Simulating Effective QED on Quantum Computers

Torin F. Stetina1,2, Anthony Ciavarella3,4, Xiaosong Li1, and Nathan Wiebe4,5,6,7

1Department of Chemistry, University of Washington, Seattle, Washington, USA
2Simons Institute for the Theory of Computation, University of California, Berkeley, California, USA
3Institute for Nuclear Theory, University of Washington, Seattle, Washington, USA
4Department of Physics, University of Washington, Seattle, Washington, USA
5Pacific Northwest National Laboratory, Richland, Washington, USA
6Department of Computer Science, University of Toronto, Toronto, Ontario, Canada
7Challenge Institute for Quantum Computation, University of Washington, Seattle, Washington, USA

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In recent years simulations of chemistry and condensed materials has emerged as one of the preeminent applications of quantum computing, offering an exponential speedup for the solution of the electronic structure for certain strongly correlated electronic systems. To date, most treatments have ignored the question of whether relativistic effects, which are described most generally by quantum electrodynamics (QED), can also be simulated on a quantum computer in polynomial time. Here we show that effective QED, which is equivalent to QED to second order in perturbation theory, can be simulated in polynomial time under reasonable assumptions while properly treating all four components of the wavefunction of the fermionic field. In particular, we provide a detailed analysis of such simulations in position and momentum basis using Trotter-Suzuki formulas. We find that the number of $T$-gates needed to perform such simulations on a $3D$ lattice of $n_s$ sites scales at worst as $O(n_s^3/\epsilon)^{1+o(1)}$ in the thermodynamic limit for position basis simulations and $O(n_s^{4+2/3}/\epsilon)^{1+o(1)}$ in momentum basis. We also find that qubitization scales slightly better with a worst case scaling of $\widetilde{O}(n_s^{2+2/3}/\epsilon)$ for lattice eQED and complications in the prepare circuit leads to a slightly worse scaling in momentum basis of $\widetilde{O}(n_s^{5+2/3}/\epsilon)$. We further provide concrete gate counts for simulating a relativistic version of the uniform electron gas that show challenging problems can be simulated using fewer than $10^{13}$ non-Clifford operations and also provide a detailed discussion of how to prepare multi-reference configuration interaction states in effective QED which can provide a reasonable initial guess for the ground state. Finally, we estimate the planewave cutoffs needed to accurately simulate heavy elements such as gold.

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[1] Heather M. Gray and Koji Terashi, "Quantum Computing Applications in Future Colliders", Frontiers in Physics 10, 864823 (2022).

[2] 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, PRX Quantum 3 2, 020324 (2022).

[3] Arpan Bhattacharyya, Lata Kh. Joshi, and Bhuvanesh Sundar, "Quantum information scrambling: from holography to quantum simulators", The European Physical Journal C 82 5, 458 (2022).

[4] Christian W. Bauer, Zohreh Davoudi, A. Baha Balantekin, Tanmoy Bhattacharya, Marcela Carena, Wibe A. de Jong, Patrick Draper, Aida El-Khadra, Nate Gemelke, Masanori Hanada, Dmitri Kharzeev, Henry Lamm, Ying-Ying Li, Junyu Liu, Mikhail Lukin, Yannick Meurice, Christopher Monroe, Benjamin Nachman, Guido Pagano, John Preskill, Enrico Rinaldi, Alessandro Roggero, David I. Santiago, Martin J. Savage, Irfan Siddiqi, George Siopsis, David Van Zanten, Nathan Wiebe, Yukari Yamauchi, Kübra Yeter-Aydeniz, and Silvia Zorzetti, "Quantum Simulation for High Energy Physics", arXiv:2204.03381.

[5] Christian W. Bauer, Benjamin Nachman, and Marat Freytsis, "Simulating Collider Physics on Quantum Computers Using Effective Field Theories", Physical Review Letters 127 21, 212001 (2021).

[6] Jue Xu, "On Lagrangian Formalism of Quantum Computation", arXiv:2112.04892.

[7] Bin Xu and Wei Xue, "3+1 Dimension Schwinger Pair Production with Quantum Computers", arXiv:2112.06863.

The above citations are from Crossref's cited-by service (last updated successfully 2022-05-29 05:41:31) and SAO/NASA ADS (last updated successfully 2022-05-29 05:41:32). The list may be incomplete as not all publishers provide suitable and complete citation data.