Suppose we have a one-qubit unitary U that maps |0⟩ to 12|0⟩+3√i2|1⟩ and |+⟩ to 1−3√i22√|0⟩+3√i−122√|1⟩. What is U?
Explicitly indicate multiplication with a * symbol. For instance, you should write 3*i+2 instead of 3i+2.
To find the matrix representation of the given unitary operator U, we need to write the given states |0⟩ and |+⟩ in the standard computational basis (|0⟩ and |1⟩).
Given:
U maps |0⟩ to 1/2 √3i|0⟩ + 3/2 √i|1⟩
U maps |+⟩ to (1 - 3/2 √i) / 2 √ |0⟩ + 3/2 √i / 2|1⟩
Let's first represent |0⟩ in the computational basis:
|0⟩ = [1, 0]^T
To find the coefficients of |0⟩ and |1⟩ in the transformed state, we equate the given expression for U (mapping of |0⟩) with the matrix representation of the transformation:
[1/2 √3i, 3/2 √i] * [ |0⟩ , |1⟩ ] = [12 |0⟩ + 3 √i/2 |1⟩]
Expanding the matrix multiplication:
(1/2 √3i) * |0⟩ + (3/2 √i) * |1⟩ = 12 |0⟩ + 3 √i/2 |1⟩
This gives us the following two equations:
1/2 √3i = 12
3/2 √i = 3 √i / 2
Solving these equations, we find:
1/2 √3i = 12 implies √3i = 24
3/2 √i = 3 √i / 2 implies √i = 1
Now let's find the coefficient of |1⟩ for the transformed |0⟩ state:
3√i/2 |1⟩ = 3 √i / 2 |1⟩
This gives us another equation:
3 √i / 2 = 3 √i / 2
Now, let's represent |+⟩ in the computational basis:
|+⟩ = (1/√2) * [1, 1]^T
To find the coefficients of |0⟩ and |1⟩ in the transformed state, we equate the given expression for U (mapping of |+⟩) with the matrix representation of the transformation:
[(1 - 3/2 √i) / 2 √, 3/2 √i / 2] * [ |0⟩ , |1⟩ ] = [(1 - 3/2 √i)/2 √ |0⟩ + 3/2 √i/2 |1⟩]
Expanding the matrix multiplication:
((1 - 3/2 √i) / 2 √) * |0⟩ + (3/2 √i / 2) * |1⟩ = (1 - 3/2 √i)/2 √ |0⟩ + 3/2 √i/2 |1⟩
This gives us the following two equations:
(1 - 3/2 √i) / 2 √ = (1 - 3/2 √i) / 2 √
3/2 √i / 2 = 3/2 √i / 2
Simplifying the equations, we find:
(1 - 3/2 √i) / 2 √ = (1 - 3/2 √i) / 2 √
3/2 √i / 2 = 3/2 √i / 2
We have obtained the same equation for both states |0⟩ and |+⟩. This means the unitary operator U is the same for both states, and we can write it in matrix form.
The matrix representation of U is:
U = [(1/2 √3i), (3/2 √i)]
Therefore, U = [1/2 √3i, 3/2 √i]