Randomized Row and Column Iterative Methods with a Quantum Computer

We consider the quantum implementations of the two classical iterative solvers for a system of linear equations, including the Kaczmarz method which uses a row of coefficient matrix in each iteration step, and the coordinate descent method which utilizes a column instead. These two methods are widely applied in big data science due to their very simple iteration schemes. In this paper we use the block-encoding technique and propose fast quantum implementations for these two approaches, under the assumption that the quantum states of each row or each column can be efficiently prepared. The quantum algorithms achieve exponential speed up at the problem size over the classical versions, meanwhile their complexity is nearly linear at the number of steps

Black-Box Quantum State Preparation with Inverse Coefficients

Black-box quantum state preparation is a fundamental building block for many higher-level quantum algorithms, which is applied to transduce the data from computational basis into amplitude. Here we present a new algorithm for performing black-box state preparation with inverse coefficients based on the technique of inequality test. This algorithm can be used as a subroutine to perform the controlled rotation stage of the Harrow-Hassidim-Lloyd (HHL) algorithm and the associated matrix inversion algorithms with exceedingly low cost. Furthermore, we extend this approach to address the general black-box state preparation problem where the transduced coefficient is a general non-linear function. The present algorithm greatly relieves the need to do arithmetic and the error is only resulted from the truncated error of binary string. It is expected that our algorithm will find wide usage both in the NISQ and fault-tolerant quantum algorithms.Comment: 11 pages, 3 figure