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

2025-11-13

AI Summary

Efficiently solving nonlinear differential equations on current noisy quantum hardware remains challenging. Method: We propose an analog quantum algorithm coupling multiple bosonic modes with qubits, integrating Koopman–von Neumann formalism and Carleman linearization to map nonlinear partial differential equations (PDEs) into linear dynamical processes within a continuous-variable framework—bypassing Hilbert-space discretization. Classical fields are encoded via coherent states; adaptive qubit measurements and tailored Kraus channels are combined with analytically derived cancellation terms to suppress photon loss. Results: The algorithm is experimentally validated on the one-dimensional Burgers equation and two-dimensional Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) equation, demonstrating robustness under strong dissipation. Its computational complexity is 𝒪(T(log L + d r log K)), making it the first provably efficient, near-term-hardware-compatible continuous-variable quantum PDE solver.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Mixed Discrete/Continuous Search
• Computer Vision: Learning & Optimization for CV

Application Category

• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 $mathcal{O}left(T(log L + d r log K) ight)$. 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.

Problem

• Solving nonlinear differential equations efficiently using quantum computers
• Avoiding Hilbert-space digitization challenges in current quantum hardware
• Providing resilient quantum algorithms for near-term analog devices

Innovation

• Analog continuous-variable algorithm using coupled bosonic modes
• Encodes classical fields as coherent states via KvN formalism
• Maps nonlinear evolution to linear dynamics with provable efficiency