Question: what happens when you apply a Hadamard gate to a qubits that's in arbitrary superposition?

A Reddit user posed a question about applying a Hadamard gate to a qubit in an arbitrary superposition, extending beyond the known cases of |0⟩ and |1⟩ states. The query focuses on how the Hadamard gate transforms a qubit in the general state α|0⟩ + β|1⟩, where α and β are complex probability amplitudes. The Hadamard gate’s standard behavior—mapping |0⟩ to |+⟩ and |1⟩ to |−⟩—serves as the baseline for understanding its effect on superpositions. Mathematically, the result is a new superposition where the amplitudes are redistributed via the Hadamard matrix, yielding (α + β)|0⟩ + (α − β)|1⟩/√2. This highlights the gate’s role in creating interference patterns, a key mechanism for quantum algorithms like Grover’s or Shor’s.