Structure-Preserving Quantum Algorithms for Sparse Polynomial Nonlinear Dynamical Systems via Compressed Carleman–Koopman Lifting and Schrödingerisation
This paper presents a rigorous derivation pipeline for the quantum simulation of sparse polynomial nonlinear ordinary differential equations. The framework proceeds in three stages: (i) mapping the nonlinear dynamics to an exact infinite-dimensional linear system via Carleman–Koopman lifting; (ii) applying a graph-theoretic truncation and compression scheme that exploits the sparsity and locality of the dependency graph; and (iii) implementing Schr\"odingerisation for the resulting class of dissipative, Hermitian positive definite generators to yield a unitary-overlap formula compatible with standard quantum primitives. The study provides a comprehensive mathematical construction of these steps, establishing the algebraic and analytical foundations required for structure-preserving quantum simulation.
This paper presents a rigorous derivation pipeline for the quantum simulation of sparse polynomial nonlinear ordinary differential equations. The framework proceeds in three stages: (i) mapping the nonlinear dynamics to an exact infinite-dimensional linear system via Carleman–Koopman lifting; (ii) applying a graph-theoretic truncation and compression scheme that exploits the sparsity and locality of the dependency graph; and (iii) implementing Schr"odingerisation for the resulting class of dissipative, Hermitian positive definite generators to yield a unitary-overlap formula compatible with standard quantum primitives. The study provides a comprehensive mathematical construction of these steps, establishing the algebraic and analytical foundations required for structure-preserving quantum simulation.
We present an efficient quantum algorithm to simulate nonlinear differential equations with polynomial vector fields of arbitrary degree on quantum platforms. Models of physical systems that are governed by ordinary differential equations (ODEs) or partial differential equation (PDEs) can be challenging to solve on classical computers due to high dimensionality, stiffness, nonlinearities, and sensitive dependence to initial conditions. For sparse n-dimensional linear ODEs, quantum algorithms have been developed which can produce a quantum state proportional to the solution in poly(log(nx)) time using the quantum linear systems algorithm (QLSA). Recently, this framework was extended to systems of nonlinear ODEs with quadratic polynomial vector fields by applying Carleman linearization that enables the embedding of the quadratic system into an approximate linear form. A detailed complexity analysis was conducted which showed significant computational advantage under certain conditions. We present an extension of this algorithm to deal with systems of nonlinear ODEs with k-th degree polynomial vector fields for arbitrary (finite) values of k. The steps involve: 1) mapping the k-th degree polynomial ODE to a higher dimensional quadratic polynomial ODE; 2) applying Carleman linearization to transform the quadratic ODE to an infinite-dimensional system of linear ODEs; 3) truncating and discretizing the linear ODE and solving using the forward Euler method and QLSA. Alternatively, one could apply Carleman linearization directly to the k-th degree polynomial ODE, resulting in a system of infinite-dimensional linear ODEs, and then apply step 3. This solution route can be computationally more efficient. We present detailed complexity analysis of the proposed algorithms, prove polynomial scaling of runtime on k and demonstrate the framework on an example.
Differential-algebraic equations (DAEs) arise naturally in constrained dynamical systems, but their algebraic constraints and hidden compatibility conditions make them more subtle than standard ordinary differential equations. This paper initiates a quantum-algorithmic study of constrained linear DAEs. We introduce a dilation framework that embeds the generally non-Hermitian constrained evolution into a projected Schrödinger-type dynamics on an enlarged Hilbert space, idtdΨ(t)=PHPΨ(t), where H is Hermitian and P is the orthogonal projector onto the lifted constraint subspace. This identifies the DAE evolution with a quantum Zeno-type dynamics and enables the use of block encodings, QSVT-based projector construction, and Hamiltonian simulation.
We apply the framework to structure-preserving discretizations of the unsteady Stokes equations, where the pressure enforces the discrete incompressibility constraint. For Stokes, the Zeno-reduced generator has the projected square factorization Sh=−ΠhΔhΠh=(GhΠh)†(GhΠh), which can be represented through a Gaussian moment dilation and implemented as a Gaussian superposition of unitary Zeno evolutions generated by a first-order square-root Hamiltonian. In the generic sparse-access model, this gives a simulation-stage cost O(h−2t), up to the usual postselection factor for preparing the normalized dissipative state. The results provide a first step toward understanding the intersection of quantum algorithms, DAEs, constrained PDE dynamics, and square-root Gaussian dilations.
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.
Discrete nonlinear dynamical systems arise in population models, iterative optimization, and numerical simulations, where computing many iterations can be costly in high dimensions. We study Carleman embedding for forced quadratic discrete systems and its application to quantum state preparation. Under explicit stability and initial-state assumptions, we prove that the truncation error decreases geometrically with the embedding degree, uniformly over all iterations. A central obstacle in discrete time is that the lifted update can become dense even when the original coefficient maps are sparse. We overcome this obstacle by constructing an exact auxiliary linear-system representation that applies the coefficient maps one tensor factor at a time, with row and column sparsities bounded independently of the truncation degree. After rescaling, the associated history system has a condition number that grows linearly with the iteration count, uniformly in the truncation degree. These results yield a quantum algorithm for preparing an approximation to the normalized nth iterate with constant success probability. Under coherent input-access assumptions, and for fixed sparsity, stability parameters, and a known positive lower bound on the final-state norm, the oracle query complexity is linear in n and polylogarithmic in the inverse accuracy. Numerical experiments in one and three dimensions illustrate the convergence of the truncated embedding.
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.
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.