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.
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 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.
Important nonlinear dynamics, such as those found in plasma and fluid systems, are typically hard to simulate on classical computers. Thus, if fault-tolerant quantum computers could efficiently solve such nonlinear problems, it would be a transformative change for many industries. In a recent breakthrough [Liu et al., PNAS 2021], the first efficient quantum algorithm for solving nonlinear differential equations was constructed, based on a single condition R<1, where R characterizes the ratio of nonlinearity to dissipation. This result, however, is limited to the class of purely dissipative systems with negative log-norm, which excludes application to many important problems. In this work, we correct technical issues with this and other prior analysis, and substantially extend the scope of nonlinear dynamical systems that can be efficiently simulated on a quantum computer in a number of ways. Firstly, we extend the existing results from purely dissipative systems to a much broader class of stable systems, and show that every quadratic Lyapunov function for the linearized system corresponds to an independent R-number criterion for the convergence of the Carleman scheme. Secondly, we extend our stable system results to physically relevant settings where conserved polynomial quantities exist. Finally, we provide extensive results for the class of non-resonant systems. With this, we are able to show that efficient quantum algorithms exist for a much wider class of nonlinear systems than previously known, and prove the BQP-completeness of nonlinear oscillator problems of exponential size. In our analysis, we also obtain several results related to the Poincaré-Dulac theorem and diagonalization of the Carleman matrix, which could be of independent interest.
Nonlinear differential equations model diverse phenomena but are notoriously difficult to solve. While there has been extensive previous work on efficient quantum algorithms for linear differential equations, the linearity of quantum mechanics has limited analogous progress for the nonlinear case. Despite this obstacle, we develop a quantum algorithm for dissipative quadratic n-dimensional ordinary differential equations. Assuming [Formula: see text], where R is a parameter characterizing the ratio of the nonlinearity and forcing to the linear dissipation, this algorithm has complexity [Formula: see text], where T is the evolution time, ϵ is the allowed error, and q measures decay of the solution. This is an exponential improvement over the best previous quantum algorithms, whose complexity is exponential in T. While exponential decay precludes efficiency, driven equations can avoid this issue despite the presence of dissipation. Our algorithm uses the method of Carleman linearization, for which we give a convergence theorem. This method maps a system of nonlinear differential equations to an infinite-dimensional system of linear differential equations, which we discretize, truncate, and solve using the forward Euler method and the quantum linear system algorithm. We also provide a lower bound on the worst-case complexity of quantum algorithms for general quadratic differential equations, showing that the problem is intractable for [Formula: see text] Finally, we discuss potential applications, showing that the [Formula: see text] condition can be satisfied in realistic epidemiological models and giving numerical evidence that the method may describe a model of fluid dynamics even for larger values of R.
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.
We study a fast-forwarded quantum algorithm for solving weakly nonlinear dissipative ordinary differential equations. Our approach is a combination of the Carleman embedding technique and the linear combination of Hamiltonian simulation algorithm for linearized systems with fast-forwarded scaling. The complexity of our algorithm does not explicitly depend on the evolution time T, thus greatly improving the previous state-of-the-art O(T) to O(1), and any remaining time dependence enters through the output norm and forcing parameters. We rigorously analyze the performance of this approach by convergence guarantees of the Carleman embedding for time-dependent coefficient matrices and detailed complexity estimates, and improve the realization of the Carleman-embedding-based algorithms by simplifying the post-selection step. In addition, we perform a numerical study on differential equations beyond the weakly nonlinear case, and identify possibility of achieving fast-forwarding scaling for systems with stronger nonlinearity or linear non-resonant effect.
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.