Source-verifiedLevel 1

Hadamard Gate

The Hadamard gate (H) is a single-qubit gate that creates an equal superposition, mapping |0⟩ to (|0⟩+|1⟩)/√2 and |1⟩ to (|0⟩−|1⟩)/√2.

What it means

The Hadamard gate is one of the most important gates in quantum computing.It performs a rotation that maps the computational basis states to the diagonal basis states: H|0⟩ = |+⟩ and H|1⟩ = |−⟩.On the Bloch sphere, it corresponds to a 180-degree rotation about the axis halfway between x and z.The Hadamard gate is its own inverse (H² = I), making it self-adjoint.It is essential for creating superpositions at the start of quantum algorithms (like Grover's and Deutsch-Jozsa), implementing quantum Fourier transforms, and converting between the Z and X measurement bases.Combined with T gates and CNOT, the Hadamard gate forms part of a universal gate set.

Everyday analogy

The Hadamard gate is like a coin flipper that takes a definite heads (|0⟩) or tails (|1⟩) and turns it into a fair spinning coin — and doing it again brings the coin back to its original definite state.
Think of it as a beam splitter in optics: a single photon entering from one port exits in a superposition of both output ports.

Common misconceptions

  • The Hadamard gate does NOT simply randomize the qubit — it creates a specific, deterministic superposition with well-defined relative phases (+1 or −1).
  • Applying the Hadamard gate twice does NOT produce a random outcome — it returns the qubit to its original state because H² = I.

Key takeaways

  • The Hadamard gate creates equal superpositions: H|0⟩ = |+⟩ = (|0⟩+|1⟩)/√2 and H|1⟩ = |−⟩ = (|0⟩−|1⟩)/√2.
  • It is self-inverse: applying H twice returns to the original state (H² = I).
  • The Hadamard gate is a cornerstone of virtually every quantum algorithm.

Check your understanding

What state does the Hadamard gate produce when applied to |1⟩?

  1. A.(|0⟩+|1⟩)/√2
  2. B.(|0⟩−|1⟩)/√2
  3. C.|0⟩
  4. D.|1⟩
Show the answer

Answer: B. (|0⟩−|1⟩)/√2

Why: H|1⟩ = (|0⟩−|1⟩)/√2 = |−⟩. The Hadamard gate maps |1⟩ to the minus state with a relative phase of −1.

Builds on

QubitSuperpositionDirac (Bra-Ket) Notation

Primary source: Barenco et al., Phys. Rev. A 52, 3457 (1995), doi:10.1103/PhysRevA.52.3457

Learn it hands-on

This concept is part of a 46-level curriculum with an interactive simulator and Lumen, a tutor whose answers are verified before you see them. Levels 1–5 are free.

Start freeSee the curriculum