# Quantum simulation of real-space dynamics

arXiv: https://arxiv.org/abs/2203.17006 · HTML: https://mlsanity.com/paper/2203.17006/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Mar 2022.

## Built on by (most cited first, 8 of 8)

- [Quantum Hamiltonian Descent](https://mlsanity.com/paper/2303.01471/) (2023)
- [Quantum Langevin Dynamics for Optimization](https://mlsanity.com/paper/2311.15587/) (2023)
- [Quantum Algorithms and Lower Bounds for Finite-Sum Optimization](https://mlsanity.com/paper/2406.03006/) (2024)
- [Wavefunction Flows: Efficient Quantum Simulation of Continuous Flow Models](https://mlsanity.com/paper/2510.08462/) (2025)
- [A quantum-classical performance separation in nonconvex optimization](https://mlsanity.com/paper/2311.00811/) (2023)
- [Quantum Non-Linear Bandit Optimization](https://mlsanity.com/paper/2503.03023/) (2025)
- [Quantum Optimization via Gradient-Based Hamiltonian Descent](https://mlsanity.com/paper/2505.14670/) (2025)
- [Scalable quantum simulation of continuous-time generative models via tensor networks](https://mlsanity.com/paper/2608.21700/) (2026)

## Abstract (arXiv)

Quantum simulation is a prominent application of quantum computers. While there is extensive previous work on simulating finite-dimensional systems, less is known about quantum algorithms for real-space dynamics. We conduct a systematic study of such algorithms. In particular, we show that the dynamics of a d-dimensional Schr"odinger equation with η particles can be simulated with gate complexity tildeObigl(η d F poly(log(g'/ε))bigr), where ε is the discretization error, g' controls the higher-order derivatives of the wave function, and F measures the time-integrated strength of the potential. Compared to the best previous results, this exponentially improves the dependence on ε and g' from poly(g'/ε) to poly(log(g'/ε)) and polynomially improves the dependence on T and d, while maintaining best known performance with respect to η. For the case of Coulomb interactions, we give an algorithm using η3(d+η)Tpoly(log(η dTg'/(Deltaε)))/Delta one- and two-qubit gates, and another using η3(4d)d/2Tpoly(log(η dTg'/(Deltaε)))/Delta one- and two-qubit gates and QRAM operations, where T is the evolution time and the parameter Delta regulates the unbounded Coulomb interaction. We give applications to several computational problems, including faster real-space simulation of quantum chemistry, rigorous analysis of discretization error for simulation of a uniform electron gas, and a quadratic improvement to a quantum algorithm for escaping saddle points in nonconvex optimization.