Provably Accurate Simulation Of Gauge Theories And Bosonic Systems

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Yu Tong^1,2, Victor V. Albert³, Jarrod R. McClean¹, John Preskill^4,5, and Yuan Su^1,4

¹Google Quantum AI, Venice, CA, USA
²Department of Mathematics, University of California, Berkeley, CA, USA
³Joint Center for Quantum Information and Computer Science, NIST and University of Maryland, College Park, MD, USA
⁴Institute for Quantum Information and Matter, Caltech, Pasadena, CA, USA
⁵AWS Center for Quantum Computing, Pasadena, CA, USA

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Abstract

Quantum many-body systems involving bosonic modes or gauge fields have infinite-dimensional local Hilbert spaces which must be truncated to perform simulations of real-time dynamics on classical or quantum computers. To analyze the truncation error, we develop methods for bounding the rate of growth of local quantum numbers such as the occupation number of a mode at a lattice site, or the electric field at a lattice link. Our approach applies to various models of bosons interacting with spins or fermions, and also to both abelian and non-abelian gauge theories. We show that if states in these models are truncated by imposing an upper limit $Lambda$ on each local quantum number, and if the initial state has low local quantum numbers, then an error at most $epsilon$ can be achieved by choosing $Lambda$ to scale polylogarithmically with $epsilon^{-1}$, an exponential improvement over previous bounds based on energy conservation. For the Hubbard-Holstein model, we numerically compute a bound on $Lambda$ that achieves accuracy $epsilon$, obtaining significantly improved estimates in various parameter regimes. We also establish a criterion for truncating the Hamiltonian with a provable guarantee on the accuracy of time evolution. Building on that result, we formulate quantum algorithms for dynamical simulation of lattice gauge theories and of models with bosonic modes; the gate complexity depends almost linearly on spacetime volume in the former case, and almost quadratically on time in the latter case. We establish a lower bound showing that there are systems involving bosons for which this quadratic scaling with time cannot be improved. By applying our result on the truncation error in time evolution, we also prove that spectrally isolated energy eigenstates can be approximated with accuracy $epsilon$ by truncating local quantum numbers at $Lambda=textrm{polylog}(epsilon^{-1})$.

Featured image: A lattice with U(1) gauge field

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► BibTeX data

@article{Tong2022provablyaccurate, doi = {10.22331/q-2022-09-22-816}, url = {https://doi.org/10.22331/q-2022-09-22-816}, title = {Provably accurate simulation of gauge theories and bosonic systems}, author = {Tong, Yu and Albert, Victor V. and McClean, Jarrod R. and Preskill, John and Su, Yuan}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{“{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {6}, pages = {816}, month = sep, year = {2022} }

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[2] Angus Kan and Yunseong Nam, “Lattice Quantum Chromodynamics and Electrodynamics on a Universal Quantum Computer”, arXiv:2107.12769.

[3] Anthony N. Ciavarella and Ivan A. Chernyshev, “Preparation of the SU(3) lattice Yang-Mills vacuum with variational quantum methods”, Physical Review D 105 7, 074504 (2022).

[4] Travis S. Humble, Andrea Delgado, Raphael Pooser, Christopher Seck, Ryan Bennink, Vicente Leyton-Ortega, C. -C. Joseph Wang, Eugene Dumitrescu, Titus Morris, Kathleen Hamilton, Dmitry Lyakh, Prasanna Date, Yan Wang, Nicholas A. Peters, Katherine J. Evans, Marcel Demarteau, Alex McCaskey, Thien Nguyen, Susan Clark, Melissa Reville, Alberto Di Meglio, Michele Grossi, Sofia Vallecorsa, Kerstin Borras, Karl Jansen, and Dirk Krücker, “Snowmass White Paper: Quantum Computing Systems and Software for High-energy Physics Research”, arXiv:2203.07091.

[5] Andrei Alexandru, Paulo F. Bedaque, Ruairí Brett, and Henry Lamm, “Spectrum of digitized QCD: Glueballs in a S (1080 ) gauge theory”, Physical Review D 105 11, 114508 (2022).

[6] A. Kan, L. Funcke, S. Kühn, L. Dellantonio, J. Zhang, J. F. Haase, C. A. Muschik, and K. Jansen, “3+1D theta-Term on the Lattice from the Hamiltonian Perspective”, The 38th International Symposium on Lattice Field Theory 112 (2022).

[7] Marius Lemm and Oliver Siebert, “Thermal Area Law for the Bose-Hubbard Model”, arXiv:2207.07760.

[8] Nhung H. Nguyen, Minh C. Tran, Yingyue Zhu, Alaina M. Green, C. Huerta Alderete, Zohreh Davoudi, and Norbert M. Linke, “Digital Quantum Simulation of the Schwinger Model and Symmetry Protection with Trapped Ions”, arXiv:2112.14262.

[9] Tomotaka Kuwahara, Tan Van Vu, and Keiji Saito, “Optimal light cone and digital quantum simulation of interacting bosons”, arXiv:2206.14736.

[10] Abhishek Rajput, Alessandro Roggero, and Nathan Wiebe, “Quantum Error Correction with Gauge Symmetries”, arXiv:2112.05186.

[11] Jiayu Shen, Di Luo, Chenxi Huang, Bryan K. Clark, Aida X. El-Khadra, Bryce Gadway, and Patrick Draper, “Simulating quantum mechanics with a θ -term and an ‘t Hooft anomaly on a synthetic dimension”, Physical Review D 105 7, 074505 (2022).

[12] Manu Mathur and Atul Rathor, “SU (N ) toric code and non-Abelian anyons”, Physical Review A 105 5, 052423 (2022).

[13] Ulysse Chabaud and Saeed Mehraban, “Holomorphic Quantum Computing”, arXiv:2111.00117.

[14] Yao Ji, Henry Lamm, and Shuchen Zhu, “Gluon Digitization via Character Expansion for Quantum Computers”, arXiv:2203.02330.

[15] Nilin Abrahamsen, Yuan Su, Yu Tong, and Nathan Wiebe, “Entanglement area law for 1D gauge theories and bosonic systems”, arXiv:2203.16012.

[16] Yonah Borns-Weil and Di Fang, “Uniform observable error bounds of Trotter formulae for the semiclassical Schrödinger equation”, arXiv:2208.07957.

The above citations are from SAO/NASA ADS (last updated successfully 2022-09-22 15:23:23). The list may be incomplete as not all publishers provide suitable and complete citation data.

Could not fetch Crossref cited-by data during last attempt 2022-09-22 15:23:21: Could not fetch cited-by data for 10.22331/q-2022-09-22-816 from Crossref. This is normal if the DOI was registered recently.

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: September 22, 2022September 22, 2022

Time Stamp: Dec 7, 2022

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Time Stamp: Aug 2, 2022

Provably accurate simulation of gauge theories and bosonic systems

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