Universal Superposition of Orthogonal States: Difference between revisions

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*'''Pure Output Case:''' First, an [[ancillary]] qubit in standard basis must be prepared with the desired weights of the final superposition. Then the [[quantum adder]] circuit must be operated on the two arbitrary and this ancilary qubit. The circuit's gates are as follows: A [[controlled swap gate]] acting on input qubits and its control qubit is the ancillary qubit. Then a CNOT gate (controlled X gate) is performed on the first input qubit with the same control qubit. The next step is the measurement. The ancillary qubit is measured in the X basis and the first input state is measured in the Z basis. If the output of the X measurement is the state with positive eigenvalue (or plus state) and the result of the Z measurement is the state with negative eigenvalue (the 1 state) the protocol is successful and the output is the superposition of the two input states with the desired weights. Otherwise, the round should be ignored.
*'''Pure Output Case:''' First, an [[ancillary]] qubit in standard basis must be prepared with the desired weights of the final superposition. Then the [[quantum adder]] circuit must be operated on the two arbitrary and this ancilary qubit. The circuit's gates are as follows: A [[controlled swap gate]] acting on input qubits and its control qubit is the ancillary qubit. Then a CNOT gate (controlled X gate) is performed on the first input qubit with the same control qubit. The next step is the measurement. The ancillary qubit is measured in the X basis and the first input state is measured in the Z basis. If the output of the X measurement is the state with positive eigenvalue (or plus state) and the result of the Z measurement is the state with negative eigenvalue (the 1 state) the protocol is successful and the output is the superposition of the two input states with the desired weights. Otherwise, the round should be ignored.
* '''Mixed Output Case:''' In the second case, the circuit is the same as for the pure output case. The only difference is that at the measurement step, regardless of the outcome of the measurements, we will not ignore any rounds and all the outcome states are valid superposition but they differ by a relative phase with each other (with negative or positive sign). As a result, the output of the circuit will be always a desired superposed state. In the cases which the relative phase of the superposition can be ignored, this case can be considered as a deterministic protocol.
\item \textbf{mixed output case:} In the second case, the circuit is the same as for the pure output case. The only difference is that at the measurement step, regardless of the outcome of the measurements, we will not ignore any rounds and all the outcome states are valid superposition but they differ by a relative phase with each other (with negative or positive sign). As a result, the output of the circuit will be always a desired superposed state. In the cases which the relative phase of the superposition can be ignored, this case can be considered as a deterministic protocol.
[[File:Universal_Superposition.jpg|right|thumb|1000px|The quantum circuit presentation of the protocol]]
\end{itemize}
\section{Figure}
The quantum circuit presentation of the protocol \\
\Qcircuit @C=.7em @R=.4em @! {
&&&&&\lstick{\alpha\ket{1}+\beta\ket{0}} & \ctrl{1} & \ctrl{1} & \measureD{X} & \rstick{\ket{\mu}} \qw \\
&&&&&\lstick{\ket{\psi}} & \multigate{1}{S} & \gate{X} &\measureD{Z} &  \rstick{\ket{n}} \qw\\
&&&&&\lstick{\ket{\psi^\perp}} & \ghost{S} & \qw & \qw & \rstick{\ket{\Psi^{\mu,n}}} \qw
}


\section{Notations}
\section{Notations}
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$e^{i\eta_{\mu,\nu}}$ is a relative phase which depends on the outputs of the measurments but in all cases, the superposition has the desired form and weights. \\
$e^{i\eta_{\mu,\nu}}$ is a relative phase which depends on the outputs of the measurments but in all cases, the superposition has the desired form and weights. \\
\end{enumerate}
\end{enumerate}
==Further Information==
==Further Information==
# [https://arxiv.org/abs/1708.04360 DKK(2017)] The above protocol
# [https://arxiv.org/abs/1708.04360 DKK(2017)] The above protocol
# [https://arxiv.org/abs/1505.04955 OGHW(2016)] The first paper that talks about and proves the no-superposition theorem. Also in this paper, they present a probabilistic protocol for superposing two arbitrary (but not completely unknown) states where we know the overlaps of them with a fixed reference state. this protocol, is also restricted to a set of input states.
# [https://arxiv.org/abs/1505.04955 OGHW(2016)] The first paper that talks about and proves the no-superposition theorem. Also in this paper, they present a probabilistic protocol for superposing two arbitrary (but not completely unknown) states where we know the overlaps of them with a fixed reference state. this protocol, is also restricted to a set of input states.
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