HHL gives exponential speedup for solving linear systems

The HHL speedup comes with severe conditions that eliminate practical utility in most cases. First, the input vector b must be loaded into quantum memory in O(log N) time — but reading N classical data points to prepare this state takes O(N) time, erasing the speedup. Second, the output is a quantum state — reading out all N components takes O(N) measurements. Third, the matrix A must be sparse and well-conditioned. Fourth, you need quantum RAM (not yet built). Under these conditions, HHL gives exponential speedup only in a narrow regime unlikely to match real-world problems.

HHL requires phase estimation and amplitude amplification circuits far beyond 4 qubits to demonstrate meaningfully. The key lesson is about reading fine print in speedup claims, not circuit construction.