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

TL;DR

This paper introduces an analog quantum algorithm using coupled bosonic modes and qubits to efficiently solve nonlinear differential equations, avoiding digital truncation and fault-tolerance issues, and demonstrating practical potential on near-term hardware.

Contribution

It presents a provably efficient continuous-variable quantum algorithm for nonlinear PDEs that bypasses Hilbert-space digitization, enabling practical simulation on current analog quantum devices.

Findings

Successfully simulated Burgers' and Fisher-KPP equations

Demonstrated resilience to photon loss and noise

Achieved logarithmic scaling with problem size

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.