Hadamard Gate

Overview

The Hadamard gate (H) is one of the most important single-qubit gates. It is the cleanest way to:

• create superposition from a basis state,
• undo that superposition, and
• switch between measurement bases (Z-basis ↔ X-basis).

Even when we avoid circuits, H is central because it teaches the core idea behind quantum behavior:

• probabilities come from amplitudes, and amplitudes can interfere.

Intuition

On the Bloch sphere:

• the Z-basis states $|0\rangle$ and $|1\rangle$ are the north and south poles (+$z$ and −$z$).
• the X-basis states are $$|+\rangle = \tfrac{1}{\sqrt{2}}(|0\rangle + |1\rangle),\qquad | - \rangle = \tfrac{1}{\sqrt{2}}(|0\rangle - |1\rangle).$$ These lie on the equator at +$x$ and −$x$.

The Hadamard gate maps poles to equator points:

• $|0\rangle \mapsto |+\rangle$
• $|1\rangle \mapsto | - \rangle$

So H is a geometric “tilt” that changes which basis looks like “up/down.”

One more crucial intuition:

• Applying H twice gets you back to where you started.

So H is its own inverse. That “do it again to undo it” property makes H a natural tool for creating and then later removing superposition.

Formal Description

Action on basis states

The defining behavior is

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

This already tells you what H does to any superposition by linearity.

Matrix form (with meaning)

In the computational basis, H is represented by

$$H = \tfrac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1\\ 1 & -1 \end{pmatrix}.$$

Let’s interpret the entries rather than “dumping a matrix”:

• The factor $\tfrac{1}{\sqrt{2}}$ keeps the output normalized.
• The first column describes what happens to $|0\rangle$.
• The second column describes what happens to $|1\rangle$.

You can check this by multiplying:

• $|0\rangle \equiv \begin{pmatrix}1\\0\end{pmatrix}$ gives the first column: $$H|0\rangle = \tfrac{1}{\sqrt{2}}\begin{pmatrix}1\\1\end{pmatrix} = \tfrac{1}{\sqrt{2}}(|0\rangle + |1\rangle) = |+\rangle.$$
• $|1\rangle \equiv \begin{pmatrix}0\\1\end{pmatrix}$ gives the second column: $$H|1\rangle = \tfrac{1}{\sqrt{2}}\begin{pmatrix}1\\-1\end{pmatrix} = \tfrac{1}{\sqrt{2}}(|0\rangle - |1\rangle) = | - \rangle.$$

A key identity (stated conceptually) is:

$$H^2 = I.$$

Meaning: applying H twice returns the original state.

Worked Example

Start with $|0\rangle$.

1. Apply H:

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

Geometrically: north pole (+$z$) moves to +$x$ on the equator.

2. Apply H again:

$$|+\rangle \xrightarrow{\;H\;} |0\rangle.$$

So H both creates and undoes superposition.

Interpretation:

• H is a basis change between the Z and X viewpoints.
• “Superposition” is not mystical; it is often just “a basis state in a different basis.”

Turtle Tip

Common Pitfalls

Quick Check

What’s Next

H makes superposition and changes basis. Next we study gates that mainly change phase rather than probabilities: the phase gates S and T, which rotate around the $z$ axis by smaller angles than Z.