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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.
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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
On the Bloch sphere the Hadamard gate is a half-turn (π rotation) about the axis halfway between x and z, (x+z)/√2: it swings the arrow from the north pole |0⟩ to the equator point |+⟩, and a second half-turn brings it back exactly (H² = I). Nothing is randomized; the result is a 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⟩?
- A.(|0⟩+|1⟩)/√2
- B.(|0⟩−|1⟩)/√2
- C.|0⟩
- 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.
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Primary source: Barenco et al., Phys. Rev. A 52, 3457 (1995), doi:10.1103/PhysRevA.52.3457
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