HHL Algorithm: Quantum Linear System Solver

The Harrow–Hassidim–Lloyd (HHL) algorithm solves the linear system Ax = b exponentially faster than classical Gaussian elimination for sparse, well-conditioned Hermitian matrices — by encoding b's eigenbasis weights into quantum amplitudes, reading each eigenvalue through phase estimation, and inverting it with a controlled ancilla rotation. This simulation lets you tune a 3-eigenvalue matrix A and an input vector b, watch each eigenmode's phase-estimation clock spin at a rate set by its eigenvalue, and see the ancilla's success probability and the resulting solution vector x update live — including the condition-number blowup that makes near-singular matrices brutal for HHL in practice.