The power of block-encoded matrix powers: improved regression techniques via faster Hamiltonian simulation
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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 learning algorithms using quantum accessible data structures. We develop several tools within the block-encoding framework, including quantum linear system solvers using block-encodings. Our results give new techniques for Hamiltonian simulation of non-sparse matrices, which could be relevant for certain quantum chemistry applications, and which in turn imply an exponential improvement in the dependence on precision in quantum linear systems solvers for non-sparse matrices. In addition, we develop a technique of variable-time amplitude estimation, based on Ambainis' variable-time amplitude amplification technique, which we are also able to apply within the framework. As applications, we design the following algorithms: (1) a quantum algorithm for the quantum weighted least squares problem, exhibiting a 6-th power improvement in the dependence on the condition number and an exponential improvement in the dependence on the precision over the previous best algorithm of Kerenidis and Prakash; (2) the first quantum algorithm for the quantum generalized least squares problem; and (3) quantum algorithms for estimating electrical-network quantities, including effective resistance and dissipated power, improving upon previous work in other input models.
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