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Supplementary Information: Difference between revisions
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→Gate Teleportation
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[[File:One Bit Teleportation.jpg|right|thumb|1000px|Figure 1: One Bit Teleportation]] | [[File:One Bit Teleportation.jpg|right|thumb|1000px|Figure 1: One Bit Teleportation]] | ||
</div>(H | </div><math>(H\otimes I)(CZ_{12})\ket{\psi}_1|+\rangle_2</math> | ||
<math>(H\otimes I)(CZ_{12})(a|0\rangle_1+b|1\rangle_1)|+\rangle_2</math> | |||
<math> (H\otimes I)(a|0\rangle_1|+\rangle_2+b|1\rangle_1|-\rangle_2)</math> | |||
<math>a|+\rangle_1|+\rangle_2+b|-\rangle_1|-\rangle_2</math> | |||
<math>|0\rangle_1\otimes(a|+\rangle_2+b|-\rangle_2)+|1\rangle_1\otimes(a|+\rangle_2-b|-\rangle_2)</math> | |||
<math>|0\rangle_1\otimes(a|+\rangle_2+b|-\rangle_2)+|1\rangle_1\otimes X(a|+\rangle_2+b|-\rangle_2)</math> | |||
<math>|0\rangle_1\otimes H(a\rangle 0\rangle _2+b\rangle 1\rangle _2)+|1\rangle _1\otimes XH(a|0\rangle _2+b|1\rangle _2)</math> | |||
<math>|0\rangle _1\otimes H|\psi\rangle _2+|1\rangle _1\otimes X|\psi\rangle _2</math> | |||
<math>|m\rangle \otimes X^mH|\psi\rangle</math> | |||
Similarly if we have the input state rotated by a Z(θ) gate the circuit would look like [[Supplementary Information#2a|Figure 2a]]. As the rotation gate Z(θ) commutes with Controlled-Phase gate. Hence, [[Supplementary Information#2b|Figure 2b]] is justified.<br/> | Similarly if we have the input state rotated by a Z(θ) gate the circuit would look like [[Supplementary Information#2a|Figure 2a]]. As the rotation gate Z(θ) commutes with Controlled-Phase gate. Hence, [[Supplementary Information#2b|Figure 2b]] is justified.<br/> | ||
<div id="2"><div id="2a"><div id="2b"><ul> | <div id="2"><div id="2a"><div id="2b"><ul> | ||
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</ul></div></div></div><br/> | </ul></div></div></div><br/> | ||
This shows that for a pair of C-Z entangled qubits, if the second qubit is in |+i state (not an eigen value of Z) then one can teleport (transfer) the first qubit state operated by any unitary gate U to the second qubit by performing operations only on the first qubit and measuring it. Next, we would need to make certain Pauli corrections (in this case Xm) to obtain U |ψi. In other words, we can say the operated state is teleported to the second qubit by a rotated basis measurement of the first qubit with additional Pauli corrections. | This shows that for a pair of C-Z entangled qubits, if the second qubit is in |+i state (not an eigen value of Z) then one can teleport (transfer) the first qubit state operated by any unitary gate U to the second qubit by performing operations only on the first qubit and measuring it. Next, we would need to make certain Pauli corrections (in this case Xm) to obtain U |ψi. In other words, we can say the operated state is teleported to the second qubit by a rotated basis measurement of the first qubit with additional Pauli corrections. | ||
===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 HZ(θ), thus leaving qubit 2 in desired state and Pauli Correction Xs1HZ(θ1)|ψi, where s1 is the measurement outcome of qubit 1.[[Appendix#3|Figure 3]]<br/> | 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 HZ(θ), thus leaving qubit 2 in desired state and Pauli Correction Xs1HZ(θ1)|ψi, where s1 is the measurement outcome of qubit 1.[[Appendix#3|Figure 3]]<br/> | ||