arXiv:1908.06229v7 [quant-ph] 13 Sep 2021 Quantum solvability of noisy linear systems of equations by divide-and-conquer strategy Wooyeong Song 1,2 , Youngrong Lim 3 , Kabgyun Jeong 4,3 , Yun-Seong Ji 4 , Jinhyoung Lee 2 , Jaewan Kim 3 , M. S. Kim 5,3 , and Jeongho Bang 6 1 Center for Quantum Information, Korea Institute of Science and Technology, Seoul, 02792, Korea 2 Department of Physics, Hanyang University, Seoul 04763, Korea 3 School of Computational Sciences, Korea Institute for Advanced Study, Seoul 02455, Korea 4 Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea 5 QOLS, Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom 6 Electronics and Telecommunications Research Institute, Daejeon 34129, Korea The first three authors (W.S., Y.L., and K.J.) contributed equally to this study and can be regarded as the main authors. Correspondence and requests for materials should be addressed to M.S.K. and J.B. E-mail: m.kim@imperial.ac.uk and jbang@etri.re.kr Abstract. Noisy linear problems have been studied in various science and engineering disciplines. A class of “hard” noisy linear problems can be formulated as follows: Given a matrix ˆ A and a vector b constructed using a finite set of samples, a hidden vector or structure involved in b is obtained by solving a noise- corrupted linear equation ˆ Ax ≈ b + η, where η is a noise vector that cannot be identified. For solving such a noisy linear problem, we consider a quantum algorithm based on a divide-and-conquer strategy, wherein a large core process is divided into smaller subprocesses. The algorithm appropriately reduces both the computational complexities and size of a quantum sample. More specifically, if a quantum computer can access a particular reduced form of the quantum samples, polynomial quantum- sample and time complexities are achieved in the main computation. The size of a quantum sample and its executing system can be reduced, e.g., from exponential to sub-exponential with respect to the problem length, which is better than other results we are aware. We analyse the noise model conditions for such a quantum advantage, and show when the divide-and-conquer strategy can be beneficial for quantum noisy linear problems.