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 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 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(ηdTg′/(Δϵ)))/Δ one- and two-qubit gates, and another using η3(4d)d/2Tpoly(log(ηdTg′/(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.

10aData Structures and Algorithms (cs.DS)10aFOS: Computer and information sciences10aFOS: Physical sciences10aQuantum Physics (quant-ph)1 aChilds, Andrew, M.1 aLeng, Jiaqi1 aLi, Tongyang1 aLiu, Jin-Peng1 aZhang, Chenyi uhttps://arxiv.org/abs/2203.1700601323nas a2200145 4500008004100000020002200041245005700063210005600120260008600176520080000262100002301062700001901085700001701104856005601121 2021 eng d a978-3-95977-195-500aQuantum Query Complexity with Matrix-Vector Products0 aQuantum Query Complexity with MatrixVector Products aDagstuhl, GermanybSchloss Dagstuhl – Leibniz-Zentrum für Informatikc2/7/20213 aWe study quantum algorithms that learn properties of a matrix using queries that return its action on an input vector. We show that for various problems, including computing the trace, determinant, or rank of a matrix or solving a linear system that it specifies, quantum computers do not provide an asymptotic speedup over classical computation. On the other hand, we show that for some problems, such as computing the parities of rows or columns or deciding if there are two identical rows or columns, quantum computers provide exponential speedup. We demonstrate this by showing equivalence between models that provide matrix-vector products, vector-matrix products, and vector-matrix-vector products, whereas the power of these models can vary significantly for classical computation.

1 aChilds, Andrew, M.1 aHung, Shih-Han1 aLi, Tongyang uhttps://drops.dagstuhl.de/opus/volltexte/2021/14124