Fast quantum learning with statistical guarantees

A wide class of quantum algorithms for learning problems exp loit fast quantum linear algebra subroutines to achieve runtimes that are exponentially faster than their classical counterparts [ Cil 18 ]. Examples of these algorithms are quantum support vector m achines [ RML14 ], quantum linear regression [ WBL12; SSP16 ], and quantum least squares [ KP17; CGJ18 ]. A careful analysis of these algorithms identified a number of caveats that limit their practical applicability such as the need for a strong form of quantum ac cess to the input data, restrictions on structural properties of the data matrix (such as conditi on number or sparsity), and modes of access to the output [ Aar15 ]. Furthermore, if one assumes that it is efficient to (classic ally) sample elements of the training data in a way proportional to their norm, then it is possible to show that classical algorithms are only polynomially slowe r (albeit the scaling of the quantum algorithms can be considerably better) [ Tan18; CL W18; Chi 19a; GLT18; Chi 19b ]. In this work we continue to investigate the limitations of qu antum algorithms for learning problems.