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Hadamard (H)

Hadamard gate tile from the Operations catalog

The Hadamard gate is the standard way to create superposition. Applied to \( |0\rangle \), it produces an equal mix of \( |0\rangle \) and \( |1\rangle \) - the starting point of most quantum algorithms.
Qubits 1
Parameters None
Inverse Itself (H² = I)
Also called H gate

What it does

H maps each basis state to an equal-weight superposition of both basis states, with a relative sign that depends on which one it started from. Qompile's own gate info panel (opened via the Info action described in Editing, viewing info, and other gate actions) puts it this way:

The Hadamard gate creates superposition by mapping \( |0\rangle \mapsto |{+}\rangle = \dfrac{1}{\sqrt{2}}(|0\rangle+|1\rangle) \) and \( |1\rangle \mapsto |{-}\rangle = \dfrac{1}{\sqrt{2}}(|0\rangle-|1\rangle) \).

Matrix

\[ H = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} \]

Action on basis states

  • \( |0\rangle \rightarrow \dfrac{1}{\sqrt{2}}\big(|0\rangle + |1\rangle\big) = |{+}\rangle \)
  • \( |1\rangle \rightarrow \dfrac{1}{\sqrt{2}}\big(|0\rangle - |1\rangle\big) = |{-}\rangle \)

Phase and Bloch-sphere effect

On the Bloch sphere, H swaps the Z-axis and X-axis (with a sign flip) - it's a 180° rotation about the diagonal axis halfway between X and Z. That's why applying H twice returns you to where you started.

Example

H on q[0] (starting at \( |0\rangle \)) is the first step of a Bell state: once followed by a CNOT, it produces the entangled state \( (|00\rangle+|11\rangle)/\sqrt{2} \).

Other notations

Naming conventions across ecosystems; exact syntax can vary by library version, and this does not claim to be Qompile's generated output unless otherwise noted.

Language Typical form
OpenQASM 2.0 h q[0];
Qiskit circuit.h[q[0]]
Cirq cirq.H(q[0])
Q# H(q[0]);