Abstract
Iteration method is commonly used in solving linear systems of equations. We present quantum algorithms for the relaxed row and column iteration methods by constructing unitary matrices in the iterative processes, which generalize row and column iteration methods to solve linear systems on a quantum computer. Comparing with the conventional row and column iteration methods, the convergence accelerates when appropriate parameters are chosen. Once the quantum states are efficiently prepared, the complexity of our relaxed row and column methods is improved exponentially and is linear with the number of the iteration steps. In addition, phase estimations and Hamiltonian simulations are not required in these algorithms.
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Acknowledgements
This work is supported by the Fundamental Research Funds for the Central Universities 22CX03005A, the Shandong Provincial Natural Science Foundation for Quantum Science No. ZR2020LLZ003, ZR2021LLZ002, Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology (Grant No. SIQSE202001), Beijing Natural Science Foundation (Z190005), the Academician Innovation Platform of Hainan Province, Academy for Multidisciplinary Studies, Capital Normal University, and NSFC No. 12075159, 12171044.
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Liu, XQ., Wang, J., Li, M. et al. Quantum relaxed row and column iteration methods based on block-encoding. Quantum Inf Process 21, 230 (2022). https://doi.org/10.1007/s11128-022-03569-8
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DOI: https://doi.org/10.1007/s11128-022-03569-8