Quantum Algorithm for Two-Dimensional Radiative Transfer in a Cloudy Atmosphere

Radiative transfer (RT) calculations are essential in climate and weather models. At high spatial resolution, three-dimensional (3D) radiative effects can no longer be neglected, but multi-dimensional RT is computationally expensive. One of the direct deterministic ways to treat the radiance field is to use the discrete ordinates method (DOM), which reduces RT to a large linear system. However, extending such deterministic solvers to fully 3D RT is computationally prohibitive, making the approach impractical for current models. Here we explore an approach based on quantum computing, which is expected to outperform classical computers for certain problems by exploiting quantum properties.

As the first step, we study stationary radiative transfer in a two-dimensional cloudy atmosphere and discretize the boundary-value problem with DOM. We design a quantum algorithm that combines a block-encoding of the DOM coefficient matrix and quantum singular value transformation (QSVT). This approach enables implementation of the inverse operation that is required to solve the linear system under the fault-tolerant quantum computation. We encode a group of wavelengths in a superposition state to process them in parallel. By targeting integrated quantities such as heating rates, we avoid reconstructing full radiance-field while still keeping the advantages of this wavelength-level parallelism.

We estimate the quantum resources required for our quantum algorithm and examine their dependence on the number of spatial grid points N, the number of discrete angles N_ang, and the number of wavelength bins N_λ. We count the number of quantum gates to measure the computational cost. The gate count increases with the condition number of the DOM coefficient matrix, which increases roughly linearly with N. The gate count also increases with the block-encoding overhead, which increases roughly quadratically with N_ang. On the other hand, the dependence on N_λ can be kept nearly constant under a low-parameter approximation of the scattering phase function. Our results suggest that quantum computing is a promising approach for the DOM-based radiative transfer in a cloudy atmosphere, especially in fully 3D settings.