Quantum Framework for Simulating Linear PDEs with Robin Boundary Conditions

TL;DR

This paper introduces a quantum simulation framework for linear PDEs with Robin boundary conditions, achieving polynomial speedup and exponential advantage in dimensions without relying on oracle queries.

Contribution

It extends previous quantum PDE simulation methods to include Robin boundary conditions, inhomogeneous terms, and variable coefficients, with explicit quantum operations and resource analysis.

Findings

Polylogarithmic block-encoding of Hamiltonian $H$

Polynomial scaling with grid points $N$

Linear scaling with spatial dimension $d$

Abstract

We propose an explicit, oracle-free quantum framework for numerically simulating general linear partial differential equations (PDEs), extending previous work 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 Schrodingerisation technique to transform the resulting system into one that admits unitary quantum evolution, enabling quantum simulation. For the Schrodinger 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 oracle-free nature of 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 $d$, achieving a polynomial speedup in $N$ and an exponential advantage in $d$, 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.