Provably Efficient Quantum Algorithms for Solving Nonlinear Differential Equations Using Multiple Bosonic Modes Coupled with Qubits

Abstract: Quantum computers have long been expected to efficiently solve complex classical differential equations. Most digital, fault-tolerant approaches use Carleman linearization to map nonlinear systems to linear ones and then apply quantum linear-system solvers. However, provable speedups typically require digital truncation and full fault tolerance, rendering such linearization approaches challenging to implement on current hardware. Here we present an analog, continuous-variable algorithm based on coupled bosonic modes with qubit-based adaptive measurements that avoids Hilbert-space digitization. This method encodes classical fields as coherent states and, via Kraus-channel composition derived from the Koopman-von Neumann (KvN) formalism, maps nonlinear evolution to linear dynamics. Unlike many analog schemes, the algorithm is provably efficient: advancing a first-order, $L$-grid point, $d$-dimensional, order-$K$ spatial-derivative, degree-$r$ polynomial-nonlinearity, strongly dissipative partial differential equations (PDEs) for $T$ time steps costs $\mathcal{O}\left(T(\log L + d r \log K)\right)$. The capability of the scheme is demonstrated by using it to simulate the one-dimensional Burgers' equation and two-dimensional Fisher-KPP equation. The resilience of the method to photon loss is shown under strong-dissipation conditions and an analytic counterterm is derived that systematically cancels the dominant, experimentally calibrated noise. This work establishes a continuous-variable framework for simulating nonlinear systems and identifies a viable pathway toward practical quantum speedup on near-term analog hardware.