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arXiv:2410.19628 · Physical Review Letters

Designing a Nearly Optimal Quantum Algorithm for Linear Differential Equations via Lindbladians

Zhong-Xia Shang, Naixu Guo, Dong An, Qi Zhao

Published Aug 26, 2025• 9 citations
arXivHTMLPDF

Abstract

Solving linear ordinary differential equations (ODEs) is one of the most promising applications for quantum computers to demonstrate exponential advantages. The challenge of designing a quantum ODE algorithm is how to embed nonunitary dynamics into intrinsically unitary quantum circuits. In this Letter, we propose a new quantum algorithm for solving ODEs by harnessing open quantum systems. Specifically, we propose a novel technique called nondiagonal density matrix encoding, which leverages the inherent nonunitary dynamics of Lindbladians to encode general linear ODEs into the nondiagonal blocks of density matrices. This framework enables us to design quantum algorithms with both theoretical simplicity and high performance. Combined with the state-of-the-art quantum Lindbladian simulation algorithms, our algorithm can outperform all existing quantum ODE algorithms and achieve near-optimal dependence on all parameters under a plausible input model. We also give applications of our algorithm including the Gibbs state preparations and the partition function estimations.

DOI
10.1103/cvl9-97qg

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Solving linear ordinary differential equations (ODEs) is one of the most promising applications for quantum computers to demonstrate exponential advantages. The challenge of designing a quantum ODE algorithm is how to embed nonunitary dynamics into intrinsically unitary quantum circuits. In this Letter, we propose a new quantum algorithm for solving ODEs by harnessing open quantum systems. Specifically, we propose a novel technique called nondiagonal density matrix encoding, which leverages the inherent nonunitary dynamics of Lindbladians to encode general linear ODEs into the nondiagonal blocks of density matrices. This framework enables us to design quantum algorithms with both theoretical simplicity and high performance. Combined with the state-of-the-art quantum Lindbladian simulation algorithms, our algorithm can outperform all existing quantum ODE algorithms and achieve near-optimal dependence on all parameters under a plausible input model. We also give applications of our algorithm including the Gibbs state preparations and the partition function estimations.
Nonautonomous linear ordinary differential equations of the form v˙(t)=A(t) v(t)\dot{v}(t) = A(t)\, v(t)v˙(t)=A(t)v(t), where A(t)A(t)A(t) is non-skew-symmetric, are often used to describe nonunitary dynamics in a variety of fields that range from open quantum system dynamics to economic modeling. Because quantum computing hardware is designed to natively implement unitary transformations, existing algorithms for solving such equations on quantum hardware are based on the assumption that the nonunitary propagator is known, and use dilation techniques to embed the nonunitary dynamics within the unitary dynamics of a larger system. However, with the exception of cases where the nonunitary propagator is known in closed form, it needs to be calculated and manipulated on a classical computer at each time step. In this paper, we propose a quantum algorithm that does not require a priori knowledge of the explicit nonunitary propagator and effectively performs the dilation on the quantum hardware. Our algorithm combines a dilation scheme that uses singular value decomposition (SVD) to write the nonunitary propagator as a sum of unitaries with simulating the dynamics of the SVD factors on the quantum hardware. The population-only time-convolutionless quantum master equation describing photoinduced charge transfer in a solvated molecular triad is used as a demonstrative example of the applicability of the algorithm and its sensitivity to noise.
We present an efficient, nearly optimal quantum algorithm for solving linear matrix differential equations, with applications to the simulation of open quantum systems and beyond. For unitary or dissipative dynamics, the algorithm computes an entry of the solution matrix with query complexity O~(νLt/ε)\widetilde{\mathcal{O}}(ν\mathcal{L} t/ε)O(νLt/ε), where the constant ννν depends on the problem parameters, L\mathcal{L}L involves a time integral of upper bounds on the norms of evolution operators, and εεε is the error. In particular, νLν\mathcal{L}νL is linear in ttt for unitary dynamics and can be a constant for dissipative dynamics. Our result contrasts prior quantum approaches for differential equations that typically require exponential time for this problem due to the encoding in a quantum state, which can lead to exponentially small amplitudes. We demonstrate the utility of the algorithm through an end-to-end application, namely the simulation of dissipative dynamics for non-interacting fermions, which can be extended to other quantum and classical systems. We compare with classical algorithms and give evidence of polynomial quantum speedups for systems in a lattice, which become more pronounced for systems with long-range interactions and can be shown to be exponential in general. We also provide a lower bound of Ω(νLt/ε)Ω(ν\mathcal{L} t/ε)Ω(νLt/ε) for unitary or dissipative dynamics that proves our algorithm is optimal up to logarithmic factors.
We propose a novel quantum algorithm for solving linear autonomous ordinary differential equations (ODEs) using the Padé approximation. For linear autonomous ODEs, the discretized solution can be represented by a product of matrix exponentials. The proposed algorithm approximates the matrix exponential by the diagonal Padé approximation, which is then encoded into a large, block-sparse linear system and solved via quantum linear system algorithms (QLSA). The detailed quantum circuit is given based on quantum oracle access to the matrix, the inhomogeneous term, and the initial state. The complexity of the proposed algorithm is analyzed. Compared to the method based on Taylor approximation, which approximates the matrix exponential using a k -th order Taylor series, the proposed algorithm improves the approximation order k from two perspectives: 1) the explicit complexity dependency on k is improved, and 2) a smaller k suffices for the same precision. Numerical experiments demonstrate the advantages of the proposed algorithm comparing to other related algorithms.
Existing quantum algorithms for simulating dynamical systems -- from Hamiltonian simulation to linear and nonlinear differential equations solvers -- simulate Markovian dynamics, in which the system's future evolution depends solely on its current state. We turn our attention to developing quantum algorithms for non-Markovian dynamical systems where the system's future evolution depends on its past history and thus has memory. Specifically, we develop efficient algorithms for linear Volterra integro-differential equations (VIDEs) with a convolution memory kernel that output a quantum state encoding the state description over a time interval or at a particular time. Given efficient circuits for the problem inputs, our algorithms achieve an exponential speedup in system size over existing classical algorithms. We develop an algorithm for general kernels assuming that M<1\textsf{M} < 1M<1, where M\textsf{M}M characterizes the strength of the memory term relative to the dissipation of the Markovian part of the dynamics. We complement this with lower bounds for general-kernel VIDEs when M≥1\textsf{M} \geq 1M≥1, showing that the problem becomes intractable for a family of systems. However, by specializing to structured kernels which admit concise decompositions over exponentials, we develop efficient quantum algorithms even when M≥1\textsf M \geq 1M≥1 by converting the VIDE into a larger set of ODEs, a procedure which we call Markovianization. As an application of the overall framework, we discuss the Mori-Zwanzig formalism used in open quantum systems and fluid dynamics. Overall, our results expand the range of dynamical systems that quantum computers can simulate efficiently.
As quantum hardware rapidly advances toward the early fault-tolerant era, a key challenge is to develop quantum algorithms that are not only theoretically sound but also hardware-friendly on near-term devices. In this work, we propose a quantum algorithm for solving linear ordinary differential equations (ODEs) with a provable runtime guarantee. Our algorithm uses only a single ancilla qubit, and is locality preserving, i.e., when the coefficient matrix of the ODE is kkk-local, the algorithm only needs to implement the time evolution of (k+1)(k+1)(k+1)-local Hamiltonians. We also discuss the connection between our proposed algorithm and Lindbladian simulation. By applying our algorithm to the interacting Hatano-Nelson model, a widely studied non-Hermitian model with rich phenomenology, and numerically simulating it under realistic noise models, we demonstrate its practical feasibility on near-term quantum devices.
We present substantially generalized and improved quantum algorithms over prior work for inhomogeneous linear and nonlinear ordinary differential equations (ODE). Specifically, we show how the norm of the matrix exponential characterizes the run time of quantum algorithms for linear ODEs opening the door to an application to a wider class of linear and nonlinear ODEs. In \cite{BCOW17}, a quantum algorithm for a certain class of linear ODEs is given, where the matrix involved needs to be diagonalizable. The quantum algorithm for linear ODEs presented here extends to many classes of non-diagonalizable matrices including singular matrices. The algorithm here is also exponentially faster than the bounds derived in \cite{BCOW17} for certain classes of diagonalizable matrices. Our linear ODE algorithm is then applied to nonlinear differential equations using Carleman linearization (an approach taken recently by us in \cite{Liue2026805118}). The improvement over that result is two-fold. First, we obtain an exponentially better dependence on error. This kind of logarithmic dependence on error has also been achieved by \cite{Xue_2021}, but only for homogeneous nonlinear equations. Second, the present algorithm can handle any sparse matrix (that models dissipation) if it has a negative log-norm (including non-diagonalizable matrices), whereas \cite{Liue2026805118} and \cite{Xue_2021} additionally require normality.