The cost of solving linear differential equations on a quantum computer: fast-forwarding to explicit resource counts

David Jennings; Matteo Lostaglio; Robert B. Lowrie; Sam Pallister; Andrew T. Sornborger

doi:10.22331/q-2024-12-10-1553

The cost of solving linear differential equations on a quantum computer: fast-forwarding to explicit resource counts

David Jennings¹, Matteo Lostaglio¹, Robert B. Lowrie², Sam Pallister¹, and Andrew T. Sornborger³

¹PsiQuantum, 700 Hansen Way, Palo Alto, CA 94304, USA
²Computational Physics and Methods Group (CCS-2), Computer, Computational and Statistical Sciences Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
³Information Sciences (CCS-3), Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

Published:	2024-12-10, volume 8, page 1553
Eprint:	arXiv:2309.07881v3
Doi:	https://doi.org/10.22331/q-2024-12-10-1553
Citation:	Quantum 8, 1553 (2024).

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Abstract

How well can quantum computers simulate classical dynamical systems? There is increasing effort in developing quantum algorithms to efficiently simulate dynamics beyond Hamiltonian simulation, but so far exact resource estimates are not known. In this work, we provide two significant contributions. First, we give the first non-asymptotic computation of the cost of encoding the solution to general linear ordinary differential equations into quantum states – either the solution at a final time, or an encoding of the whole history within a time interval. Second, we show that the stability properties of a large class of classical dynamics allow their fast-forwarding, making their quantum simulation much more time-efficient. From this point of view, quantum Hamiltonian dynamics is a boundary case that does not allow this form of stability-induced fast-forwarding. In particular, we find that the history state can always be output with complexity $O(T^{1/2})$ for any stable linear system. We present a range of asymptotic improvements over state-of-the-art in various regimes. We illustrate our results with a family of dynamics including linearized collisional plasma problems, coupled, damped, forced harmonic oscillators and dissipative nonlinear problems. In this case the scaling is quadratically improved, and leads to significant reductions in the query counts after inclusion of all relevant constant prefactors.

Featured image: The space of all linear ODEs can be organized by 2 Lyapunov parameters. These determine the resource costs for our quantum algorithm, and in particular we find that – if the linear system is in the left-half of the figure – ‘fast-forwarding’ is possible on a quantum computer, with complexity sublinear in time.

Popular summary

The simulation of dynamical systems is ubiquitous across the sciences, however when the dimension of the system becomes large we encounter the ‘curse of dimensionality’ that prohibits efficient simulation on classical computers. Quantum computers, on the other hand, are known to be able to circumvent this curse, and efficiently simulate the dynamics of quantum systems that evolve in an exponentially large Hilbert space. A natural question is therefore to ask how well quantum computers can simulate other high dimensional dynamical systems that arise in classical physics?

Towards this end we provide two important results. Firstly, we give the first rigorous, non-asymptotic resource counts for solving an arbitrary linear ODE system on a quantum computer. Secondly, we show that classical linear systems that are `stable' can be 'fast-forwarded' on a quantum computer, with complexity sublinear in time. The latter is realized by incorporating classical Lyapunov theory into our quantum algorithm, and elucidates the dependency of quantum speedups on physical features of classical systems.

► BibTeX data

@article{Jennings2024costofsolvinglinear,
  doi = {10.22331/q-2024-12-10-1553},
  url = {https://doi.org/10.22331/q-2024-12-10-1553},
  title = {The cost of solving linear differential equations on a quantum computer: fast-forwarding to explicit resource counts},
  author = {Jennings, David and Lostaglio, Matteo and Lowrie, Robert B. and Pallister, Sam and Sornborger, Andrew T.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1553},
  month = dec,
  year = {2024}
}

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