



@article {3187,
	title = {Quantum simulation of real-space dynamics},
	journal = {Quantum},
	volume = {6},
	year = {2022},
	month = {11/8/2022},
	pages = {860},
	abstract = {<p>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{\"o}dinger equation with η particles can be simulated with gate complexity O~(ηdFpoly(log(g\&prime;/ϵ))), where ϵ is the discretization error, g\&prime; 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\&prime; from poly(g\&prime;/ϵ) to poly(log(g\&prime;/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(ηdTg\&prime;/(Δϵ)))/Δ one- and two-qubit gates, and another using η3(4d)d/2Tpoly(log(ηdTg\&prime;/(Δϵ)))/Δ one- and two-qubit gates and QRAM operations, where T is the evolution time and the parameter Δ 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.</p>},
	doi = {https://doi.org/10.22331/q-2022-11-17-860},
	url = {https://doi.org/10.22331\%2Fq-2022-11-17-860},
	author = {Andrew M. Childs and Jiaqi Leng and Tongyang Li and Jin-Peng Liu and Chenyi Zhang}
}
