Time-marching based quantum solvers for time-dependent linear differential equations

Di Fang1,2,3, Lin Lin1,4,3, and Yu Tong5,1

1Department of Mathematics, University of California, Berkeley, CA 94720, USA
2Simons Institute for the Theory of Computing, University of California, Berkeley, CA 94720, USA
3Challenge Institute for Quantum Computation, University of California, Berkeley, CA 94720, USA
4Applied Mathematics and Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
5Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA

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Abstract

The time-marching strategy, which propagates the solution from one time step to the next, is a natural strategy for solving time-dependent differential equations on classical computers, as well as for solving the Hamiltonian simulation problem on quantum computers. For more general homogeneous linear differential equations $\frac{\mathrm{d}}{\mathrm{d} t} |\psi(t)\rangle = A(t) |\psi(t)\rangle, |\psi(0)\rangle = |\psi_0\rangle$, a time-marching based quantum solver can suffer from exponentially vanishing success probability with respect to the number of time steps and is thus considered impractical. We solve this problem by repeatedly invoking a technique called the uniform singular value amplification, and the overall success probability can be lower bounded by a quantity that is independent of the number of time steps. The success probability can be further improved using a compression gadget lemma. This provides a path of designing quantum differential equation solvers that is alternative to those based on quantum linear systems algorithms (QLSA). We demonstrate the performance of the time-marching strategy with a high-order integrator based on the truncated Dyson series. The complexity of the algorithm depends linearly on the amplification ratio, which quantifies the deviation from a unitary dynamics. We prove that the linear dependence on the amplification ratio attains the query complexity lower bound and thus cannot be improved in the worst case. This algorithm also surpasses existing QLSA based solvers in three aspects: (1) $A(t)$ does not need to be diagonalizable. (2) $A(t)$ can be non-smooth, and is only of bounded variation. (3) It can use fewer queries to the initial state $|\psi_0\rangle$. Finally, we demonstrate the time-marching strategy with a first-order truncated Magnus series, which simplifies the implementation compared to high-order truncated Dyson series approach, while retaining the aforementioned benefits. Our analysis also raises some open questions concerning the differences between time-marching and QLSA based methods for solving differential equations.

Quantum computers are naturally suited for simulating unitary dynamics, offering the prospect of simulating quantum systems that the best known classical algorithms cannot handle. Recent years have seen much effort in extending this advantage to general linear ordinary differential equations (ODEs) which are not necessarily unitary. On classical computers, these ODEs are usually solved via the time-marching strategy, i.e., dividing the time into short segments and evolving for one segment at a time. It was previously believed that implementing this strategy on a quantum computer would result in a computational cost that is exponential in the number of time steps. In this study, we present an efficient algorithm for solving linear ODEs using the time-marching strategy. Our algorithm can handle non-smooth coefficient matrices, requires fewer queries to the initial state, and is free of some of the technical constraints present in previous works. One main technical tool we develop is a "compression gadget" that enhances the success probability of a sequence of non-unitary operations, which could have independent interest.

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[7] Kaoru Mizuta and Keisuke Fujii, "Optimal Hamiltonian simulation for time-periodic systems", Quantum 7, 962 (2023).

[8] Alexander M. Dalzell, Sam McArdle, Mario Berta, Przemyslaw Bienias, Chi-Fang Chen, András Gilyén, Connor T. Hann, Michael J. Kastoryano, Emil T. Khabiboulline, Aleksander Kubica, Grant Salton, Samson Wang, and Fernando G. S. L. Brandão, "Quantum algorithms: A survey of applications and end-to-end complexities", arXiv:2310.03011, (2023).

[9] Shi Jin, Nana Liu, and Yue Yu, "Time complexity analysis of quantum algorithms via linear representations for nonlinear ordinary and partial differential equations", Journal of Computational Physics 487, 112149 (2023).

[10] Ivan Novikau, Ilya Y. Dodin, and Edward A. Startsev, "Encoding of linear kinetic plasma problems in quantum circuits via data compression", arXiv:2403.11989, (2024).

[11] Hari Krovi, "Quantum algorithms to simulate quadratic classical Hamiltonians and optimal control", arXiv:2404.07303, (2024).

[12] Hari Krovi, "Improved quantum algorithms for linear and nonlinear differential equations", Quantum 7, 913 (2023).

[13] Yulong Dong, Lin Lin, Hongkang Ni, and Jiasu Wang, "Infinite quantum signal processing", arXiv:2209.10162, (2022).

[14] Óscar Amaro and Diogo Cruz, "A Living Review of Quantum Computing for Plasma Physics", arXiv:2302.00001, (2023).

[15] Jin-Peng Liu and Lin Lin, "Dense outputs from quantum simulations", arXiv:2307.14441, (2023).

[16] Osama Muhammad Raisuddin and Suvranu De, "Quantum Relaxation for Linear Systems in Finite Element Analysis", arXiv:2308.01377, (2023).

[17] Osama Muhammad Raisuddin and Suvranu De, "Quantum Multigrid Algorithm for Finite Element Problems", arXiv:2404.07466, (2024).

[18] Nam Nguyen and Richard Thompson, "Solving Maxwells Equations using Variational Quantum Imaginary Time Evolution", arXiv:2402.14156, (2024).

The above citations are from Crossref's cited-by service (last updated successfully 2024-05-26 09:58:55) and SAO/NASA ADS (last updated successfully 2024-05-26 09:58:56). The list may be incomplete as not all publishers provide suitable and complete citation data.