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We apply the framework of block-encodings, introduced by Low and Chuang(under the name standard-form), to the study of quantum machine learningalgorithms and derive general results that are applicable to a variety of inputmodels, including sparse matrix oracles and matrices stored in a datastructure. We develop several tools within the block-encoding framework, suchas singular value estimation of a block-encoded matrix, and quantum linearsystem solvers using block-encodings. The presented results give new techniquesfor Hamiltonian simulation of non-sparse matrices, which could be relevant forcertain quantum chemistry applications, and which in turn imply an exponentialimprovement in the dependence on precision in quantum linear systems solversfor non-sparse matrices.  In addition, we develop a technique of variable-time amplitude estimation,based on Ambainis' variable-time amplitude amplification technique, which weare also able to apply within the framework.  As applications, we design the following algorithms: (1) a quantum algorithmfor the quantum weighted least squares problem, exhibiting a 6-th powerimprovement in the dependence on the condition number and an exponentialimprovement in the dependence on the precision over the previous best algorithmof Kerenidis and Prakash; (2) the first quantum algorithm for the quantumgeneralized least squares problem; and (3) quantum algorithms for estimatingelectrical-network quantities, including effective resistance and dissipatedpower, improving upon previous work.

The power of block-encoded matrix powers: improved regression techniques via faster Hamiltonian simulation