Quantum simulation of realspace dynamics
Abstract
Quantum simulation is a prominent application of quantum computers. While there is extensive previous work on simulating finitedimensional systems, less is known about quantum algorithms for realspace dynamics. We conduct a systematic study of such algorithms. In particular, we show that the dynamics of a ddimensional Schrödinger equation with η particles can be simulated with gate complexity O~(ηdFpoly(log(g′/ϵ))), where ϵ is the discretization error, g′ controls the higherorder derivatives of the wave function, and F measures the timeintegrated 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(ηdTg′/(Δϵ)))/Δ one and twoqubit gates, and another using η3(4d)d/2Tpoly(log(ηdTg′/(Δϵ)))/Δ one and twoqubit 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 realspace 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.
 Publication:

Quantum
 Pub Date:
 November 2022
 DOI:
 10.22331/q20221117860
 arXiv:
 arXiv:2203.17006
 Bibcode:
 2022Quant...6..860C
 Keywords:

 Quantum Physics;
 Computer Science  Data Structures and Algorithms
 EPrint:
 Quantum 6, 860 (2022)