33 research outputs found
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
Quantum speedup of leverage score sampling and its application
Leverage score sampling is crucial to the design of randomized algorithms for
large-scale matrix problems, while the computation of leverage scores is a
bottleneck of many applications. In this paper, we propose a quantum algorithm
to accelerate this useful method. The speedup is at least quadratic and could
be exponential for well-conditioned matrices. We also prove some quantum lower
bounds, which suggest that our quantum algorithm is close to optimal. As an
application, we propose a new quantum algorithm for rigid regression problems
with vector solution outputs. It achieves polynomial speedups over the best
classical algorithm known. In this process, we give an improved randomized
algorithm for rigid regression.Comment: 23 pages, the paper is shortened and the main results are stated more
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