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arXiv:2602.07097

Alleviating Post-Linearization Challenges for Solving Nonlinear Systems on a Quantum Computer

Tayyab Ali

quant-phSubmitted Feb 6, 2026• Announced Feb 9, 2026• 0 citations
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Abstract

The linearity inherent in quantum mechanics limits current quantum hardware from directly solving nonlinear systems governed by nonlinear differential equations. One can opt for linearization frameworks such as Carleman linearization, which provides a high dimensional infinite linear system corresponding to a finite nonlinear system, as an indirect way of solving nonlinear systems using current quantum computers. We provide an efficient data access model to load this infinite linear representation of the nonlinear system, upto truncation order $N$, on a quantum computer by decomposing the Hamiltonian into the weighted sum of non-unitary operators, namely the Sigma basis. We have shown that the Sigma basis provides an exponential reduction in the number of decomposition terms compared to the traditional decomposition, which is usually done in a linear combination of Pauli operators. Once the Hamiltonian is decomposed, we then use the concept of unitary completion to construct the circuit for the implementation of each weighted tensor product component $\mathcal{H}_{j}$ of the decomposition.

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

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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

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Herein, we present a polylogarithmic decomposition method to load the matrix from the linearized 1-dimensional Burgers' equation onto a quantum computer. First, we use the Carleman linearization method to map the nonlinear Burgers' equation into an infinite linear system of equations, which is subsequently truncated to order $α$. This new finite linear system is then embedded into a larger system of equations with the key property that its matrix can be decomposed into a linear combination of $\mathcal{O}(\log n_t + α^2\log n_x)$ terms for $n_t$ time steps and $n_x$ spatial grid points. While the terms in this linear combination are not unitary, each can be implemented using a simple block encoding procedure. A numerical simulation is performed by combining our approach with the variational quantuam linear solver demonstrating that accurate solutions are possible. Finally, a resource estimate shows that the upper bound of the Clifford and T gate counts scale like $\mathcal{O}(α(\log n_x)^2)$ and $\mathcal{O}((\log n_x)^2)$, respectively. This is therefore the first explicit polylogarithmic data loading method with respect to $n_x$ and $n_t$ for a Carleman linearized system.

arXiv:2505.00285•Physical Review A•Jan 23, 2026•2 citations

A Scalable Approach to Solve the Carleman Linearized Burgers' Equation on a Quantum Computer

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Nonlinear ordinary and partial differential equations are ubiquitous in science and engineering, yet finding their solutions is often computationally intractable for classical hardware. To determine if quantum computers can offer a practical advantage, one critical challenge that must be solved is determining how to efficiently load exponentially sized matrices onto quantum hardware. In this article, we introduce an alternative linear combination of unitaries (LCU) strategy which relies on an intermediate linear combination of non-unitaries (LCNU) and a systematic embedding procedure. One advantage of this LCU strategy is that it maintains the exact number of terms as in the LCNU. Therefore, this approach offers a data loading framework for matrices that lack an efficient decomposition using the standard LCU alone. Using this approach, we construct a generalized LCNU framework for any Carleman linearized autonomous dynamical system having a polynomial nonlinearity. To demonstrate the effectiveness of our approach, we construct an LCNU for the 3D Carleman linearized lattice Boltzmann equation (LBE). Here, we find that the number of terms in the decomposition scales like $N_s\sim\mathcal{O}(α^2Q^2)$, where $α$ is the Carleman truncation order and $Q$ is the number of discrete velocities. Importantly, $N_s$ is independent of the number of spatial and temporal discretization points. We then perform a resource estimation of our LCNU's T gate cost when combined with the (1) PREP and SELECT block encoding oracles, and (2) variational quantum linear solver. In the former, the T cost scales like $\mathcal{O}(α^3Q^2(\log_2n)^2)$, where $n$ is the total number of spatial grid points. The latter requires exactly $N_s^2(\log_2 (2n_tn^α)+1)$ circuits per iteration for $n_t$ time steps, with a worst case T gate cost of $\mathcal{O}(α(\log_2Qn)^2)$ among them.

quant-pharXiv:2605.00302•May 4, 2026•0 citations

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.

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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
The linearity inherent in quantum mechanics limits current quantum hardware from directly solving nonlinear systems governed by nonlinear differential equations. One can opt for linearization frameworks such as Carleman linearization, which provides a high dimensional infinite linear system corresponding to a finite nonlinear system, as an indirect way of solving nonlinear systems using current quantum computers. We provide an efficient data access model to load this infinite linear representation of the nonlinear system, upto truncation order NNN, on a quantum computer by decomposing the Hamiltonian into the weighted sum of non-unitary operators, namely the Sigma basis. We have shown that the Sigma basis provides an exponential reduction in the number of decomposition terms compared to the traditional decomposition, which is usually done in a linear combination of Pauli operators. Once the Hamiltonian is decomposed, we then use the concept of unitary completion to construct the circuit for the implementation of each weighted tensor product component Hj\mathcal{H}_{j}Hj​ of the decomposition.
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.
Herein, we present a polylogarithmic decomposition method to load the matrix from the linearized 1-dimensional Burgers' equation onto a quantum computer. First, we use the Carleman linearization method to map the nonlinear Burgers' equation into an infinite linear system of equations, which is subsequently truncated to order ααα. This new finite linear system is then embedded into a larger system of equations with the key property that its matrix can be decomposed into a linear combination of O(log⁡nt+α2log⁡nx)\mathcal{O}(\log n_t + α^2\log n_x)O(lognt​+α2lognx​) terms for ntn_tnt​ time steps and nxn_xnx​ spatial grid points. While the terms in this linear combination are not unitary, each can be implemented using a simple block encoding procedure. A numerical simulation is performed by combining our approach with the variational quantuam linear solver demonstrating that accurate solutions are possible. Finally, a resource estimate shows that the upper bound of the Clifford and T gate counts scale like O(α(log⁡nx)2)\mathcal{O}(α(\log n_x)^2)O(α(lognx​)2) and O((log⁡nx)2)\mathcal{O}((\log n_x)^2)O((lognx​)2), respectively. This is therefore the first explicit polylogarithmic data loading method with respect to nxn_xnx​ and ntn_tnt​ for a Carleman linearized system.
Efficiently solving nonlinear ordinary and partial differential equations using a quantum computer is a major challenge due its inherent linearity. To circumvent this challenge, the Carleman linearization method has been proposed to transform a nonlinear ordinary differential equation into a linear system of equations, the primary advantage being that existing quantum linear systems algorithms may then be applied to obtain a solution. However, this methodology also brings forth several major challenges that must be addressed to attain a quantum advantage. Herein, we address several of these challenges enabling us to solve the Carleman linearized one-dimensional Burgers' equation on real and simulated quantum hardware. All simulations were performed on BlueQubit's platform allowing for quantum circuits to be run on GPU or QPU's seamlessly. We first demonstrate that the Carleman linearized Burgers' equation can be efficiently loaded onto a quantum computer using the linear combination of non-unitaries method, an alternative to the linear combintaiton of unitaries approach. Once loaded, the linear system is then solved using the variational quantum linear solver. Since a naive implementation of this solver is hindered by the barren plateau phenomenon, we introduce a multigridding method to solve the problem in a series of stages with the solution of the previous stage acting as a warm start for the next stage. This approach is found to significantly improve the accuracy of the solution compared with a naive cold start. Finally, circuits with a combined number of spatial and temporal discretization points totaling up to 280≈10242^{80} \approx 10^{24}280≈1024 are transpiled onto real quantum hardware demonstrating that the proposed methodology could feasibly produce a quantum advantage on future hardware.
Nonlinear ordinary and partial differential equations are ubiquitous in science and engineering, yet finding their solutions is often computationally intractable for classical hardware. To determine if quantum computers can offer a practical advantage, one critical challenge that must be solved is determining how to efficiently load exponentially sized matrices onto quantum hardware. In this article, we introduce an alternative linear combination of unitaries (LCU) strategy which relies on an intermediate linear combination of non-unitaries (LCNU) and a systematic embedding procedure. One advantage of this LCU strategy is that it maintains the exact number of terms as in the LCNU. Therefore, this approach offers a data loading framework for matrices that lack an efficient decomposition using the standard LCU alone. Using this approach, we construct a generalized LCNU framework for any Carleman linearized autonomous dynamical system having a polynomial nonlinearity. To demonstrate the effectiveness of our approach, we construct an LCNU for the 3D Carleman linearized lattice Boltzmann equation (LBE). Here, we find that the number of terms in the decomposition scales like Ns∼O(α2Q2)N_s\sim\mathcal{O}(α^2Q^2)Ns​∼O(α2Q2), where ααα is the Carleman truncation order and QQQ is the number of discrete velocities. Importantly, NsN_sNs​ is independent of the number of spatial and temporal discretization points. We then perform a resource estimation of our LCNU's T gate cost when combined with the (1) PREP and SELECT block encoding oracles, and (2) variational quantum linear solver. In the former, the T cost scales like O(α3Q2(log⁡2n)2)\mathcal{O}(α^3Q^2(\log_2n)^2)O(α3Q2(log2​n)2), where nnn is the total number of spatial grid points. The latter requires exactly Ns2(log⁡2(2ntnα)+1)N_s^2(\log_2 (2n_tn^α)+1)Ns2​(log2​(2nt​nα)+1) circuits per iteration for ntn_tnt​ time steps, with a worst case T gate cost of O(α(log⁡2Qn)2)\mathcal{O}(α(\log_2Qn)^2)O(α(log2​Qn)2) among them.
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.