Quantum algorithms offer an exponential advantage with respect to the number of dependent variables for solving certain nonlinear ordinary differential equations (ODEs). These algorithms typically begin by transforming the original nonlinear ODE into a higher-dimensional linear ODE using a linearization technique, most commonly Carleman linearization. Existing works restrict their analysis to ODEs where the nonlinearities are polynomial functions of the dependent variables, significantly limiting their applicability. In this work we construct an efficient quantum algorithm for solving ODEs with `Fourier' nonlinear terms expressible as
du/dt=G0+G1eiu, where
u denotes a vector of
n complex variables evolving with
t,
G0 is an
n-dimensional complex vector,
G1 is an
n×n complex matrix and
eiu denotes the vector with entries
{eiuj}. To tackle the Fourier nonlinear term, which is not expressible as a finite sum of polynomials of
u, our algorithm employs a generalization of the Carleman linearization technique known as Koopman linearization. We also make other methodological advances towards relaxing the stringent dissipativity condition required for efficient solution extraction and towards integrated readout of classical quantities from the solution state. Our results open avenues to the development of efficient quantum algorithms for a significantly wider class of high-dimensional nonlinear ODEs, thereby broadening the scope of their applications.
This paper explores the integration of Carleman linearization and quantum annealing to solve nonlinear ordinary differential equations (ODEs) under the Chebyshev spectral method. Using Carleman linearization, the computation of optimal Chebyshev coefficients is reformulated as a quadratic program and further transformed into a Quadratic Unconstrained Binary Optimization (QUBO) problem via binary expansion. This QUBO formulation is well-suited for quantum annealers, where solutions are obtained by minimizing the Hamiltonian of the corresponding Ising model. To address hardware limitations, we employ an iterative quantum annealing strategy with dynamically updated parameters. Case studies on the D-Wave platform demonstrate the potential of our approach to harness the strengths of Carleman linearization and Chebyshev approximation theory, offering continuous and differentiable solutions to nonlinear ODEs, especially promising as quantum resources become more economically viable.
We develop a pivot-shifted Carleman linearization framework for quantum algorithms solving quadratic nonlinear ordinary differential equations. By shifting the dynamics by a pivot state prior to Carleman lifting, and combining this with a Lyapunov transform and rescaling, we enlarge the class of nonlinear systems that can be efficiently simulated on quantum computers. For systems that exhibit stability in the shifted coordinates, we establish long time convergence of the truncated Carleman embedding. We prove that the truncation order scales only logarithmically with the simulation time and target precision, and we derive end-to-end quantum query complexity bounds for preparing a state proportional to the final solution. By introducing a modified nonlinearity condition, this framework entirely removes the conventional lower bound requirement on the initial condition. For more general systems that remain unstable after shifting, we provide short time convergence guarantees that are similarly free from the initial condition constraints. Numerical experiments on the logistic and the Lotka-Volterra equations demonstrate that an appropriate pivot choice improves stability and accuracy, and yields exponential error decay with truncation order. These results show that pivot shifting provides a practical and theoretically justified route for extending Carleman-based quantum algorithms to a broader class of nonlinear dynamical systems.
Addressing nonlinear numerical problems in quantum computing remains challenging, but several approaches, including Carleman linearization (CL), have been proposed to construct quantum algorithms for such problems. In this work, we develop a CL-based method for a high-dimensional nonlinear stochastic differential equation (SDE)
dXt=(A(t)Xt+F(t)Xt⊗s)dt+B(t)dWt for an
RN-valued random process
Xt, aiming at quantum speed-up with respect to the dimension
N. We introduce auxiliary processes
Ytj that approximate
Xt⊗j, derive a linearized system of SDEs for them, and obtain the linear ordinary differential equation (ODE) governing their expectations. We then apply a quantum ODE solver to it and, from the resulting quantum state, extract an estimate for the expectation of a linear function of
Xt by overlap estimation. We show that, provided the nonlinear and noise terms in the SDE are sufficiently small relative to the linear term, the CL error decreases exponentially with the truncation order
J in CL, and we can estimate the target expectation with a
polylog(N) number of queries to unitaries encoding coefficient matrices
A,
F, and
B. Finally, we also run a numerical demonstration for an SDE related to Langevin dynamics, observing the exponential decay of the CL error as
J increases.
Abstract. In this paper, we explore the embedding of nonlinear dynamical systems into linear ordinary differential equations (ODEs) via the Carleman linearization method. Under strong dissipative conditions, numerous previous works have established rigorous error bounds and linear convergence for Carleman linearization, which have facilitated the identification of quantum advantages in simulating large-scale dynamical systems. Our analysis extends these findings by exploring error bounds beyond the traditional dissipative condition, thereby broadening the scope of quantum computational benefits to a new class of dynamical regimes. This novel regime is defined by a resonance condition, and we prove how this resonance condition leads to a linear convergence with respect to the truncation level [Formula: see text] in Carleman linearization. We support our theoretical advancements with numerical experiments on a variety of models, including the Burgers’ equation, Fermi–Pasta–Ulam (FPU) chains, and the Korteweg-de Vries (KdV) equations, to validate our analysis and demonstrate the practical implications.
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.