HW 3 - Bloch Sphere Problems

5.5 (Kasirajan)

Question

A quantum gate performs the transformation $| - ⟩ \to | 0 ⟩$ and $| + ⟩ \to | 1 ⟩$ . Write the corresponding unitary matrix.

We assume the standard basis $| 0 ⟩, | 1 ⟩$ for this unitary matrix $G$ .

Define the gate $G$ here. Recall that:

| + ⟩ = \frac{1}{\sqrt{2}} (| 0 ⟩ + | 1 ⟩), | - ⟩ = \frac{1}{\sqrt{2}} (| 0 ⟩ - | 1 ⟩)

Thus since the mappings are defined as they are above, notice that:

| + ⟩ + | - ⟩ = \sqrt{2} | 0 ⟩

Thus since $G$ is linear:

G (| + ⟩ + | - ⟩) = G | + ⟩ + G | - ⟩ = | 1 ⟩ + | 0 ⟩ = G (\sqrt{2} | 0 ⟩) = \sqrt{2} G | 0 ⟩

As such:

G | 0 ⟩ = \frac{1}{\sqrt{2}} (| 0 ⟩ + | 1 ⟩)

In a similar manner, we have:

| + ⟩ - | - ⟩ = \sqrt{2} | 1 ⟩

So:

G (| + ⟩ - | - ⟩) = G | + ⟩ - G | - ⟩ = | 1 ⟩ - | 0 ⟩ = \sqrt{2} G | 1 ⟩

Thus:

G | 1 ⟩ = \frac{1}{\sqrt{2}} (- | 0 ⟩ + | 1 ⟩)

Plug these as the columns for our matrix $G$ :

G = \frac{1}{\sqrt{2}} [\begin{matrix} 1 & - 1 \\ 1 & 1 \end{matrix}]

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