# Quantum simulation in the semi-classical regime

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

### 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 $\mathcal{O}\Big({ (\varepsilon \hbar)^{-\frac12} \mathrm{polylog}(\varepsilon^{-\frac{3}{2\ell}} \hbar^{-1-\frac{1}{2\ell}})}\Big),$ 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 $\mathcal{O}\Big({\varepsilon^{-\frac12} \mathrm{polylog}( \varepsilon^{-\frac3{2\ell}} \hbar^{-1} )}\Big),$ with $\ell$ again indicating the smoothness of the solution.

### ► References

[1] Shi Jin, Peter Markowich, and Christof Sparber. Mathematical and computational methods for semiclassical Schrödinger equations''. Acta Numerica 20, 121–209 (2011). doi: https:/​/​doi.org/​10.1017/​S0962492911000031.
https:/​/​doi.org/​10.1017/​S0962492911000031

[2] Max Born and Robert Oppenheimer. Zur quantentheorie der molekeln''. Annalen der physik 389, 457–484 (1927). doi: https:/​/​doi.org/​10.1002/​andp.19273892002.
https:/​/​doi.org/​10.1002/​andp.19273892002

[3] John C Tully. Molecular dynamics with electronic transitions''. The Journal of Chemical Physics 93, 1061–1071 (1990). doi: https:/​/​doi.org/​10.1063/​1.459170.
https:/​/​doi.org/​10.1063/​1.459170

[4] Clarence Zener. Non-adiabatic crossing of energy levels''. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 137, 696–702 (1932). doi: https:/​/​doi.org/​10.1098/​rspa.1932.0165.
https:/​/​doi.org/​10.1098/​rspa.1932.0165

[5] Folkmar A Bornemann, Peter Nettesheim, and Christof Schütte. Quantum-classical molecular dynamics as an approximation to full quantum dynamics''. The Journal of chemical physics 105, 1074–1083 (1996). doi: https:/​/​doi.org/​10.1063/​1.471952.
https:/​/​doi.org/​10.1063/​1.471952

[6] Karen Drukker. Basics of surface hopping in mixed quantum/​classical simulations''. Journal of Computational Physics 153, 225–272 (1999). doi: https:/​/​doi.org/​10.1006/​jcph.1999.6287.
https:/​/​doi.org/​10.1006/​jcph.1999.6287

[7] Juergen Hinze. MC-SCF. I. the multi-configuration self-consistent-field method''. The Journal of Chemical Physics 59, 6424–6432 (1973). doi: https:/​/​doi.org/​10.1063/​1.1680022.
https:/​/​doi.org/​10.1063/​1.1680022

[8] Weizhu Bao, Shi Jin, and Peter A Markowich. On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime''. Journal of Computational Physics 175, 487–524 (2002). doi: https:/​/​doi.org/​10.1006/​jcph.2001.6956.
https:/​/​doi.org/​10.1006/​jcph.2001.6956

[9] Andrew M Childs, Yuan Su, Minh C Tran, Nathan Wiebe, and Shuchen Zhu. Theory of Trotter error with commutator scaling''. Physical Review X 11, 011020 (2021). doi: https:/​/​doi.org/​10.1103/​PhysRevX.11.011020.
https:/​/​doi.org/​10.1103/​PhysRevX.11.011020

[10] François Golse, Shi Jin, and Thierry Paul. On the convergence of time splitting methods for quantum dynamics in the semiclassical regime''. Foundations of Computational Mathematics 21, 613–647 (2021). doi: https:/​/​doi.org/​10.1007/​s10208-020-09470-z.
https:/​/​doi.org/​10.1007/​s10208-020-09470-z

[11] Caroline Lasser and Christian Lubich. Computing quantum dynamics in the semiclassical regime''. Acta Numerica 29, 229–401 (2020). doi: https:/​/​doi.org/​10.1017/​S0962492920000033.
https:/​/​doi.org/​10.1017/​S0962492920000033

[12] Haruo Yoshida. Construction of higher order symplectic integrators''. Physics letters A 150, 262–268 (1990). doi: https:/​/​doi.org/​10.1016/​0375-9601(90)90092-3.
https:/​/​doi.org/​10.1016/​0375-9601(90)90092-3

[13] Ryan Babbush, Nathan Wiebe, Jarrod McClean, James McClain, Hartmut Neven, and Garnet Kin-Lic Chan. Low-depth quantum simulation of materials''. Physical Review X 8, 011044 (2018). doi: https:/​/​doi.org/​10.1103/​PhysRevX.8.011044.
https:/​/​doi.org/​10.1103/​PhysRevX.8.011044

[14] Christof Zalka. Efficient simulation of quantum systems by quantum computers''. Fortschritte der Physik: Progress of Physics 46, 877–879 (1998). doi: https:/​/​doi.org/​10.1098/​rspa.1998.0162.
https:/​/​doi.org/​10.1098/​rspa.1998.0162

[15] Stephen Wiesner. Simulations of many-body quantum systems by a quantum computer'' (1996).

[16] Nicholas J Ward, Ivan Kassal, and Alán Aspuru-Guzik. Preparation of many-body states for quantum simulation''. The Journal of chemical physics 130, 194105 (2009). doi: https:/​/​doi.org/​10.1063/​1.3115177.
https:/​/​doi.org/​10.1063/​1.3115177

[17] Vivek V Shende, Stephen S Bullock, and Igor L Markov. Synthesis of quantum-logic circuits''. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). doi: https:/​/​doi.org/​10.1109/​TCAD.2005.855930.

[18] Stephen S Bullock and Igor L Markov. Asymptotically optimal circuits for arbitrary n-qubit diagonal computations''. Quantum Information & Computation 4, 27–47 (2004).

[19] Norbert Schuch and Jens Siewert. Programmable networks for quantum algorithms''. Physical review letters 91, 027902 (2003). doi: https:/​/​doi.org/​10.1103/​PhysRevLett.91.027902.
https:/​/​doi.org/​10.1103/​PhysRevLett.91.027902

[20] Guang Hao Low and Nathan Wiebe. Hamiltonian simulation in the interaction picture'' (2018). https:/​/​arxiv.org/​abs/​1805.00675v2.
arXiv:1805.00675v2

[21] Ivan Kassal, Stephen P Jordan, Peter J Love, Masoud Mohseni, and Alán Aspuru-Guzik. Polynomial-time quantum algorithm for the simulation of chemical dynamics''. Proceedings of the National Academy of Sciences 105, 18681–18686 (2008). doi: https:/​/​doi.org/​10.1073/​pnas.0808245105.
https:/​/​doi.org/​10.1073/​pnas.0808245105

[22] Yu Tong, Dong An, Nathan Wiebe, and Lin Lin. Fast inversion, preconditioned quantum linear system solvers, fast Green's-function computation, and fast evaluation of matrix functions''. Physical Review A 104, 032422 (2021). doi: https:/​/​doi.org/​10.1103/​PhysRevA.104.032422.
https:/​/​doi.org/​10.1103/​PhysRevA.104.032422

[23] Dominic W Berry, Andrew M Childs, Richard Cleve, Robin Kothari, and Rolando D Somma. Exponential improvement in precision for simulating sparse Hamiltonians''. In Forum of Mathematics, Sigma. Volume 5. Cambridge University Press (2017). doi: https:/​/​doi.org/​10.1017/​fms.2017.2.
https:/​/​doi.org/​10.1017/​fms.2017.2

[24] Stéphane Descombes and Mechthild Thalhammer. An exact local error representation of exponential operator splitting methods for evolutionary problems and applications to linear Schrödinger equations in the semi-classical regime''. BIT Numerical Mathematics 50, 729–749 (2010). doi: https:/​/​doi.org/​10.1007/​s10543-010-0282-4.
https:/​/​doi.org/​10.1007/​s10543-010-0282-4

[25] Peter A Markowich, Paola Pietra, and Carsten Pohl. Numerical approximation of quadratic observables of schrödinger-type equations in the semi-classical limit''. Numerische Mathematik 81, 595–630 (1999). doi: https:/​/​doi.org/​10.1007/​s002110050406.
https:/​/​doi.org/​10.1007/​s002110050406

[26] William J Huggins, Kianna Wan, Jarrod McClean, Thomas E O'Brien, Nathan Wiebe, and Ryan Babbush. Nearly optimal quantum algorithm for estimating multiple expectation values'' (2021). Nearly optimal quantum algorithm for estimating multiple expectation values.

[27] Joseph E Pasciak. Spectral and pseudospectral methods for advection equations''. Mathematics of Computation 35, 1081–1092 (1980). doi: https:/​/​doi.org/​10.1090/​S0025-5718-1980-0583488-0.
https:/​/​doi.org/​10.1090/​S0025-5718-1980-0583488-0

[28] Daniel Koch, Laura Wessing, and Paul M Alsing. Introduction to coding quantum algorithms: A tutorial series using Qiskit'' (2019). https:/​/​arxiv.org/​abs/​2111.09283.
arXiv:2111.09283

[29] Robert Wille, Rod Van Meter, and Yehuda Naveh. IBM’s Qiskit tool chain: Working with and developing for real quantum computers''. In 2019 Design, Automation & Test in Europe Conference & Exhibition (DATE). Pages 1234–1240. IEEE (2019). doi: https:/​/​doi.org/​10.23919/​DATE.2019.8715261.
https:/​/​doi.org/​10.23919/​DATE.2019.8715261

[30] David S Sholl and John C Tully. A generalized surface hopping method''. The Journal of chemical physics 109, 7702–7710 (1998). doi: https:/​/​doi.org/​10.1063/​1.477416.
https:/​/​doi.org/​10.1063/​1.477416

[31] Shi Jin, Peng Qi, and Zhiwen Zhang. An Eulerian surface hopping method for the Schrödinger equation with conical crossings''. Multiscale Modeling & Simulation 9, 258–281 (2011). doi: https:/​/​doi.org/​10.1137/​090774185.
https:/​/​doi.org/​10.1137/​090774185

[32] Atsushi Ishikawa, Hiroyuki Nakashima, and Hiroshi Nakatsuji. Accurate solutions of the Schrödinger and dirac equations of h2+, hd+, and ht+: With and without Born–Oppenheimer approximation and under magnetic field''. Chemical Physics 401, 62–72 (2012). doi: https:/​/​doi.org/​10.1016/​j.chemphys.2011.09.013.
https:/​/​doi.org/​10.1016/​j.chemphys.2011.09.013

[33] Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd. Quantum algorithm for solving linear systems of equations''. Physical Review Letters 103, 150502 (2009). doi: https:/​/​doi.org/​10.1103/​PhysRevLett.103.150502.
https:/​/​doi.org/​10.1103/​PhysRevLett.103.150502

[34] Andrew M Childs, Robin Kothari, and Rolando D Somma. Quantum algorithm for systems of linear equations with exponentially improved dependence on precision''. SIAM Journal on Computing 46, 1920–1950 (2017). doi: https:/​/​doi.org/​10.1137/​16M1087072.
https:/​/​doi.org/​10.1137/​16M1087072

[35] Andrew M Childs and Jin-Peng Liu. Quantum spectral methods for differential equations''. Communications in Mathematical Physics 375, 1427–1457 (2020). doi: https:/​/​doi.org/​10.1007/​s00220-020-03699-z.
https:/​/​doi.org/​10.1007/​s00220-020-03699-z

[36] Jean-Pierre Petit. An interpretation of cosmological model with variable light velocity''. Modern Physics Letters A 3, 1527–1532 (1988). doi: https:/​/​doi.org/​10.1142/​S0217732388001823.
https:/​/​doi.org/​10.1142/​S0217732388001823

### 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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