Hadamard Gate
Puts a qubit into an equal superposition — the most common single-qubit gate.
Formula
Variables
Example
H|0⟩ = (|0⟩ + |1⟩)/√2
How it works
The Hadamard gate is a single-qubit operation that mixes the amplitudes of |0⟩ and |1⟩ so that a definite basis state can become an equal quantum superposition. The factor 1/\sqrt{2} is there to keep the total probability equal to 1, because the squared magnitudes of the two output amplitudes must add up to 1. The sign flip in the lower-right entry creates a relative minus sign, which is what makes the gate useful for interference in quantum algorithms.
When to use it
Use the Hadamard gate when you want to create superposition before measurement, such as in quantum search, interference-based circuits, and Bell-state preparation. It is also a standard first step in many algorithms that need to spread probability across multiple computational paths.
Worked example, step by step
Start with the qubit |1⟩, written as the column vector \begin{pmatrix}0\\1\end{pmatrix} Apply the Hadamard matrix: H|1⟩ = \frac{1}{\sqrt{2}}\begin{pmatrix}1 & 1\\1 & -1\end{pmatrix}\begin{pmatrix}0\\1\end{pmatrix} Multiply the matrix by the vector: \begin{pmatrix}1\cdot 0 + 1\cdot 1\\1\cdot 0 + (-1)\cdot 1\end{pmatrix} = \begin{pmatrix}1\\-1\end{pmatrix} Include the scale factor: \frac{1}{\sqrt{2}}\begin{pmatrix}1\\-1\end{pmatrix} Write in ket notation: H|1⟩ = (|0⟩ - |1⟩)/\sqrt{2} Check normalization: (1/\sqrt{2})^2 + (-1/\sqrt{2})^2 = 1/2 + 1/2 = 1
Common mistakes to avoid
- Forgetting the 1/\sqrt{2} factor and producing a non-normalized state
- Thinking the Hadamard always gives the same output for |0⟩ and |1⟩; the relative sign changes
- Treating the minus sign as a probability subtraction instead of a phase difference that affects interference
Did You Know?
Almost every quantum algorithm begins with Hadamard gates to create the superposition needed for quantum parallelism.
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More in Quantum Computing
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IntermediateGeneral state of a single qubit as a superposition of the computational basis states.
Normalisation Constraint
BasicThe squared amplitudes of a qubit must sum to 1, ensuring total probability is 100 %.
Pauli-X Gate (Quantum NOT)
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CNOT Gate
IntermediateTwo-qubit gate that flips the target qubit only when the control qubit is |1⟩.
Bell State (EPR Pair)
IntermediateMaximally entangled two-qubit state — measuring one qubit instantly determines the other.
Born Rule (Measurement Probability)
IntermediateProbability of measuring outcome |k⟩ from a quantum state |ψ⟩.