Pauli-Y Gate

Overview

You’ve met two Pauli gates:

• X swaps $|0\rangle$ and $|1\rangle$ (bit-flip-like).
• Z keeps $|0\rangle$ and flips the sign of $|1\rangle$ (phase-flip-like).

The Pauli-Y gate (Y) completes the trio. It’s important because it shows how amplitude swapping and phase naturally combine into a single operation. Geometrically, it gives us the missing axis: a clean rotation about the $y$-axis on the Bloch sphere.

Intuition

On the Bloch sphere:

• X corresponds to a 180° rotation about the $x$ axis.
• Z corresponds to a 180° rotation about the $z$ axis.
• Y corresponds to a 180° rotation about the $y$ axis.

So Y is not “new randomness.” It’s a different kind of flip: it changes the Bloch vector by rotating around a different axis.

Why do we need Y in addition to X and Z?

• With only X and Z, you can talk about flips about the $x$ and $z$ axes.
• But the Bloch sphere is 3D. To describe rotations naturally, you want all three axes.

Another key intuition: Y includes a built-in phase factor. Compared to X (which swaps amplitudes), Y swaps amplitudes and introduces a $\pm i$ phase. That phase is not cosmetic—it changes interference behavior in later steps.

Formal Description

Action on basis states

The defining action of Y is

$$Y|0\rangle = i|1\rangle,\qquad Y|1\rangle = -i|0\rangle.$$

Let’s unpack what that means:

• Like X, Y swaps the roles of $|0\rangle$ and $|1\rangle$.
• Unlike X, it multiplies the swapped component by $\pm i$ (a 90° phase shift in the complex plane).

Matrix form (introduced carefully)

In the computational basis, Y is represented by the 2×2 matrix

$$Y = \begin{pmatrix} 0 & -i\\ i & 0 \end{pmatrix}.$$

Every symbol here has a purpose:

• $i$ is the imaginary unit, with $i^2 = -1$.
• The off-diagonal entries mean “swap components” (like X).
• The factors $\pm i$ mean “add a phase while swapping.”

Now apply it to a general single-qubit state

$$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle.$$

Using the basis-state actions above (and linearity),

$$Y|\psi\rangle = \alpha\,Y|0\rangle + \beta\,Y|1\rangle = \alpha\,i|1\rangle + \beta\,(-i)|0\rangle = -i\beta|0\rangle + i\alpha|1\rangle.$$

So:

• the $|0\rangle$ amplitude becomes $-i\beta$,
• the $|1\rangle$ amplitude becomes $i\alpha$.

That is “swap + phase.”

Worked Example

Start with the state

$$|+\rangle = \tfrac{1}{\sqrt{2}}(|0\rangle + |1\rangle).$$

Apply Y:

$$Y|+\rangle = \tfrac{1}{\sqrt{2}}(Y|0\rangle + Y|1\rangle) = \tfrac{1}{\sqrt{2}}(i|1\rangle - i|0\rangle) = \tfrac{i}{\sqrt{2}}(|1\rangle - |0\rangle).$$

Up to an overall (global) phase factor $i$, this is

$$\tfrac{1}{\sqrt{2}}(|0\rangle - |1\rangle) = | - \rangle.$$

Geometric interpretation:

• $|+\rangle$ points along +$x$ on the Bloch sphere.
• $| - \rangle$ points along −$x$.
• Y rotates the Bloch vector by 180° about the $y$ axis, taking +$x$ to −$x$.

Turtle Tip

Common Pitfalls

Quick Check

What’s Next

Now we have the three Pauli gates X, Y, and Z. Next we introduce a gate that’s central for basis changes: the Hadamard (H) gate, which connects the Z-basis to the X-basis and makes interference extremely visible.