Question

Suppose we have a quantum circuit that takes the input |0⟩ and outputs |+⟩, and also takes the input |1⟩ and outputs −|−⟩. If we input √2*i/√3|+⟩ + 1/√3|−⟩, what does the circuit output?

Answer 1

I'm going to assuming that:

|+> = 1/sqrt(2) [|0> + |1>]

|-> = 1/sqrt(2) [|0> - |1>]

Then there is a unitary operator U that relates output to input as:

|output> = U|input>

You can then calculate the output without having to expand everything in one basis, you can simplify things by invoking the fact that U is unitary.

We want to compute:

|s> = U|k>

with |k> = i sqrt(2/3)|+> + 1/sqrt(3)|->

Note that if:

|b> = U|a>

then

<b| = <a|U-dagger ----->

<b|U = <a|U-dagger U = <a|

So, we have:

<+|s> = <+|U|k> = <0|k> =

i/sqrt(3) + 1/sqrt(6)

<-|s> = <-|U|k> = -<1|k> =

-i/sqrt(3) + 1/sqrt(6)


Therefore:

|s> = [i/sqrt(3) + 1/sqrt(6)]|+>
+ [-i/sqrt(3) + 1/sqrt(6)] |->