Quantum Simulation In The Semi-classical Regime

Republished By Plato

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Shi Jin¹, Xiantao Li², and Nana Liu³

¹School of Mathematical Sciences, Institute of Natural Sciences, MOE-LSEC and SHL-MAC, Shanghai Jiao Tong University, Shanghai, China
²Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA
³Institute of Natural Sciences, University of Michigan-Shanghai Jiao Tong University Joint Institute, MOE-LSEC, Shanghai Jiao Tong University, Shanghai, China

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Abstract

Solving the time-dependent Schrödinger equation is an important application area for quantum algorithms. We consider Schrödinger’s equation in the semi-classical regime. Here the solutions exhibit strong multiple-scale behavior due to a small parameter $hbar$, in the sense that the dynamics of the quantum states and the induced observables can occur on different spatial and temporal scales. Such a Schrödinger equation finds many applications, including in Born-Oppenheimer molecular dynamics and Ehrenfest dynamics. This paper considers quantum analogues of pseudo-spectral (PS) methods on classical computers. Estimates on the gate counts in terms of $hbar$ and the precision $varepsilon$ are obtained. It is found that the number of required qubits, $m$, scales only logarithmically with respect to $hbar$. When the solution has bounded derivatives up to order $ell$, the symmetric Trotting method has gate complexity $mathcal{O}Big({ (varepsilon hbar)^{-frac12} mathrm{polylog}(varepsilon^{-frac{3}{2ell}} hbar^{-1-frac{1}{2ell}})}Big),$ provided that the diagonal unitary operators in the pseudo-spectral methods can be implemented with $mathrm{poly}(m)$ operations. When physical observables are the desired outcomes, however, the step size in the time integration can be chosen independently of $hbar$. The gate complexity in this case is reduced to $mathcal{O}Big({varepsilon^{-frac12} mathrm{polylog}( varepsilon^{-frac3{2ell}} hbar^{-1} )}Big),$ with $ell$ again indicating the smoothness of the solution.

► BibTeX data

@article{Jin2022quantumsimulationin, doi = {10.22331/q-2022-06-17-739}, url = {https://doi.org/10.22331/q-2022-06-17-739}, title = {Quantum simulation in the semi-classical regime}, author = {Jin, Shi and Li, Xiantao and Liu, Nana}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{“{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {6}, pages = {739}, month = jun, year = {2022} }

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Cited by

[1] Andrew M. Childs, Jiaqi Leng, Tongyang Li, Jin-Peng Liu, and Chenyi Zhang, “Quantum simulation of real-space dynamics”, arXiv:2203.17006.

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On Crossref’s cited-by service no data on citing works was found (last attempt 2022-07-16 22:40:51).

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.

Time Stamp: June 17, 2022July 16, 2022

Time Stamp: Oct 6, 2022

Quantum simulation in the semi-classical regime

Republished By Plato

Abstract

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