Editing Universal Superposition of Orthogonal States
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* '''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. | * '''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. | ||
== | ==Notations== | ||
*<math>\alpha</math> , <math>\beta</math>: weights of the superposition (<math>|\alpha|^2 + |\beta|^2 = 1</math>) | *<math>\alpha</math> , <math>\beta</math>: weights of the superposition (<math>|\alpha|^2 + |\beta|^2 = 1</math>) | ||
*<math>|\psi\rangle , |\psi^\perp\rangle</math>: initial states | *<math>|\psi\rangle , |\psi^\perp\rangle</math>: initial states | ||
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**'''mixed output case:''' In this case, $P_{succ} = 1$ and the described protocol is perfect. | **'''mixed output case:''' In this case, $P_{succ} = 1$ and the described protocol is perfect. | ||
== | ==Pseudo Code== | ||
'''<u>Stage 1</u>''' Preparation and operation</br></br> | '''<u>Stage 1</u>''' Preparation and operation</br></br> | ||
*'''Input:''' The ancillary state <math>|a\rangle = \alpha|1\rangle + \beta |0\rangle</math>, <math>|\psi\rangle</math>, <math>|\psi^\perp\rangle</math> | *'''Input:''' The ancillary state <math>|a\rangle = \alpha|1\rangle + \beta |0\rangle</math>, <math>|\psi\rangle</math>, <math>|\psi^\perp\rangle</math> | ||
*'''Output:''' <math>|\Psi_2\rangle = \alpha|1\rangle X|\psi^\perp\rangle |\psi\rangle + \beta |0\rangle |\psi\rangle |\psi^\perp\rangle</math> | *'''Output:''' <math>|\Psi_2\rangle = \alpha|1\rangle X|\psi^\perp\rangle |\psi\rangle + \beta |0\rangle |\psi\rangle |\psi^\perp\rangle</math> | ||
*At this stage the following gates will be performed respectively: | *At this stage the following gates will be performed respectively: | ||
#'''S gate (Controled Swap):''' Performing this gate will interchange the input qubits. The total states before and after performing this gate are as follows: | #'''S gate (Controled Swap):''' Performing this gate will interchange the input qubits. The total states before and after performing this gate are as follows:</br> | ||
<math>|\Psi_0\rangle = \alpha|1\rangle |\psi\rangle |\psi^\perp\rangle + \beta |0\rangle |\psi\rangle |\psi^\perp\rangle</math></br> | |||
<math>|\Psi_1\rangle = \alpha|1\rangle |\psi^\perp\rangle |\psi\rangle + \beta |0\rangle |\psi\rangle |\psi^\perp\rangle</math></br> | |||
#'''CNOT gate:''' The total state after this step is as follows: | #'''CNOT gate:''' The total state after this step is as follows:</br> | ||
<math>|\Psi_2\rangle = \alpha|1\rangle X|\psi^\perp\rangle |\psi\rangle + \beta |0\rangle |\psi\rangle |\psi^\perp\rangle</math> | |||
'''<u>Stage 2</u>''' Measurements</br></br> | '''<u>Stage 2</u>''' Measurements</br></br> | ||
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*'''Output:''' The superposed state <math>|\Psi\rangle</math> | *'''Output:''' The superposed state <math>|\Psi\rangle</math> | ||
#Measure qubit 1 (the ancilary qubit) in X basis and the qubit 2 in Z basis. | #Measure qubit 1 (the ancilary qubit) in X basis and the qubit 2 in Z basis. | ||
*'''Pure Output Case:''' | |||
##''if:''' The output of the X measurement is 0 AND the output of Z measurement is 1 | |||
##'''Then:''' Accept the round | |||
##'''Else:''' Reject. | |||
* The successful output is in the form: </br> | |||
<math>|\Psi\rangle = C(\alpha |\psi\rangle + \beta e^{i\eta} |\psi^\perp\rangle)</math></br> | <math>|\Psi\rangle = C(\alpha |\psi\rangle + \beta e^{i\eta} |\psi^\perp\rangle)</math></br> | ||
<math>e^{i\eta}</math> is a relative phase which is <math>\frac{\langle1|\psi\rangle}{\langle0|\psi^\perp\rangle}</math></br> | <math>e^{i\eta}</math> is a relative phase which is <math>\frac{\langle1|\psi\rangle}{\langle0|\psi^\perp\rangle}</math></br> | ||
*'''Mixed Output Case:''' Always accept. The protocol is perfect. | |||
* The output will be: | |||
<math>|\Psi^{\mu,\nu} \rangle = C (\alpha |\psi\rangle + \beta e^{i\eta_{\mu,\nu}} |\psi^\perp\rangle) </math></br> | |||
<math>e^{i\eta_{\mu,\nu}}</math> is a relative phase which depends on the outputs of the measurements but in all cases, the superposition has the desired form and weights. | |||
==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. | ||