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Quantum Algorithm for Linear Differential Equations with Exponentially Improved Dependence on Precision

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Abstract

We present a quantum algorithm for systems of (possibly inhomogeneous) linear ordinary differential equations with constant coefficients. The algorithm produces a quantum state that is proportional to the solution at a desired final time. The complexity of the algorithm is polynomial in the logarithm of the inverse error, an exponential improvement over previous quantum algorithms for this problem. Our result builds upon recent advances in quantum linear systems algorithms by encoding the simulation into a sparse, well-conditioned linear system that approximates evolution according to the propagator using a Taylor series. Unlike with finite difference methods, our approach does not require additional hypotheses to ensure numerical stability.

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Authors and Affiliations

  1. Department of Physics and Astronomy, Macquarie University, Sydney, Australia

    Dominic W. Berry

  2. Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD, USA

    Andrew M. Childs

  3. Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD, USA

    Andrew M. Childs, Aaron Ostrander & Guoming Wang

  4. Department of Physics, University of Maryland, College Park, MD, USA

    Aaron Ostrander

Authors
  1. Dominic W. Berry
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  2. Andrew M. Childs
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  3. Aaron Ostrander
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  4. Guoming Wang
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Corresponding author

Correspondence to Aaron Ostrander.

Additional information

Communicated by M. M. Wolf

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Berry, D.W., Childs, A.M., Ostrander, A. et al. Quantum Algorithm for Linear Differential Equations with Exponentially Improved Dependence on Precision. Commun. Math. Phys. 356, 1057–1081 (2017). https://doi.org/10.1007/s00220-017-3002-y

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  • DOI: https://doi.org/10.1007/s00220-017-3002-y

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