Time-dependent Hamiltonian Simulation of Highly Oscillatory Dynamics and Superconvergence for Schrödinger Equation

Dong An; Di Fang; Lin Lin

doi:10.22331/q-2022-04-15-690

Time-dependent Hamiltonian Simulation of Highly Oscillatory Dynamics and Superconvergence for Schrödinger Equation

Dong An¹, Di Fang^2,3,5, and Lin Lin^2,4,5

¹Joint Center for Quantum Information and Computer Science (QuICS), University of Maryland, College Park, MD 20742, USA
²Department of Mathematics, University of California, Berkeley, CA 94720, USA
³Simons Institute for the Theory of Computing, University of California, Berkeley, CA 94720, USA
⁴Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
⁵Challenge Institute for Quantum Computation, University of California, Berkeley, CA 94720, USA

Published:	2022-04-15, volume 6, page 690
Eprint:	arXiv:2111.03103v2
Doi:	https://doi.org/10.22331/q-2022-04-15-690
Citation:	Quantum 6, 690 (2022).

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Abstract

We propose a simple quantum algorithm for simulating highly oscillatory quantum dynamics, which does not require complicated quantum control logic for handling time-ordering operators. To our knowledge, this is the first quantum algorithm that is both insensitive to the rapid changes of the time-dependent Hamiltonian and exhibits commutator scaling. Our method can be used for efficient Hamiltonian simulation in the interaction picture. In particular, we demonstrate that for the simulation of the Schrödinger equation, our method exhibits superconvergence and achieves a surprising second order convergence rate, of which the proof rests on a careful application of pseudo-differential calculus. Numerical results verify the effectiveness and the superconvergence property of our method.

► BibTeX data

@article{An2022timedependent,
  doi = {10.22331/q-2022-04-15-690},
  url = {https://doi.org/10.22331/q-2022-04-15-690},
  title = {Time-dependent {H}amiltonian {S}imulation of {H}ighly {O}scillatory {D}ynamics and {S}uperconvergence for {S}chr{\"{o}}dinger {E}quation},
  author = {An, Dong and Fang, Di and Lin, Lin},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {690},
  month = apr,
  year = {2022}
}

► References

[1] T. Albash and D. A. Lidar. Adiabatic quantum computation. Rev. Mod. Phys., 90:015002, 2018. doi:10.1103/RevModPhys.90.015002.
https://doi.org/10.1103/RevModPhys.90.015002

[2] D. An, D. Fang, and L. Lin. Time-dependent unbounded Hamiltonian simulation with vector norm scaling. Quantum, 5:459, may 2021. doi:10.22331/q-2021-05-26-459.
https://doi.org/10.22331/q-2021-05-26-459

[3] D. W. Berry, G. Ahokas, R. Cleve, and B. C. Sanders. Efficient quantum algorithms for simulating sparse Hamiltonians. Commun. Math. Phys., 270(2):359–371, 2007. doi:10.1007/s00220-006-0150-x.
https://doi.org/10.1007/s00220-006-0150-x

[4] D. W. Berry and A. M. Childs. Black-box Hamiltonian simulation and unitary implementation. Quantum Information & Computation, 12(1-2):29–62, 2012. doi:10.26421/QIC12.1-2.
https://doi.org/10.26421/QIC12.1-2

[5] D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma. Exponential improvement in precision for simulating sparse Hamiltonians. In Proceedings of the forty-sixth annual ACM symposium on Theory of computing, pages 283–292, 2014. doi:10.1145/2591796.2591854.
https://doi.org/10.1145/2591796.2591854

[6] D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma. Simulating Hamiltonian dynamics with a truncated Taylor series. Phys. Rev. Lett., 114:090502, 2015. doi:10.1103/PhysRevLett.114.090502.
https://doi.org/10.1103/PhysRevLett.114.090502

[7] D. W. Berry, A. M. Childs, and R. Kothari. Hamiltonian simulation with nearly optimal dependence on all parameters. Proceedings of the 56th IEEE Symposium on Foundations of Computer Science, pages 792–809, 2015. doi:10.1109/FOCS.2015.54.
https://doi.org/10.1109/FOCS.2015.54

[8] D. W. Berry, A. M. Childs, Y. Su, X. Wang, and N. Wiebe. Time-dependent Hamiltonian simulation with $l^{1}$-norm scaling. Quantum, 4:254, 2020. doi:10.22331/q-2020-04-20-254.
https://doi.org/10.22331/q-2020-04-20-254

[9] D. W. Berry, R. Cleve, and S. Gharibian. Gate-efficient discrete simulations of continuous-time quantum query algorithms. Quantum Information and Computation, 14(1-2):1–30, 2014. doi:10.26421/QIC14.1-2-1.
https://doi.org/10.26421/QIC14.1-2-1

[10] S. Blanes, F. Casas, and M. Thalhammer. High-order commutator-free quasi-magnus exponential integrators for non-autonomous linear evolution equations. Computer Physics Communications, 220:243–262, 2017. URL: https://www.sciencedirect.com/science/article/pii/S0010465517302357, doi:https://doi.org/10.1016/j.cpc.2017.07.016.
https://doi.org/10.1016/j.cpc.2017.07.016
https://www.sciencedirect.com/science/article/pii/S0010465517302357

[11] A. Boulkhemair. L2 Estimates for Weyl Quantization. Journal of Functional Analysis, 165(1):173–204, 1999. doi:10.1006/jfan.1999.3423.
https://doi.org/10.1006/jfan.1999.3423

[12] R. L. Burden, J. D. Faires, and A. C. Reynolds. Numerical analysis. Brooks Cole, 2000.

[13] E. Campbell. Random compiler for fast Hamiltonian simulation. Phys. Rev. Lett., 123(7):070503, 2019. doi:10.1103/PhysRevLett.123.070503.
https://doi.org/10.1103/PhysRevLett.123.070503

[14] S. Chakraborty, A. Gilyén, and S. Jeffery. The power of block-encoded matrix powers: improved regression techniques via faster hamiltonian simulation. arXiv:1804.01973, 2018. 10.4230/LIPIcs.ICALP.2019.33.
https://doi.org/10.4230/LIPIcs.ICALP.2019.33
arXiv:1804.01973

[15] C.-F. Chen, H.-Y. Huang, R. Kueng, and J. A. Tropp. Quantum simulation via randomized product formulas: Low gate complexity with accuracy guarantees. 2020. arXiv:2008.11751.
https://doi.org/10.1103/PRXQuantum.2.040305
arXiv:2008.11751

[16] Y.-H. Chen, A. Kalev, and I. Hen. Quantum algorithm for time-dependent hamiltonian simulation by permutation expansion. PRX Quantum, 2:030342, Sep 2021. URL: https://link.aps.org/doi/10.1103/PRXQuantum.2.030342, doi:10.1103/PRXQuantum.2.030342.
https://doi.org/10.1103/PRXQuantum.2.030342

[17] A. M. Childs, D. Maslov, Y. Nam, N. J. Ross, and Y. Su. Toward the first quantum simulation with quantum speedup. Proc. Nat. Acad. Sci., 115:9456–9461, 2018. doi:10.1073/pnas.1801723115.
https://doi.org/10.1073/pnas.1801723115

[18] A. M. Childs, A. Ostrander, and Y. Su. Faster quantum simulation by randomization. Quantum, 3:182, 2019. doi:10.22331/q-2019-09-02-182.
https://doi.org/10.22331/q-2019-09-02-182

[19] A. M. Childs and Y. Su. Nearly optimal lattice simulation by product formulas. Phys. Rev. Lett., 123(5):050503, 2019. doi:10.1103/PhysRevLett.123.050503.
https://doi.org/10.1103/PhysRevLett.123.050503

[20] A. M. Childs, Y. Su, M. C. Tran, N. Wiebe, and S. Zhu. Theory of trotter error with commutator scaling. Phys. Rev. X, 11:011020, 2021. doi:10.1103/PhysRevX.11.011020.
https://doi.org/10.1103/PhysRevX.11.011020

[21] A. M. Childs and N. Wiebe. Hamiltonian simulation using linear combinations of unitary operations. Quantum Information and Computation, 12, Nov 2012. URL: http://dx.doi.org/10.26421/QIC12.11-12, doi:10.26421/qic12.11-12.
https://doi.org/10.26421/QIC12.11-12

[22] H. O. Cordes. On compactness of commutators of multiplications and convolutions, and boundedness of pseudodifferential operators. Journal of Functional Analysis, 18(2):115–131, 1975. doi:10.1016/0022-1236(75)90020-8.
https://doi.org/10.1016/0022-1236(75)90020-8

[23] D. Dong and I. R. Petersen. Quantum control theory and applications: a survey. IET Control Theory & Applications, 4(12):2651–2671, 2010. doi:10.1049/iet-cta.2009.0508.
https://doi.org/10.1049/iet-cta.2009.0508

[24] E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser. Quantum computation by adiabatic evolution. 2000. arXiv:quant-ph/0001106.
arXiv:quant-ph/0001106

[25] A. Gilyén, S. Arunachalam, and N. Wiebe. Optimizing quantum optimization algorithms via faster quantum gradient computation. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1425–1444, 2019. doi:10.1137/1.9781611975482.87.
https://doi.org/10.1137/1.9781611975482.87

[26] A. Gilyén, Y. Su, G. H. Low, and N. Wiebe. Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 193–204, 2019. doi:10.1145/3313276.3316366.
https://doi.org/10.1145/3313276.3316366

[27] E. Hairer, S. P. Nørsett, and G. Wanner. Solving ordinary differential equation I: nonstiff problems, volume 8. Springer, 1987. doi:10.1007/978-3-540-78862-1.
https://doi.org/10.1007/978-3-540-78862-1

[28] M. Hochbruck and C. Lubich. On Magnus integrators for time-dependent Schrödinger equations. SIAM J. Numer. Anal., 41(3):945–963, 2003. doi:10.1137/S0036142902403875.
https://doi.org/10.1137/S0036142902403875

[29] J. Huyghebaert and H. De Raedt. Product formula methods for time-dependent Schrödinger problems. J. Phys. A, 23(24):5777–5793, 1990. doi:10.1088/0305-4470/23/24/019.
https://doi.org/10.1088/0305-4470/23/24/019

[30] A. Iserles and S. P. Nørsett. On the solution of linear differential equations in lie groups. Phil. Trans. R. Soc. A., 357:983–1019, 1999. doi:10.1098/rsta.1999.0362.
https://doi.org/10.1098/rsta.1999.0362

[31] M. Kieferová, A. Scherer, and D. W. Berry. Simulating the dynamics of time-dependent hamiltonians with a truncated dyson series. Physical Review A, 99(4), Apr 2019. URL: http://dx.doi.org/10.1103/PhysRevA.99.042314, doi:10.1103/physreva.99.042314.
https://doi.org/10.1103/PhysRevA.99.042314

[32] I. Kivlichan, J. McClean, N. Wiebe, C. Gidney, A. Aspuru-Guzik, G.-L. Chan, and R. Babbush. Quantum Simulation of Electronic Structure with Linear Depth and Connectivity. Phys. Rev. Lett., 120(11):110501, 2018. doi:10.1103/PhysRevLett.120.110501.
https://doi.org/10.1103/PhysRevLett.120.110501

[33] I. D. Kivlichan, N. Wiebe, R. Babbush, and A. Aspuru-Guzik. Bounding the costs of quantum simulation of many-body physics in real space. J. Phys. A Math. Theor., 50:305301, 2017. doi:10.1088/1751-8121/aa77b8.
https://doi.org/10.1088/1751-8121/aa77b8

[34] A. W. Knapp. Basic Real Analysis. Springer Science & Business Media, 2005. doi:10.1007/0-8176-4441-5.
https://doi.org/10.1007/0-8176-4441-5

[35] G. H. Low. Hamiltonian simulation with nearly optimal dependence on spectral norm. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 491–502, 2019. doi:10.1145/3313276.3316386.
https://doi.org/10.1145/3313276.3316386

[36] G. H. Low and I. L. Chuang. Optimal Hamiltonian simulation by quantum signal processing. Phys. Rev. Lett., 118:010501, 2017. doi:10.1103/PhysRevLett.118.010501.
https://doi.org/10.1103/PhysRevLett.118.010501

[37] G. H. Low and I. L. Chuang. Hamiltonian simulation by qubitization. Quantum, 3:163, Jul 2019. URL: http://dx.doi.org/10.22331/q-2019-07-12-163, doi:10.22331/q-2019-07-12-163.
https://doi.org/10.22331/q-2019-07-12-163

[38] G. H. Low and N. Wiebe. Hamiltonian simulation in the interaction picture. 2019. arXiv:1805.00675.
arXiv:1805.00675

[39] A. Ma, A. B. Magann, T.-S. Ho, and H. Rabitz. Optimal control of coupled quantum systems based on the first-order magnus expansion: Application to multiple dipole-dipole-coupled molecular rotors. Physical Review A, 102(1), Jul 2020. URL: http://dx.doi.org/10.1103/PhysRevA.102.013115, doi:10.1103/physreva.102.013115.
https://doi.org/10.1103/PhysRevA.102.013115

[40] W. Magnus. On the exponential solution of differential equations for a linear operator. Comm. Pure Appl. Math., 7:649–673, 1954. doi:10.1002/cpa.3160070404.
https://doi.org/10.1002/cpa.3160070404

[41] A. Martinez. An introduction to semiclassical and microlocal analysis, volume 994. Springer, 2002. doi:10.1007/978-1-4757-4495-8.
https://doi.org/10.1007/978-1-4757-4495-8

[42] J. Mizrahi, B. Neyenhuis, K. G. Johnson, W. C. Campbell, C. Senko, D. Hayes, and C. Monroe. Quantum control of qubits and atomic motion using ultrafast laser pulses. Applied Physics B, 114(1-2):45–61, Nov 2013. URL: http://dx.doi.org/10.1007/s00340-013-5717-6, doi:10.1007/s00340-013-5717-6.
https://doi.org/10.1007/s00340-013-5717-6

[43] M. A. Nielsen, M. R. Dowling, M. Gu, and A. C. Doherty. Optimal control, geometry, and quantum computing. Phys. Rev. A, 73(6):062323, 2006. doi:10.1103/PhysRevA.73.062323.
https://doi.org/10.1103/PhysRevA.73.062323

[44] D. Poulin, A. Qarry, R. Somma, and F. Verstraete. Quantum simulation of time-dependent Hamiltonians and the convenient illusion of Hilbert space. Phys. Rev. Lett., 106(17):170501, 2011. doi:10.1103/PhysRevLett.106.170501.
https://doi.org/10.1103/PhysRevLett.106.170501

[45] A. Rajput, A. Roggero, and N. Wiebe. Hybridized methods for quantum simulation in the interaction picture, 2021. arXiv:2109.03308.
arXiv:2109.03308

[46] B. Şahinoğlu and R. D. Somma. Hamiltonian simulation in the low energy subspace. 2020. arXiv:2006.02660.
https://doi.org/10.1038/s41534-021-00451-w
arXiv:2006.02660

[47] E. M. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, volume 3. Princeton Univ. Pr., 1993.

[48] E. M. Stein and R. Shakarchi. Functional analysis. Princeton University Press, 2011. doi:10.1515/9781400840557.
https://doi.org/10.1515/9781400840557

[49] Y. Su, D. W. Berry, N. Wiebe, N. Rubin, and R. Babbush. Fault-tolerant quantum simulations of chemistry in first quantization, 2021. arXiv:2105.12767, doi:10.1103/PRXQuantum.2.040332.
https://doi.org/10.1103/PRXQuantum.2.040332
arXiv:2105.12767

[50] M. Thalhammer. A fourth-order commutator-free exponential integrator for nonautonomous differential equations. SIAM Journal on Numerical Analysis, 44(2):851–864, 2006. URL: http://www.jstor.org/stable/40232777, doi:10.1137/05063042.
https://doi.org/10.1137/05063042
http://www.jstor.org/stable/40232777

[51] M. C. Tran, S.-K. Chu, Y. Su, A. M. Childs, and A. V. Gorshkov. Destructive error interference in product-formula lattice simulation. Phys. Rev. Lett., 124(22):220502, 2020. doi:10.1103/PhysRevLett.124.220502.
https://doi.org/10.1103/PhysRevLett.124.220502

[52] L. Wahlbin. Superconvergence in Galerkin finite element methods. Springer, 2006. doi:10.1007/BFb0096835.
https://doi.org/10.1007/BFb0096835

[53] D. Wecker, M. B. Hastings, N. Wiebe, B. K. Clark, C. Nayak, and M. Troyer. Solving strongly correlated electron models on a quantum computer. Phys. Rev. A, 92:062318, 2015. doi:10.1103/PhysRevA.92.062318.
https://doi.org/10.1103/PhysRevA.92.062318

[54] N. Wiebe, D. Berry, P. Høyer, and B. C. Sanders. Higher order decompositions of ordered operator exponentials. J. Phys. A, 43(6):065203, 2010. doi:10.1088/1751-8113/43/6/065203.
https://doi.org/10.1088/1751-8113/43/6/065203

[55] M. Zworski. Semiclassical Analysis. American Mathematical Society, 2012. doi:10.1090/gsm/138.
https://doi.org/10.1090/gsm/138

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