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What would be the outcome of this circuit?

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The outcome is $-i\left| 11 \right\rangle $. Here is how I obtained it.

$$ \left| 01 \right\rangle \xrightarrow{\text{X}} \left| 11 \right\rangle \xrightarrow{\text{Y}} -i\left| 10 \right\rangle \xrightarrow{\text{CNOT}} -i\left| 11 \right\rangle \xrightarrow{\text{SWAP}} -i\left| 11 \right\rangle $$

$X$ gate changes 0 to 1, and 1 to 0:

$$ X \left| 0 \right\rangle = \left| 1 \right\rangle \qquad X \left| 1 \right\rangle = \left| 0 \right\rangle $$

$Y$ gate also changes 0 to 1, and 1 to 0, but also changes the phase of the qubit:

$$ Y \left| 0 \right\rangle = i\left| 1 \right\rangle \qquad Y \left| 1 \right\rangle = -i\left| 0 \right\rangle $$

CNOT changes the second qubit if the first qubit is in $\left| 1 \right\rangle$ state and does nothing if the first qubit is in $\left| 0 \right\rangle$ state:

$$ CNOT \left| 00 \right\rangle = \left| 00 \right\rangle \qquad CNOT \left| 01 \right\rangle = \left| 01 \right\rangle $$

$$ CNOT \left| 10 \right\rangle = \left| 11 \right\rangle \qquad CNOT \left| 11 \right\rangle = \left| 10 \right\rangle $$

SWAP gate changes the qubits, first qubit's state becomes the state of the second qubit and vice versa:

$$ SWAP \left| 00 \right\rangle = \left| 00 \right\rangle \qquad SWAP \left| 01 \right\rangle = \left| 10 \right\rangle $$

$$ SWAP \left| 10 \right\rangle = \left| 01 \right\rangle \qquad SWAP \left| 11 \right\rangle = \left| 11 \right\rangle $$

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  • $\begingroup$ what would be the input of this circuit and why? $\endgroup$ – toaster_fan Jan 13 at 17:37
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    $\begingroup$ The input is given in your example: in the picture after q[0] it is written | 0> and after q[1] it is written |1>. So, the input state is |0>|1> = |01> $\endgroup$ – Davit Khachatryan Jan 13 at 17:41
  • $\begingroup$ and for that particular input state (|01>) the circuit produces -i|11> state $\endgroup$ – Davit Khachatryan Jan 13 at 17:45
  • $\begingroup$ I just applied the gates to the input state by taking into account that X|0> = |1>, Y|1> = -i|0>, CNOT |1> |0> = |1> |1> and SWAP |11> = |11> $\endgroup$ – Davit Khachatryan Jan 13 at 17:51
  • $\begingroup$ I have edited the answer, hope now it is more clear how I obtained the result :) $\endgroup$ – Davit Khachatryan Jan 13 at 18:08

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