Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2212.10775

An efficient quantum algorithm for simulating polynomial differential equations

Amit Surana, Abeynaya Gnanasekaran, Tuhin Sahai

Published Dec 21, 2022• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2212.10775

Related papers

Efficient quantum algorithm for solving differential equations with Fourier nonlinearity via Koopman linearization

Judd Katz, Gopikrishnan Muraleedharan, Abhijeet Alase

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 $d{\bf u}/dt = G_0 + G_1 e^{i{\bf u}}$, where ${\bf u}$ denotes a vector of $n$ complex variables evolving with $t$, $G_0$ is an $n$-dimensional complex vector, $G_1$ is an $n \times n$ complex matrix and $e^{i{\bf u}}$ denotes the vector with entries $\{e^{iu_j}\}$. To tackle the Fourier nonlinear term, which is not expressible as a finite sum of polynomials of ${\bf u}$, 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.

quant-pharXiv:2512.06488•Dec 9, 2025•0 citations

Quantum Algorithms for Nonlinear Differential Equations via Pivot-Shifted Carleman Linearization

Ke Wang, Zikang Jia, Shravan Veerapaneni +1 more

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.

quant-pharXiv:2605.20071•May 20, 2026•1 citation

Efficient quantum algorithm for dissipative nonlinear differential equations

Jin-Peng Liu, Herman Øie Kolden, Hari Krovi +3 more

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.

arXiv:2011.03185•Proceedings of the National Academy of Sciences•Aug 26, 2021•238 citations

Reduced basis algorithm for solving nonlinear differential equations on quantum computers

Monica Lăcătuş, Matthias Möller, Sauro Succi

As quantum computing moves toward scientific computing applications, nonlinear differential equations remain a central challenge since quantum evolution is intrinsically linear. In this work, we introduce a reduced basis algorithm (RBA) for polynomial nonlinear ordinary differential equations (ODEs) and spatially discretized partial differential equations (PDEs). After time discretization, the method composes the resulting polynomial update map over $m$ timesteps, identifies the reduced monomial basis appearing in this composed map, and constructs a linear RBA operator whose action recovers the exact $m$-timestep nonlinear dynamics. Thus, at the level of the chosen discrete update rule, the method introduces no additional approximation error beyond the time discretization error. The qubit number requirement is governed by the size of the reduced monomial basis. For an $n$-dimensional polynomial ODE system of degree $p>1$, the lifted register requires at most $q_m^{\mathrm{ODE}} = O(nm\log p)$ qubits in the full basis scenario. For PDEs discretized on $N^D$ grid points, a locality-based construction requires at most $q_m^{\mathrm{PDE}} = O(D\log N + n m^{D+1}\log p)$ qubits. Hence, the dependence on the grid size remains logarithmic, while the nonlinear overhead is controlled by local reduced basis size. The main computational burden is moved from the quantum computer to a classical preprocessing step, where the reduced monomial basis and RBA operator are constructed for the chosen timestep window. Through numerical tests on the Lorenz system and the one-dimensional Burgers equation, we verify that the RBA reproduces the corresponding discrete time nonlinear dynamics exactly, while exposing the trade-off between timestep composition, reduced basis growth, and locality.

math.NACross-listedarXiv:2606.13457•Jun 12, 2026•0 citations

Quantum algorithm for high-dimensional nonlinear stochastic differential equations based on Carleman linearization

Koichi Miyamoto

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) $\mathrm{d} \mathbf{X}_t = \left(A(t) \mathbf{X}_t + F(t) \mathbf{X}_t^{\otimes s}\right) \mathrm{d} t + B(t) \mathrm{d} \mathbf{W}_t$ for an $\mathbb{R}^N$-valued random process $\mathbf{X}_t$, aiming at quantum speed-up with respect to the dimension $N$. We introduce auxiliary processes $\mathbf{Y}^j_t$ that approximate $\mathbf{X}_t^{\otimes j}$, 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 $\mathbf{X}_t$ 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 ${\rm polylog}(N)$ 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.

quant-pharXiv:2610.27416•Oct 21, 2026•0 citations

A Polynomial Time Quantum Algorithm for Exponentially Large Scale Nonlinear Differential Equations via Hamiltonian Simulation

Yu Tanaka, Keisuke Fujii

Quantum computers have the potential to efficiently solve a system of nonlinear ordinary differential equations (ODEs), which play a crucial role in various industries and scientific fields. However, it remains unclear which system of nonlinear ODEs, and under what assumptions, can achieve exponential speedup using quantum computers. In this work, we introduce a class of systems of nonlinear ODEs that can be efficiently solved on quantum computers, where the efficiency is defined as solving the system with computational complexity of $O(T {\rm log}(N) {\rm polylog}(1/ε))$, where $T$ is the evolution time, $ε$ is the allowed error, and $N$ is the number of variables in the system. Specifically, we employ the Koopman-von Neumann linearization to map the system of nonlinear ODEs to Hamiltonian dynamics and find conditions where the norm of the mapped Hamiltonian is preserved and the Hamiltonian is sparse. This allows us to use the optimal Hamiltonian simulation technique for solving the nonlinear ODEs with $O({\rm log}(N))$ overhead. Furthermore, we show that the nonlinear ODEs include a wide range of systems of nonlinear ODEs, such as the nonlinear harmonic oscillators and the short-range Kuramoto model. This is the first concrete example of solving systems of nonlinear ODEs with exponential quantum speedup by the Koopman-von Neumann linearization, although it is noted that this assumes efficient preparation of the initial state and computation of the output. These findings contribute significantly to the application of quantum computers in solving nonlinear problems.

arXiv:2305.00653•May 1, 2023•0 citations
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 nnn-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 kkk-th degree polynomial vector fields for arbitrary (finite) values of kkk. The steps involve: 1) mapping the kkk-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 kkk-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 kkk and demonstrate the framework on an example.
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+G1eiud{\bf u}/dt = G_0 + G_1 e^{i{\bf u}}du/dt=G0​+G1​eiu, where u{\bf u}u denotes a vector of nnn complex variables evolving with ttt, G0G_0G0​ is an nnn-dimensional complex vector, G1G_1G1​ is an n×nn \times nn×n complex matrix and eiue^{i{\bf u}}eiu denotes the vector with entries {eiuj}\{e^{iu_j}\}{eiuj​}. To tackle the Fourier nonlinear term, which is not expressible as a finite sum of polynomials of u{\bf u}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 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.
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.
As quantum computing moves toward scientific computing applications, nonlinear differential equations remain a central challenge since quantum evolution is intrinsically linear. In this work, we introduce a reduced basis algorithm (RBA) for polynomial nonlinear ordinary differential equations (ODEs) and spatially discretized partial differential equations (PDEs). After time discretization, the method composes the resulting polynomial update map over mmm timesteps, identifies the reduced monomial basis appearing in this composed map, and constructs a linear RBA operator whose action recovers the exact mmm-timestep nonlinear dynamics. Thus, at the level of the chosen discrete update rule, the method introduces no additional approximation error beyond the time discretization error. The qubit number requirement is governed by the size of the reduced monomial basis. For an nnn-dimensional polynomial ODE system of degree p>1p>1p>1, the lifted register requires at most qmODE=O(nmlog⁡p)q_m^{\mathrm{ODE}} = O(nm\log p)qmODE​=O(nmlogp) qubits in the full basis scenario. For PDEs discretized on NDN^DND grid points, a locality-based construction requires at most qmPDE=O(Dlog⁡N+nmD+1log⁡p)q_m^{\mathrm{PDE}} = O(D\log N + n m^{D+1}\log p)qmPDE​=O(DlogN+nmD+1logp) qubits. Hence, the dependence on the grid size remains logarithmic, while the nonlinear overhead is controlled by local reduced basis size. The main computational burden is moved from the quantum computer to a classical preprocessing step, where the reduced monomial basis and RBA operator are constructed for the chosen timestep window. Through numerical tests on the Lorenz system and the one-dimensional Burgers equation, we verify that the RBA reproduces the corresponding discrete time nonlinear dynamics exactly, while exposing the trade-off between timestep composition, reduced basis growth, and locality.
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\mathrm{d} \mathbf{X}_t = \left(A(t) \mathbf{X}_t + F(t) \mathbf{X}_t^{\otimes s}\right) \mathrm{d} t + B(t) \mathrm{d} \mathbf{W}_tdXt​=(A(t)Xt​+F(t)Xt⊗s​)dt+B(t)dWt​ for an RN\mathbb{R}^NRN-valued random process Xt\mathbf{X}_tXt​, aiming at quantum speed-up with respect to the dimension NNN. We introduce auxiliary processes Ytj\mathbf{Y}^j_tYtj​ that approximate Xt⊗j\mathbf{X}_t^{\otimes j}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\mathbf{X}_tXt​ 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 JJJ in CL, and we can estimate the target expectation with a polylog(N){\rm polylog}(N)polylog(N) number of queries to unitaries encoding coefficient matrices AAA, FFF, and BBB. Finally, we also run a numerical demonstration for an SDE related to Langevin dynamics, observing the exponential decay of the CL error as JJJ increases.
Quantum computers have the potential to efficiently solve a system of nonlinear ordinary differential equations (ODEs), which play a crucial role in various industries and scientific fields. However, it remains unclear which system of nonlinear ODEs, and under what assumptions, can achieve exponential speedup using quantum computers. In this work, we introduce a class of systems of nonlinear ODEs that can be efficiently solved on quantum computers, where the efficiency is defined as solving the system with computational complexity of O(Tlog(N)polylog(1/ε))O(T {\rm log}(N) {\rm polylog}(1/ε))O(Tlog(N)polylog(1/ε)), where TTT is the evolution time, εεε is the allowed error, and NNN is the number of variables in the system. Specifically, we employ the Koopman-von Neumann linearization to map the system of nonlinear ODEs to Hamiltonian dynamics and find conditions where the norm of the mapped Hamiltonian is preserved and the Hamiltonian is sparse. This allows us to use the optimal Hamiltonian simulation technique for solving the nonlinear ODEs with O(log(N))O({\rm log}(N))O(log(N)) overhead. Furthermore, we show that the nonlinear ODEs include a wide range of systems of nonlinear ODEs, such as the nonlinear harmonic oscillators and the short-range Kuramoto model. This is the first concrete example of solving systems of nonlinear ODEs with exponential quantum speedup by the Koopman-von Neumann linearization, although it is noted that this assumes efficient preparation of the initial state and computation of the output. These findings contribute significantly to the application of quantum computers in solving nonlinear problems.