Gates & Circuits
Hadamard Gate
A fundamental single-qubit gate that creates an equal superposition of |0⟩ and |1⟩.
Definition
The Hadamard gate is one of the most widely used quantum gates, transforming a qubit in a definite state (|0⟩ or |1⟩) into an equal superposition of both. It's typically the very first gate applied in many quantum algorithms, including Grover's Algorithm and Shor's Algorithm.
Technical Definition
Mathematically, the Hadamard gate is represented by the matrix H = (1/√2)[[1,1],[1,-1]]. Applied to |0⟩, it produces (1/√2)(|0⟩ + |1⟩); applied to |1⟩, it produces (1/√2)(|0⟩ - |1⟩).
Visual Explanation: An Analogy
Think of the Hadamard gate like flipping a fair coin that starts perfectly balanced on one edge — after the 'flip,' it has an exactly equal chance of landing heads or tails, representing the equal superposition created.
Real-World Use Cases
- The standard way to initialize qubits into superposition at the start of an algorithm
- Used repeatedly throughout the Quantum Fourier Transform
- A key building block in creating Bell states when combined with a CNOT gate