Analog photonic simulator for large-scale transport

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.