Resource-efficient quantum simulation of transport phenomena via Hamiltonian embedding

TL;DR

This paper introduces a resource-efficient quantum simulation framework for transport phenomena using Hamiltonian embedding, achieving theoretical speedups and practical reductions in quantum circuit depth, demonstrated on a 2D advection PDE.

Contribution

The authors develop a Hamiltonian embedding technique for efficient quantum simulation of transport equations with theoretical guarantees and hardware-friendly implementation.

Findings

Achieves exponential speedups in specific transport problems.

Provides an order-of-magnitude reduction in circuit depth.

Demonstrates a 2D advection PDE simulation on a trapped-ion quantum computer.

Abstract

Transport phenomena play a key role in a variety of application domains, and efficient simulation of these dynamics remains an outstanding challenge. While quantum computers offer potential for significant speedups, existing algorithms either lack rigorous theoretical guarantees or demand substantial quantum resources, preventing scalable and efficient validation on realistic quantum hardware. To address this gap, we develop a comprehensive framework for simulating classes of transport equations, offering both rigorous theoretical guarantees -- including exponential speedups in specific cases -- and a systematic, hardware-efficient implementation. Central to our approach is the Hamiltonian embedding technique, a white-box approach for end-to-end simulation of sparse Hamiltonians that avoids abstract query models and retains near-optimal asymptotic complexity. Empirical resource estimates indicate that our approach can yield an order-of-magnitude (e.g., $42\times$) reduction in circuit depth given favorable problem structures. We then apply our framework to solve linear and nonlinear transport PDEs, including the first experimental demonstration of a 2D advection equation on a trapped-ion quantum computer.