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qRLS: quantum relaxation for linear systems in finite element analysis

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Abstract

Quantum linear system algorithms (QLSAs) for gate-based quantum computing can provide exponential speedups for solving linear systems but face challenges when applied to finite element problems due to the growth of the condition number with problem size. Furthermore, QLSAs cannot use an approximate solution or initial guess to output an improved solution. Here, we present quantum relaxation for linear system (qRLS), as an iterative approach for gate-based quantum computers by embedding linear stationary iterations into a larger block linear system. The condition number of the block linear system scales linearly with the number of iterations independent of the size and condition number of the original system. The well-conditioned system enables a practical iterative solution of finite element problems using the state-of-the-art quantum signal processing (QSP) variant of QLSAs, for which we provide numerical results using a quantum computer simulator. The iteration complexity demonstrates favorable scaling relative to classical architectures, as the solution time is independent of system size and requires O(log(N)) qubits. This represents an exponential efficiency gain, offering a new approach for iterative finite element problem-solving on quantum hardware.

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Acknowledgements

The authors would like to acknowledge the support of this work by NIBIB R01EB0005807, R01EB25241, R01EB033674 and R01EB032820 grants.

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Authors and Affiliations

  1. Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Institute, 110 8th St., Troy, NY, 12180, USA

    Osama Muhammad Raisuddin

  2. Florida Agricultural and Mechanical University-Florida State University College of Engineering, 2525 Pottsdamer St., Tallahassee, FL, 32310, USA

    Suvranu De

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  1. Osama Muhammad Raisuddin
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Correspondence to Osama Muhammad Raisuddin.

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Raisuddin, O., De, S. qRLS: quantum relaxation for linear systems in finite element analysis. Engineering with Computers (2024). https://doi.org/10.1007/s00366-024-01975-3

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