The Hadamard gate, written H, is the most-used gate in quantum computing. It takes a qubit in a definite state (say |0⟩) and puts it into an even mix of |0⟩ and |1⟩. That mix is what people mean by superposition.
Almost every quantum algorithm starts by applying a Hadamard to every qubit. That move sets up the system to consider every possible input at once.
Visualize what H does
The clearest way to see what the Hadamard gate does is to watch it on the Bloch sphere. Pick a starting state, then click Hadamard (H). The arrow rotates 180° about the diagonal x+z axis.
The four key facts:
- H sends
|0⟩→|+⟩(equator, +x direction). - H sends
|1⟩→|−⟩(equator, −x direction). - H sends
|+⟩→|0⟩(back to the top pole). - H sends
|−⟩→|1⟩(back to the bottom pole).
Apply H twice and the qubit returns to its original state.
The math
The Hadamard gate is described by a 2×2 matrix:
Applied to |0⟩ = [1, 0], it produces (1/√2)[1, 1]: equal weight on |0⟩ and |1⟩. The 1/√2 keeps the squared amplitudes summing to 1, which is what makes it a valid quantum state.
Why it matters
Quantum algorithms get their speedups from interference: amplitudes that add up for the right answer and cancel out for the wrong ones. To set up interference, you first need a superposition, and the Hadamard gate is how you make one.
That's why almost every quantum algorithm opens with H on every qubit: Shor's factoring algorithm, Grover's search, Deutsch–Jozsa, the QFT. Without H, the qubits would just sit at |0⟩ and the algorithm would have nothing to work with.
Related concepts
- Pauli gates (X, Y, Z): the other fundamental single-qubit gates.
- What is a quantum gate? The umbrella concept.
- Quantum measurement: what happens when you measure a qubit in superposition.
- The Bloch sphere: the visualization H is rotating.
Frequently asked questions
What does the Hadamard gate do?
The Hadamard gate (H) puts a qubit into a superposition. Applied to |0⟩ it produces |+⟩ = (|0⟩ + |1⟩)/√2, an equal-weight mix. Applied to |1⟩ it produces |−⟩. On the Bloch sphere it's a 180° rotation about the diagonal axis halfway between x and z.
Why is it called the Hadamard gate?
It's named after the French mathematician Jacques Hadamard, whose Hadamard matrix construction (1893) is the same operator, just scaled to qubits. The matrix has the elegant property that H · H = I: apply H twice and you're back where you started.
What is the Hadamard gate matrix?
It's a 2×2 matrix: H = (1/√2) · [[1, 1], [1, -1]]. The 1/√2 normalization keeps the result a valid quantum state: the squared amplitudes still sum to 1.
Is the Hadamard gate the same as a quantum NOT?
No. The quantum NOT gate is Pauli-X, which flips |0⟩ ↔ |1⟩. The Hadamard gate creates superposition; it doesn't swap the basis states, it mixes them.
How is the Hadamard gate used in quantum algorithms?
Most quantum algorithms start with a Hadamard on every qubit, putting the whole system into a uniform superposition over every possible input. The algorithm then evaluates them all at once. You see this opening move in Shor's algorithm, Grover's search, the Deutsch–Jozsa algorithm, and many others.
Is the Hadamard gate reversible?
Yes. Every quantum gate is reversible (they're all unitary matrices). H is its own inverse: apply it twice and you're back to where you started.
Get Qubi
Run a Hadamard gate with your hands.
Qubi is a real model qubit you can hold. Apply H, X, Y, Z, S, T, and see the state change in your hand.