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?
In the form a│0> + b│1>
To find the output of the quantum circuit for the given input, we need to apply the circuit's transformations to the input state. Let's break down the problem step by step.
Step 1: Expand the input state using the distributive property:
√2 * i/√3 |+⟩ + 1/√3 |-⟩
Step 2: Apply the given circuit transformations to the corresponding basis states.
For input state |0⟩:
The circuit maps |0⟩ to |+⟩,
Therefore, √2 * i/√3 |+⟩ + 1/√3 |-⟩
becomes
√2 * i/√3 |+⟩ + 0 |-⟩
For input state |1⟩:
The circuit maps |1⟩ to −|−⟩,
Therefore, √2 * i/√3 |+⟩ + 1/√3 |-⟩
becomes
0 |+⟩ − 1/√3 |−⟩
Step 3: Normalize the state
To express the state in the required form a|0⟩ + b|1⟩, we need to normalize it by dividing each coefficient by the magnitude of the state.
The magnitude of the state is given by:
√(|a|^2 + |b|^2)
So, let's calculate the magnitude:
Magnitude = √(|0|^2 + |-1/√3|^2) = √(0 + 1/3) = √(1/3)
Magnitude = 1/√3
Now, we can divide each coefficient by the magnitude to normalize the state:
Normalized state = (1/√3) * (0 |+⟩ − 1/√3 |−⟩)
Normalized state = -1/3√3 |−⟩
Therefore, the output of the given quantum circuit for the input √2*i/√3|+> + 1/√3|−> is -1/3√3 |−⟩, or in the required form, 0│0⟩ + (-1/3√3)│1⟩.