> will operate <math>\mathrm{X}^{\mathrm{s1}}\mathrm{HZ}(\theta_1)|\psi\rangle\otimes I</math> on qubits <math>\mathrm{2}</math> and <math>\mathrm{3}</math>. If qubit <math>\mathrm{2}</math> when measured in the given basis yields outcome <math>\mathrm{s2}</math>, qubit <math>\mathrm{3}</math> results in the following state <math>{\mathrm{X}}^{\mathrm{s2}}\mathrm{HZ}(\theta_2){\mathrm{X}}^{\mathrm{s1}}\mathrm{HZ}(\theta_1)|\psi\rangle</math>. Using the relation we shift all the Pauli corrections to one end i.e. qubit <math>\mathrm{3}</math> becomes <math>{\mathrm{X}}^{\mathrm{s2}}\mathrm{HZ}(\theta_2)\mathrm{HZ}(\theta_1)|\psi\rangle</math> (<math>{\mathrm{Z}}^{\mathrm{s1}}\mathrm{H}={\mathrm{HX}}^{\mathrm{s1}}</math>). This method of computation requires sequential measurement of states i.e. all the states should not be measured simultaneously. As outcome of qubit <math>\mathrm{1}</math> can be used to choose sign of <math>\pm\theta_2</math>. This technique is also known as adaptive measurement. With each measurement, the qubits before the one measured at present have been destroyed by measurement. It is a feed-forward mechanism, hence known as one way quantum computation. | The measurement on qubit <math>\mathrm{1}</math> will operate <math>\mathrm{X}^{\mathrm{s1}}\mathrm{HZ}(\theta_1)|\psi\rangle\otimes I</math> on qubits <math>\mathrm{2}</math> and <math>\mathrm{3}</math>. If qubit <math>\mathrm{2}</math> when measured in the given basis yields outcome <math>\mathrm{s2}</math>, qubit <math>\mathrm{3}</math> results in the following state <math>{\mathrm{X}}^{\mathrm{s2}}\mathrm{HZ}(\theta_2){\mathrm{X}}^{\mathrm{s1}}\mathrm{HZ}(\theta_1)|\psi\rangle</math>. Using the relation we shift all the Pauli corrections to one end i.e. qubit <math>\mathrm{3}</math> becomes <math>{\mathrm{X}}^{\mathrm{s2}}\mathrm{HZ}(\theta_2)\mathrm{HZ}(\theta_1)|\psi\rangle</math> (<math>{\mathrm{Z}}^{\mathrm{s1}}\mathrm{H}={\mathrm{HX}}^{\mathrm{s1}}</math>). This method of computation requires sequential measurement of states i.e. all the states should not be measured simultaneously. As outcome of qubit <math>\mathrm{1}</math> can be used to choose sign of <math>\pm\theta_2</math>. This technique is also known as adaptive measurement. With each measurement, the qubits before the one measured at present have been destroyed by measurement. It is a feed-forward mechanism, hence known as one way quantum computation. | ||