Quantum Differential Equation Solvers with Low State Preparation Cost: Eliminating the Time Dependence in Dissipative Equations

TL;DR

This paper introduces quantum algorithms for simulating linear dissipative differential equations that significantly reduce the simulation time by focusing only on the effective period before substantial dissipation, achieving exponential improvements.

Contribution

The work presents a novel approach to quantum simulation that eliminates time dependence in dissipative equations while maintaining low state preparation costs, surpassing previous methods.

Findings

Algorithms eliminate the entire time dependence in simulations.

Achieve more than exponential speedup over prior quantum algorithms.

Maintain low state preparation cost during simulation.

Abstract

Linear dissipative differential equation is a fundamental model for a large number of physical systems, such as quantum dynamics with non-Hermitian Hamiltonian, open quantum system dynamics, diffusion process and damped system. In this work, we propose efficient quantum algorithms for simulating linear dissipative differential equations. The key idea of our algorithms is to perform the simulation only over an effective time period when the dynamics has not significantly dissipated yet, rather than over the entire physical evolution period. We conduct detailed analysis on the complexity of our algorithms and show that, while maintaining low state preparation cost, our algorithms can completely eliminate the time dependence. This is a more than exponential improvement compared to the previous state-of-the-art quantum algorithms.