Quantum Principal Component Analysis Only Achieves an Exponential Speedup Because of Its State Preparation Assumptions

A central roadblock to analyzing quantum algorithms on states is the lack of a comparable input model for classical algorithms. Inspired by recent work author [E. Tang, STOC 2019.], we introduce such model, where assume can efficiently perform ${\ensuremath{\ell}}^{2}$-norm samples data, natural analog that efficient state preparation data. Though this produces less practical than (stronger) standard computation, it captures versions many features and nuances linear algebra With describe analogs Lloyd, Mohseni, Rebentrost's principal component analysis [S. M. P. Rebentrost, Nat. Phys. 10, 631 (2014).] nearest-centroid clustering Quantum supervised unsupervised machine learning]. Since they are only polynomially slower, these suggest exponential speedups their counterparts simply an artifact assumptions.