Hadamardportti

Hadamardportti, known in English as the Hadamard gate, is a fundamental single-qubit quantum operation that implements the Hadamard transform. It converts basis states into equal superpositions and enables quantum interference necessary for many algorithms.

In the computational basis, H|0> = (|0> + |1>)/√2 and H|1> = (|0> - |1>)/√2. The matrix form is (1/√2) times the 2x2 matrix with entries [[1, 1], [1, -1]]. The gate is real and symmetric.

Properties of the Hadamard portti include unitary and Hermitian nature, with H^2 = I, making it its own inverse. It interchanges the computational basis with the Hadamard basis, whose states are |+> = (|0> + |1>)/√2 and |-> = (|0> - |1>)/√2. Its action creates equal superpositions of basis states, enabling interference patterns that are central to quantum algorithms.

Role in quantum circuits: the Hadamard portti is used to prepare uniform superpositions over basis states and to initiate interference effects. It appears as a common first step in many algorithms, such as Deutsch–Jozsa and Grover’s search, and is a standard component in various implementations of the quantum Fourier transform.

Physical implementation: realized across multiple platforms, including superconducting qubits, trapped ions, and photonic systems. Implementations typically use calibrated pulses or optical elements to realize the 1/√2 rotation with the correct phase relationship. In software, it is a standard single-qubit gate in quantum circuit libraries.