



02005nas a2200181   4500008004100000245004600041210004500087260001400132300000800146490000600154520152300160100002301683700001601706700001701722700001801739700001801757856004801775       2022                        eng d00aQuantum simulation of real-space dynamics0 aQuantum simulation of realspace dynamics  c11/8/2022  a8600 v63 a<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 Schrödinger 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>1 aChilds, Andrew, M.1 aLeng, Jiaqi1 aLi, Tongyang1 aLiu, Jin-Peng1 aZhang, Chenyi  uhttps://doi.org/10.22331%2Fq-2022-11-17-86001482nas a2200145   4500008004100000245005500041210005500096260001400151490000600165520107700171100001801248700001601266700001701282856003701299       2021                        eng d00aQuantum Algorithms for Escaping from Saddle Points0 aQuantum Algorithms for Escaping from Saddle Points  c8/19/20210 v53 a<p>We initiate the study of quantum algorithms for escaping from saddle points with provable guarantee. Given a function f:Rn&rarr;R, our quantum algorithm outputs an ϵ-approximate second-order stationary point using O~(log2n/ϵ1.75) queries to the quantum evaluation oracle (i.e., the zeroth-order oracle). Compared to the classical state-of-the-art algorithm by Jin et al. with O~(log6n/ϵ1.75) queries to the gradient oracle (i.e., the first-order oracle), our quantum algorithm is polynomially better in terms of n and matches its complexity in terms of 1/ϵ. Our quantum algorithm is built upon two techniques: First, we replace the classical perturbations in gradient descent methods by simulating quantum wave equations, which constitutes the polynomial speedup in n for escaping from saddle points. Second, we show how to use a quantum gradient computation algorithm due to Jordan to replace the classical gradient queries by quantum evaluation queries with the same complexity. Finally, we also perform numerical experiments that support our quantum speedup.</p>1 aZhang, Chenyi1 aLeng, Jiaqi1 aLi, Tongyang  uhttps://arxiv.org/abs/2007.10253