Lindbladian Homotopy Analysis Method to Solve Nonlinear Partial Differential Equations

Abstract: Quantum scientific computing is to solve engineering and science problems such as simulation and optimization on quantum computerss. Solving ordinary and partial differential equations (PDEs) is essential in simulations. However, existing quantum approaches to solve nonlinear PDEs suffer from the issues of curse of dimensionality and convergence during the linearization process. In this paper, a Lindbladian homotopy analysis method (LHAM) is proposed as a quantum differential equation solver to simulate nonunitary and nonlinear dynamics. The original nonlinear problem is first converted to a recursive sequence of linear PDEs with the homotopy analysis and reformulated as a higher-dimensional lower block triangular linear homogeneous system. The solution is then embedded in the density matrix and obtained through the Lindbladian dynamics simulation. Compared to other methods such as the Carleman linearization and Koopman-von Neumann approach where the dimension of Hilbert space increases polynomially with the inverse of truncation error, the Hilbert space in LHAM increases only logarithmically. LHAM is demonstrated nonlinear PDEs including Burgers' equation and magnetohydrodynamics equations.

• The paper introduces LHAM to recast nonlinear PDEs as a sequence of linearized equations embedded in Lindbladian dynamics.
• It avoids Hilbert space explosion by reducing dimensionality from exponential or quadratic to O(D log(1/ε)) scaling.
• Numerical demonstrations on Burgers’ and MHD equations validate LHAM’s accuracy with RMS errors around 1% and controlled convergence.

Lindbladian Homotopy Analysis Method for Nonlinear PDEs

Context and Motivation

The solution of nonlinear partial differential equations (PDEs) represents a central challenge for scientific computing and quantum information science, especially given their ubiquity in fluid dynamics, plasma physics, and related domains. Existing quantum algorithms addressing nonlinear ODEs and PDEs—such as Carleman linearization, the Koopman-von Neumann approach, and recent quantum homotopy or variational solvers—suffer from severe scaling limitations due to dimensionality expansion, convergence instability, or state-preparation bottlenecks. These approaches typically encode the nonlinearity via embedding into exponentially or combinatorially larger Hilbert spaces, with the associated quantum resource costs rapidly overwhelming any polynomial or even logarithmic scaling in physical parameters.

The present work introduces the Lindbladian Homotopy Analysis Method (LHAM), a quantum-centric approach for nonlinear and nonunitary differential equations. LHAM orchestrates a homotopy transform to recast the nonlinear PDE as a recursive sequence of nonhomogeneous linear PDEs, and then lifts the entire recursion into a lower block triangular homogeneous linear system. Crucially, the solution trajectory is embedded in a density matrix and realized as Lindbladian (GKSL) dynamics, circumventing the state space explosion and affording direct simulation of dissipative and nonunitary processes.

Derivation and Structure of LHAM

Homotopy Analysis–based Linearization

Given a nonlinear PDE:

$$\frac{\partial}{\partial t} \boldsymbol{u} = \mathcal{M} \boldsymbol{u} + \mathcal{N}(\boldsymbol{u}),$$

where $\mathcal{M}$ is a linear differential operator and $\mathcal{N}$ is nonlinear, LHAM applies a homotopy with embedding parameter $q$. The homotopy-Maclaurin series expansion parameterizes the solution as

$$\boldsymbol{\Phi}(\boldsymbol{x}, t; q) = \sum_{m=0}^\infty \boldsymbol{u}^{(m)}(\boldsymbol{x}, t) q^m,$$

where each homotopy order $m$ yields a nonhomogeneous linear equation for $\boldsymbol{u}^{(m)}$, recursively coupled through nonlinear source terms derived from the previous orders.

Importantly, this process does not require auxiliary variable embedding or ‘lifting’ in the style of Carleman or Koopman-based quantum encodings, and the homotopy truncation order $M$ required for a target accuracy $\epsilon$ only scales logarithmically: $M = O(\log(1/\epsilon))$ for convergent problems.

Autonomous Linear System Lifting

To simulate the entire recursive process as a single autonomous linear system, LHAM couples each nonhomogeneous equation to auxiliary channel variables with exponential decay, forming a lower block-triangular system:

$\mathcal{M}$0

where $\mathcal{M}$1 stacks all physical and auxiliary variables, and the system matrix $\mathcal{M}$2 is of size $\mathcal{M}$3 (with $\mathcal{M}$4 the spatial basis size, and $\mathcal{M}$5 the homotopy/truncation order). This avoids exponential blowup: the dimension is reduced from $\mathcal{M}$6 (Carleman, Koopman) to $\mathcal{M}$7.

Lindbladian Dynamics Simulation

Standard quantum simulation algorithms implement only unitary (Schrödinger) dynamics, and nonunitary evolution necessitates costly Hamiltonian dilation, LCUSH (Linear Combination of Unitaries+State Preparation), or repeated projective measurements. LHAM, by contrast, encodes both the Hermitian and anti-Hermitian parts of the generator in the Lindblad master equation, utilizing jump operators aligned with the dissipative structure:

$\mathcal{M}$8

This formalism exploits structure-preserving Lindbladian simulation, and only two ancilla qubits are needed, regardless of system size or time discretization—substantially reducing the ancilla overhead compared to Schrödingerization or LCUSH. The physical solution is recovered from an off-diagonal block of the evolved density matrix via straightforward extraction, and the state-preparation complexity is polylogarithmic in the precision.

Comparative Complexity and Scaling

LHAM achieves asymptotically exponential improvement over Carleman linearization and Koopman-von Neumann approaches in terms of Hilbert space dimension as a function of target error. Let $\mathcal{M}$9 denote the spatial basis size, and $\mathcal{N}$0 the desired truncation error:

• LHAM: Hilbert space dimension scales as $\mathcal{N}$1.
• Carleman: $\mathcal{N}$2 with $\mathcal{N}$3, so $\mathcal{N}$4.
• Koopman-von Neumann: $\mathcal{N}$5, i.e., $\mathcal{N}$6 if $\mathcal{N}$7 scales with $\mathcal{N}$8.

Therefore, for high-dimensional nonlinear PDEs where accuracy requirements push $\mathcal{N}$9 large, only LHAM remains tractable at scale on quantum hardware.

Numerical Demonstrations

Burgers’ Equation

Burgers’ equation, a testbed for shockwave formation and nonlinear advection-diffusion, was solved with the LHAM. Using a truncated Fourier basis ($q$0), and fourth-order homotopy truncation ($q$1), the LHAM solution achieved RMS error of $q$2 and relative $q$3 norm error of $q$4 with respect to classical finite difference ground truth. The convergence with order was consistent, and the acquired solution profiles matched classical approximations from homotopy order $q$5 onward.

Reduced Two-Dimensional Magnetohydrodynamics (MHD)

A nonlinear coupled MHD PDE system—comprising vorticity and magnetic potential—was solved using LHAM with a modest Fourier truncation ($q$6) and homotopy order $q$7. RMS and $q$8 errors were substantially lower than linearized (non-homotopic) solutions, with field profiles for vorticity and magnetic potential closely shadowing those of classical pseudo-spectral codes. The main limitation was the low spatial resolution; higher $q$9 would further suppress errors, indicating scalability for higher-fidelity physics when quantum resources allow.

Implications and Future Directions

LHAM’s design recasts the tradeoff between algorithmic expressivity for nonlinear systems and quantum resource requirements. The method is applicable to a broad class of nonlinear dissipative systems, including those not naturally amenable to Hamiltonian simulation techniques, and allows direct encoding of both dissipative and unitary dynamics in a hardware tractable formalism.

• Theoretic implications: LHAM resolves a significant bottleneck in quantum PDE simulation—the curse of dimensionality induced by linearization schemes for nonlinear ODE/PDEs. By leveraging the recursive, block-structured homotopy and dissipative-level Lindbladian embedding, exponential space blowup is avoided, and convergence is provably controlled.
• Practical implications: The reduction in both qubit number (physical and ancilla) and gate complexity makes previously infeasible quantum scientific computing tasks plausible, especially as mid-circuit measurement and reset become standard features of NISQ and future quantum architectures.

Further Developments: The extension of LHAM to strongly nonlinear or high-polynomial-degree systems is conditioned on the growth of the auxiliary variable count; however, even under worst-case combinatorics, symmetry and commutativity present in physically meaningful PDEs mitigate space requirements compared to Carleman-style lifting. The framework is compatible with functional/basis expansions and, through integration with weak/generic solvers [sul2025generic], may serve as a universal quantum solver for nonlinear dynamical systems.

Conclusion

The Lindbladian Homotopy Analysis Method constitutes a significant advance in quantum algorithms for nonlinear, nonunitary dynamics. By combining the recursive and dimension-efficient structure of homotopy with dissipative quantum channel simulation, LHAM dramatically compresses the quantum resource cost for solving nonlinear PDEs, while maintaining strong control over accuracy and convergence. Demonstrations on classical fluid and plasma model equations confirm practical performance and scalability potential. Continued analysis of complexity, robustness for highly nonlinear systems, and implementation optimizations for quantum hardware are the next critical steps.