The Hadamard transform is a mathematical operation used in various fields, including quantum computing, signal processing, and information theory. It is a specific kind of unitary transformation that takes an input vector and transforms it into another vector of the same dimension. The Hadamard transform is particularly useful because it creates superposition states in quantum computing and can be implemented efficiently.
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The normalized Hadamard transform maps a vector indexed by binary strings through entries , where the dot product is evaluated modulo two. It is its own inverse and an orthogonal transformation, hence also a unitary operator on complex input vectors. The Hadamard gate is the quantum case, and the Hadamard basis is the image of the computational basis. The Walsh-Hadamard transform gives the same tensor product construction.
For two qubits, applying one Hadamard gate to each prepares the uniform quantum superpositionEach probability amplitude is , so each computational-basis measurement outcome has probability . Applying the Hadamard transform again reconstructs : signed contributions cancel at the other three outputs. Its usefulness lies in combining simple local quantum gates into an operation on exponentially many amplitudes. Extracting useful information still requires an appropriate subsequent transformation and measurement.