Exponential Reduction of Mesh Dependence in Quantum Estimation of Parabolic PDE Observables

Quantum algorithms for partial differential equations (PDEs) often suffer from a "mesh dependence" problem: as the spatial resolution $h$ increases to improve accuracy, the computational cost grows polynomially. For parabolic equations, this is compounded by the decay of the solution norm, which makes preparing a normalized final state prohibitively expensive. This paper asks whether one can estimate linear and quadratic observables (such as heat flux or dissipation) directly, without preparing the normalized final state, and whether this can be done with only polylogarithmic dependence on the mesh size.

The author develops a multilevel quantum algorithm that operates on a nested hierarchy of Galerkin discretizations. Instead of block-encoding fine and coarse operators separately, the algorithm encodes their difference directly using a shifted Ritz-Schur factorization. This construction exposes the $O(h^2)$ normalization of the two-grid correction, allowing the fine-coarse cancellation to occur coherently within the quantum circuit. The algorithm uses a contour-based linear combination of unitaries (LCU) to reconstruct the target-time solution, and optimized amplitude estimation to extract the final observable.

The study demonstrates that by targeting observables directly—rather than the normalized solution state—the exponential overhead associated with postselection is removed. Furthermore, the multilevel corrected-resolvent estimator reduces the end-to-end complexity to $\widetilde{\mathcal{O}}(1 + (T\epsilon)^{-1})$, where $\epsilon$ is the target precision and $T$ is the time. This result holds for linear and quadratic observables with derivative orders up to $\chi=2$, effectively removing all polynomial dependence on the mesh size $h^{-1}$ for these classes of problems.

This work provides a significant advancement in the efficiency of quantum PDE solvers. By shifting the focus from state preparation to observable estimation and utilizing multilevel structures, the algorithm achieves a complexity that is essentially independent of the spatial resolution. This makes quantum simulation of parabolic systems significantly more practical for high-precision engineering applications where fine meshes are required.