Applied Physics CSIE II: UNIT IV: Quantum Computing
Hadamard Gate (H GATE)
Definition, Action, Matrix form, Operation, Bloch Sphere action | Quantum gates
The Hadamard gate (H) is a single‒qubit quantum gate that creates and removes superposition.
HADAMARD GATE (H‒GATE)
Definition
The
Hadamard gate (H) is a single‒qubit quantum gate that creates and removes
superposition.
Or
The
Hadamard gate is a single‒qubit gate that transforms basis states into equal
superpositions. Its matrix is 
and it creates states 
.
Action
It
transforms the basis states |0> and |1> into equal superpositions.
Matrix form
The
matrix for the Hadamard gate (H) can be defined as
Operation
The
Hadamard gate is self‒inverse. It maps input |m> to
The
effect of the Hadamard gate Action on the basis states are given by, assuming
m=0 and 1, respectively
(i) On |0>
This
is an equal superposition of |0> and |1>.
(ii) On |1>
Another
balanced superposition but with a relative phase of −1.
Thus,
we notice that a Hadamard gate converts a |0> state and a |1> to a
superposition state of |0> and |1>. When the superposed qubit is
measured, there is an equal probability of it being in the state |1> and
|0>.
Bloch Sphere action
The
H‒gate performs a 180° rotation around the axis at 45° between the X and Z
axes.
Maps
the Z‒basis states (|0>, |1>) to X‒basis states
Maps
X‒basis states back to Z‒basis
Properties of H Gate
(i)
Creates Superposition
H
is the main gate used to put qubits into superposition.
(ii)
Self‒Inverse (H2 = I)
Applying
Hadamard twice returns the qubit to the original state:
H(H|Ψ>) = |Ψ>
(i)
Basis Change Gate
Converts
computational basis (|0>, |1>) to superposition basis Converts
superposition basis back to computational basis
Applied Physics CSIE II: UNIT IV: Quantum Computing : Tag: Applied Physics : Definition, Action, Matrix form, Operation, Bloch Sphere action | Quantum gates - Hadamard Gate (H GATE)
Applied Physics CSIE II: UNIT IV: Quantum Computing
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