Quantum simulation of real-space dynamics

Andrew M. Childs1,2, Jiaqi Leng1,3, Tongyang Li4,5,6, Jin-Peng Liu1,3, and Chenyi Zhang7

1Joint Center for Quantum Information and Computer Science, University of Maryland
2Department of Computer Science, University of Maryland
3Department of Mathematics, University of Maryland
4Center on Frontiers of Computing Studies, Peking University
5School of Computer Science, Peking University
6Center for Theoretical Physics, Massachusetts Institute of Technology
7Institute for Interdisciplinary Information Sciences, Tsinghua University

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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 $\eta$ particles can be simulated with gate complexity $\tilde{O}\bigl(\eta d F \text{poly}(\log(g'/\epsilon))\bigr)$, where $\epsilon$ 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 $\epsilon$ and $g'$ from $\text{poly}(g'/\epsilon)$ to $\text{poly}(\log(g'/\epsilon))$ and polynomially improves the dependence on $T$ and $d$, while maintaining best known performance with respect to $\eta$. For the case of Coulomb interactions, we give an algorithm using $\eta^{3}(d+\eta)T\text{poly}(\log(\eta dTg'/(\Delta\epsilon)))/\Delta$ one- and two-qubit gates, and another using $\eta^{3}(4d)^{d/2}T\text{poly}(\log(\eta dTg'/(\Delta\epsilon)))/\Delta$ 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.

We develop quantum algorithms for simulating the dynamics of interacting quantum particles in $d$ dimensions. Compared to the best previous results, our algorithm is exponentially better in terms of the discretization error $\epsilon$ and polynomially better in terms of the simulation time $T$ and the dimension $d$. 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.

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