Block-encoding based quantum algorithm for linear systems with displacement structures.

Matrices with the displacement structures of circulant, Toeplitz, and Hankel types as well as matrices with structures generalizing these types are omnipresent in computations of sciences and engineering. In this paper, we present efficient and memory-reduced quantum algorithms for solving linear systems with such structures by devising a new approach to implement the block-encodings of these structured matrices. More specifically, by decomposing $n\times n$ dense matrices into linear combinations of displacement matrices, we first deduce the parameterized representations of the matrices with displacement structures so that they can be treated similarly. With such representations, we then construct $\epsilon$-approximate block-encodings of these structured matrices in two different data access models, i.e., the black-box model and the QRAM data structure model. It is shown the quantum linear system solvers based on the proposed block-encodings provide a quadratic speedup with respect to the dimension over classical algorithms in the black-box model and an exponential speedup in the QRAM data structure model. In particular, these linear system solvers subsume known results with significant improvements and also motivate new instances where there was no specialized quantum algorithm before. As an application, one of the quantum linear system solvers is applied to the linear prediction of time series, which justifies the claimed quantum speedup is achievable for problems of practical interest.