nLab Hadamard gate



Quantum systems

quantum logic

quantum physics

quantum probability theoryobservables and states

quantum information

quantum computation


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Constructivism, Realizability, Computability



In quantum computing/quantum information theory, by the Hadamard gate one means the quantum logic gate which acts on a single qbit by

q 0|0+q 1|112(q 0+q 1)|0+12(q 0q 1)|1, q_0 \cdot \vert 0 \rangle + q_1 \cdot \vert 1\rangle \;\;\; \mapsto \;\;\; \frac{1}{\sqrt{2}} \big( q_0 + q_1 \big) \cdot \vert 0 \rangle + \frac{1}{\sqrt{2}} \big( q_0 - q_1 \big) \cdot \vert 1 \rangle \,,

hence which is the linear map H: 2 2H \,\colon\, \mathbb{C}^2 \longrightarrow \mathbb{C}^2 which in the standard linear basis is given by the matrix

12(+1 +1 +1 1). \frac{1}{\sqrt{2}} \left( \array{ +1 & +1 \\ +1 & -1 } \right) \,.

Simple as it is, the Hadamard gate plays a central role in theory and practice of quantum circuits, notably where in its combination with the CNOT gate it is used to produce Bell states (such as in the quantum teleportation protocol).


See also:

Last revised on October 26, 2022 at 08:16:12. See the history of this page for a list of all contributions to it.