Quantum Computing: The Walsh-Hadamard Matrix — Backbone of Grover's Diffusion Operator
The post walks through building the 8×8 Walsh-Hadamard matrix by tensoring three single-qubit Hadamard gates. Starting with one qubit, H = (1/√2)[[1,1],[1,-1]], it extends to two qubits via tensor product, producing a 4×4 matrix, then to three qubits for an 8×8 matrix. The key property is that H is its own inverse (H² = I). The matrix is used in Grover's algorithm where the diffusion operator applies H⊗³ twice per iteration. The Hadamard Reference table is shown to be the full Walsh-Hadamard matrix. The article emphasizes that the circuit uses only parallel Hadamard gates, no entangling gates, yet the tensor product structure yields the necessary sign pattern for amplitude amplification.
Understanding the Walsh-Hadamard matrix is essential for implementing Grover's search algorithm on quantum computers.