Hamiltonian Simulation for Advection-Diffusion Equation with arbitrary transport field

TL;DR

This paper introduces a quantum-inspired Hamiltonian simulation method for solving the advection-diffusion equation with complex, spatially varying transport fields, demonstrating effectiveness on classical benchmarks and real quantum hardware.

Contribution

It develops a novel quantum algorithm for arbitrary transport fields in PDEs, combining upwinding and central differencing for stability, and validates it on IBM Quantum hardware.

Findings

Successful simulation of 2D advection-diffusion on 16 qubits

Effective handling of complex transport scenarios

Validation on real quantum hardware

Abstract

We present a novel approach to solve the advection-diffusion equation under arbitrary transporting fields using a quantum-inspired 'Schrodingerisation' technique for Hamiltonian simulation. Although numerous methods exist for solving partial differential equations (PDEs), Hamiltonian simulation remains a relatively underexplored yet promising direction-particularly in the context of long-term, fault-tolerant quantum computing. Building on this potential, our quantum algorithm is designed to accommodate non-trivial, spatially varying transport fields and is applicable to both 2D and 3D advection-diffusion problems. To ensure numerical stability and accuracy, the algorithm combines an upwinding discretization scheme for the advective component and the central differencing for diffusion, adapted for quantum implementation through a tailored mix of approximation and optimization techniques. We demonstrate the algorithm's effectiveness on benchmark scenarios involving coupled rotational, shear, and diffusive transport in two and three dimensions. Additionally, we implement the 2D advection-diffusion equation using 16 qubits on IBM Quantum hardware, validating our method and highlighting its practical applicability and robustness.