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arXiv:2606.00581

Analog photonic simulator for large-scale transport

Mengyu Zhao, Xuezhi Zhu, Nikita Guseynov, Yewei Yuan, Na Wang, Meihong Wang, Yunyun Cao, Shi Jin, Nana Liu, Changde Xie, Kunchi Peng, Xiaolong Su

quant-phphysics.opticsSubmitted May 30, 2026• Announced Jun 2, 2026• 0 citations
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Abstract

Transport equations describe how physical quantities -- such as mass, energy, momentum, concentration, probability, or fields -- are carried, propagated, or redistributed through space and time, forming a foundational class of partial differential equations across science and engineering. However, high-dimensional partial differential equations are difficult to represent on digital grids because the number of degrees of freedom grows exponentially with dimension. Continuous-variable quantum photonics on the other hand can represent and evolve these large-scale fields without first discretizing space into a discrete grid. We demonstrate a large-scale analog photonic simulator for the constant-coefficient advection equation, a transport equation that is a fundamental benchmark for scientific computing. The solution of a $d$-variable advection equation is encoded into $d$ optical modes, so that the partial differential equation evolution maps directly to programmable phase-space displacements generated by optical quadrature momenta. Using a time-domain continuous-variable quantum photonic platform, we validate programmable control with $20,000$ single-mode squeezed states and $20,000$ two-mode squeezed states, and implement transport dynamics on a $20,000$-mode cluster-state resource. Homodyne measurements then verifies mode-resolved displacement control, which can provide first and second-order moment information of the solution to the advection equation, with final achievable relative error as low as $0.8\%$ and $0.92\%$ for first and second-order moment observables respectively. Our results establish continuous-variable photonics as a suitable programmable analog platform for large-scale advection equations.

Comments
47 pages, 22 figures, 14 tables

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Abstract Quantum simulators were originally proposed for simulating one partial differential equation (PDE) in particular—Schrödinger’s equation. Can quantum simulators also efficiently simulate other PDEs? While most computational methods for PDEs—both classical and quantum—are digital (they must be discretised first), PDEs have continuous degrees of freedom. This suggests that an analog representation can be more natural. While digital quantum degrees of freedom are usually described by qubits, the analog or continuous quantum degrees of freedom can be captured by qumodes. Based on a method called Schrödingerisation, we show how to directly map D-dimensional linear PDEs onto a ( D + 1 ) -qumode quantum system where analog or continuous-variable (CV) Hamiltonian simulation on D + 1 qumodes can be used. This very simple methodology does not require one to discretise PDEs first, and it is not only applicable to linear PDEs but also to some nonlinear PDEs and systems of nonlinear ordinary differential equations. We show some examples using this method, including the Liouville equation, heat equation, Fokker–Planck equation, Black–Scholes equations, wave equation and Maxwell’s equations. We also devise new protocols for linear PDEs with random coefficients, important in uncertainty quantification, where it is clear how the analog or CV framework is most natural. This also raises the possibility that some PDEs may be simulated directly on analog quantum systems by using Hamiltonians natural for those quantum systems.
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