The Hadamard Gate

Quantum computers use qubits instead of classical bits. While regular bits are either 0 or 1, qubits can exist in a “mix” of both states simultaneously, thanks to a phenomenon called superposition.

What Are Quantum Gates?

Just like classical computers use logic gates (AND, OR, NOT) to process bits, quantum computers use quantum gates to manipulate qubits. These gates perform operations that change the probabilities of a qubit’s state.

Hadamard Gate (H gate)

The Hadamard Gate is one of the most important quantum gates. It creates superposition, a fundamental quantum property that allows qubits to exist in multiple states simultaneously.

The Hadamard Gate (H gate)  is like a “quantum coin flip.” If a qubit is in state |0〉 (like “heads”), the gate transforms it into a superposition: 50% chance of 0, 50% chance of 1. Mathematically, it’s represented as:

H = 1/√2 [1  1
1 -1]

For example, applying it to |0〉 gives (|0〉 + |1〉)/√2, a perfect balance between both states.

Qubit states are described using vectors in a Hilbert space (a quantum version of 3D space). For example:

• |0〉 = [1, 0] (like pointing “up”)
• |1〉 = [0, 1] (like pointing “down”)

 

A superposition like (|0〉 + |1〉)/√2 is a vector pointing diagonally. The Hadamard Gate “rotates” the vector’s direction to create this mix. When applied to a qubit, it transforms the basis states as follows:

• |0⟩ → (|0⟩ + |1⟩)/√2
• |1⟩ → (|0⟩ – |1⟩)/√2

 

This means that if a qubit starts in state |0⟩ or |1⟩, applying the Hadamard gate places it into an equal superposition of both states.

 

More information:

• https://www.pennylane.ai/