Computing Library › Quantum Logic Gates
Quantum Logic Gates
Hadamard Gate (H)
The Hadamard gate creates superposition — it turns a definite |0⟩ or |1⟩ into an equal blend, and is the doorway to nearly every quantum algorithm.
Matrix
Unitary matrix
1/√21/√21/√2−1/√2
Circuit symbol
Action on basis states
| input | output |
|---|---|
| |0⟩ | |+⟩ = (|0⟩+|1⟩)/√2 |
| |1⟩ | |−⟩ = (|0⟩−|1⟩)/√2 |
| |+⟩ | |0⟩ |
| |−⟩ | |1⟩ |
How it works
H maps the computational basis to the ± basis and back — it is its own inverse. Geometrically it is a 180° rotation about the diagonal (x+z)/√2 axis. A layer of H on every qubit builds the uniform superposition that Grover, Deutsch–Jozsa, and phase estimation all begin from.
In code (Qiskit)
python
qc = QuantumCircuit(1)
qc.h(0) # |0> -> (|0>+|1>)/sqrt(2)single-qubitCliffordHermitiansuperposition