Hadamard Gate

The gateway to superposition, transforms basis states into equal superpositions.

The Hadamard gate (H gate) is perhaps the most important single-qubit gate. It creates superposition from definite states and is the starting point for most quantum algorithms.

Definition

$$H=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)$$

Action on Basis States

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

$$H|1⟩=\frac{1}{\sqrt{2}}(|0⟩-|1⟩)=|-⟩$$

The Hadamard creates an equal superposition with 50/50 measurement probabilities.

Key Properties

Self-Inverse

$$H^2=I$$

Applying Hadamard twice returns to the original state.

Basis Change

Hadamard transforms between two important bases:

• Computational basis: $\{|0⟩,|1⟩\}$ (Z eigenstates)
• Hadamard basis: $\{|+⟩,|-⟩\}$ (X eigenstates)

$$H:|0⟩\leftrightarrow |+⟩,|1⟩\leftrightarrow |-⟩$$

Relation to Paulis

$$H=\frac{1}{\sqrt{2}}(X+Z)$$

Also: $HXH=Z$ and $HZH=X$

Bloch Sphere

Hadamard is a 180° rotation around the axis halfway between X and Z.

Hadamard on Multiple Qubits

Applying H to all $n$ qubits creates a uniform superposition over all $2^n$ states:

$$H^{\otimes n}|0{⟩}^{\otimes n}=\frac{1}{\sqrt{2^n}}\sum _{x=0}^{2^n-1}|x⟩$$

This is the starting point for many quantum algorithms (Grover’s, Shor’s, etc.).

Walsh-Hadamard Transform

The $n$-qubit Hadamard operation:

$$H^{\otimes n}|x⟩=\frac{1}{\sqrt{2^n}}\sum _{y=0}^{2^n-1}(-1{)}^{x\cdot y}|y⟩$$

where $x\cdot y$ is the bitwise inner product. This is the quantum Walsh-Hadamard transform.

In Quantum Circuits

The Hadamard is typically drawn as a box with “H”:

     ┌───┐
|0⟩──┤ H ├──|+⟩
     └───┘

Why It’s Important

Almost every quantum algorithm starts with Hadamards:

1. Initialize qubits in $|0⟩$
2. Apply Hadamard to create superposition
3. Perform computation
4. Extract answer

Without Hadamard, there’s no superposition. Without superposition, there’s no quantum advantage.

See also: Superposition, Quantum Gate, Pauli Gates, Quantum Algorithm