Toward Fault-Tolerant Quantum Methods for Climate-Relevant Nonlinear Dynamics: Carleman versus Liouville/Fokker–Planck Reformulations

To explore fault-tolerant quantum approaches for nonlinear differential equations, we compare two preprocessing routes that recast nonlinear dynamics as linear problems that can be solved via quantum linear-systems algorithms. We start from a system of nonlinear ordinary differential equations. The key step is the linearization: the nonlinear system is reformulated as a linear, or truncated linear, set of evolution equations. These can in principle be mapped to a matrix problem of the form Ax = y which represents the discrete-time update [9,10,13–18]. This mapping is only useful if the resulting operator remains sparse and well-conditioned and if state preparation and readout are not prohibitively expensive. 

The focus is on two prominent reformulation techniques. Carleman linearization is trajectory-based: it lifts a nonlinear system into a truncated set of linear equations and therefore targets single-trajectory evolution [12–16]. In contrast, the Liouville formulation rewrites nonlinear dynamics as a linear continuity equation for a probability density and therefore targets ensemble evolution directly [17,18]. With stochastic forcing, this becomes a Fokker–Planck equation [19,20]. We compare these routes because it remains open which type of linearization is better suited for climate-relevant nonlinear dynamics and for future quantum acceleration. Our earlier comparison showed that Liouville-based reformulations can remain informative in regimes where Carleman accuracy decreases because of truncation growth [17].

The ensemble route is then extended to stochastic dynamics. Here, noise is seen as a potential modelling choice that changes the represented dynamics and may become relevant both for numerical regularization and for describing unresolved subgrid-scale dynamics in climate models. As a chaotic test case with climate relevance, the Lorenz-63 system [21] is studied. 

This work focuses on validating the linearization step and assessing its usefulness as quantum-compatible preprocessing, rather than on a full quantum implementation. We do not simulate a quantum circuit. Instead, we implement each classical linearization and compare its predictions against reference solutions of the original nonlinear systems obtained from higher-accuracy classical discrete time integration. In our earlier study, this comparison was carried out for Carleman and Liouville reformulations on nonlinear toy models [17]. In the present extension, we further investigate Liouville versus Fokker–Planck dynamics for the two-fixed-point toy model and for the Lorenz-63 system. The stochastic ensemble evolution is sampled by shot-based Monte Carlo simulations, while time evolution is computed via forward Euler time.

For the toy model, the deterministic Liouville formulation remains accurate in regimes where Carleman degrades because of truncation error. In this setting, its statistical observables agreed with the exact trajectory. Moreover, the ensemble evolution remains accurate over time up to finite-size effects. When additive noise is introduced (see Fig. 1), the transient ensemble evolution changes. Here, stochastic jumps allow ensemble members to switch between the deterministic phase-space paths, which can shorten convergence times (see Fig. 1 (iii)). Depending on the initial condition, this alters the intermediate dynamics of the ensemble evolution and can also change the long-term distribution. This effect seems to be relevant in systems with multiple basins of attraction and with absorbing fixed-points. For Lorenz-63, the effect of noise is different (see Fig. 2 and 3). The intermediate evolution is modified, whereas the long-time distribution remains similar in the tested cases. We interpret this difference as a consequence of the Lorenz attractor structure, where trajectories can repeatedly return between regions of phase space rather than becoming irreversibly trapped in fixed points. As both Fokker–Planck simulations accurately capture the ensemble evolution of their respective noisy systems, this highlights the importance of the noise model whenever transient dynamics, rather than only the steady state solution, are of interest.

In parallel, the linearized problems are assessed in terms of their general future quantum requirements based on [16-19], including the scaling behaviour, sparsity, conditioning, and qubit requirements. 

The main value of this work is to assess realistic long-term quantum routes for climate-relevant nonlinear dynamics, and their current limitations. Carleman linearization preserves a trajectory viewpoint, but in strongly nonlinear or chaotic regimes the required truncation order can grow quickly, increasing the dimension of the lifted system and limiting practical usefulness [13-17]. Liouville and Fokker–Planck reformulations are more naturally aligned with uncertainty quantification and statistical observables, which are often the quantities of interest in climate science. The main drawback is the encoding cost. For a discretization with G grid points, V variables which are discretized into B value bins, the quantum state for the density requires q = GV log2(B) qubits. This is more compact than the full classical state dimension scaling exponentially with GV, but it still grows linearly with the number of discretized degrees of freedom and can therefore become very large even for fault-tolerant hardware [16,17].

Whether these approaches are useful in practice depends on conditioning, state preparation and readout overhead, oracle assumptions, and the specific QLSA routine used [9,10,16-19]. A further challenge is stochastic modelling itself. Small artificial white-noise terms are not automatically sufficient, because smaller diffusion coefficients can require finer discretization and therefore larger qubit counts. For climate applications, the next step is therefore to consider non-white noise models that better reflect unresolved subgrid processes. These can affect scaling, conditioning, and any eventual quantum advantage.