Quantum framework for simulating linear PDEs with Robin boundary conditions

Abstract

We propose an explicit quantum framework for numerically simulating general linear partial differential equations (PDEs), extending previous work (Guseynov et al 2025 Phys. Rev. Res. 7 033100) to incorporate (a) Robin boundary conditions—which include Neumann and Dirichlet conditions as special cases–(b) inhomogeneous terms, and (c) variable coefficients in space and time. Our approach begins with a general finite-difference discretization and applies the Schrödingerisation technique to transform the resulting system into one that admits unitary quantum evolution, enabling quantum simulation. For the Schrödinger equation corresponding to the discretized PDE, we construct an efficient block-encoding of the Hamiltonian H that scales polylogarithmically with the number of grid points N. This encoding is compatible with quantum signal processing and allows for the implementation of the evolution operator e−iHt. The explicit circuit construction in our method permits complexity to be measured in fundamental gate units–namely, CNOT gates and single-qubit rotations–bypassing the inefficiencies of oracle queries. Consequently, the overall algorithm scales polynomially with N and linearly with the spatial dimension d. Under certain input/output assumptions our method achieves a polynomial speedup in N and an exponential advantage in d for a wide class of PDEs, thereby mitigating the classical curse of dimensionality. The validity and efficiency of the proposed approach are further substantiated by numerical simulations. By explicitly defining the quantum operations and quantifying their resource requirements, our approach offers a practical alternative for numerically solving PDEs, distinct from others that rely on oracle queries and purely asymptotic scaling methods.