Refined Quantum Algorithms for Principal Component Analysis and Solving Linear System

We outline refined versions of two major quantum algorithms for performing principal component analysis and solving linear equations. Our methods are exponentially faster than their classical counterparts and even previous quantum algorithms/dequantization algorithms. Oracle/black-box access to classical data is not required, thus implying great capacity for near-term realization. Several applications and implications of these results are discussed. First, we show that a Hamiltonian $H$ with classically known rows/columns can be efficiently simulated, adding another model in addition to the well-known sparse access and linear combination of unitaries models. Second, we provide a simpler proof of the known result that quantum matrix inversion cannot achieve sublinear complexity $\kappa^{1-\gamma}$ where $\kappa$ is the conditional number of the inverted matrix.