Time-dependent Hamiltonian simulation with $L^1$-norm scaling

Dominic W. Berry; Andrew M. Childs; Yuan Su; Xin Wang; Nathan Wiebe

doi:10.22331/q-2020-04-20-254

Time-dependent Hamiltonian simulation with $L^1$-norm scaling

Dominic W. Berry¹, Andrew M. Childs^2,3, Yuan Su^2,3, Xin Wang^3,4, and Nathan Wiebe^5,6,7

¹Department of Physics and Astronomy, Macquarie University, Sydney, NSW 2109, Australia
²Department of Computer Science, University of Maryland, College Park, MD 20742, USA
³Institute for Advanced Computer Studies and Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD 20742, USA
⁴Institute for Quantum Computing, Baidu Research, Beijing 100193, China
⁵Department of Physics, University of Washington, Seattle, WA 98195, USA
⁶Pacific Northwest National Laboratory, Richland, WA 99354, USA
⁷Google Inc., Venice, CA 90291, USA

Published:	2020-04-20, volume 4, page 254
Eprint:	arXiv:1906.07115v2
Doi:	https://doi.org/10.22331/q-2020-04-20-254
Citation:	Quantum 4, 254 (2020).

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Abstract

The difficulty of simulating quantum dynamics depends on the norm of the Hamiltonian. When the Hamiltonian varies with time, the simulation complexity should only depend on this quantity instantaneously. We develop quantum simulation algorithms that exploit this intuition. For sparse Hamiltonian simulation, the gate complexity scales with the $L^1$ norm $\int_{0}^{t}\mathrm{d}\tau\lVert{H(\tau)}\rVert_{\max}$, whereas the best previous results scale with $t\max_{\tau\in[0,t]}\lVert{H(\tau)}\rVert_{\max}$. We also show analogous results for Hamiltonians that are linear combinations of unitaries. Our approaches thus provide an improvement over previous simulation algorithms that can be substantial when the Hamiltonian varies significantly. We introduce two new techniques: a classical sampler of time-dependent Hamiltonians and a rescaling principle for the Schrödinger equation. The rescaled Dyson-series algorithm is nearly optimal with respect to all parameters of interest, whereas the sampling-based approach is easier to realize for near-term simulation. These algorithms could potentially be applied to semi-classical simulations of scattering processes in quantum chemistry.

► BibTeX data

@article{Berry2020timedependent,
  doi = {10.22331/q-2020-04-20-254},
  url = {https://doi.org/10.22331/q-2020-04-20-254},
  title = {Time-dependent {H}amiltonian simulation with {$L^1$}-norm scaling},
  author = {Berry, Dominic W. and Childs, Andrew M. and Su, Yuan and Wang, Xin and Wiebe, Nathan},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {4},
  pages = {254},
  month = apr,
  year = {2020}
}

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