A Resource Efficient Approach For Quantum And Classical Simulations Of Gauge Theories In Particle Physics

Abstract

Gauge theories establish the standard model of particle physics, and lattice gauge theory (LGT) calculations employing Markov Chain Monte Carlo (MCMC) methods have been pivotal in our understanding of fundamental interactions. The present limitations of MCMC techniques may be overcome by Hamiltonian-based simulations on classical or quantum devices, which further provide the potential to address questions that lay beyond the capabilities of the current approaches. However, for continuous gauge groups, Hamiltonian-based formulations involve infinite-dimensional gauge degrees of freedom that can solely be handled by truncation. Current truncation schemes require dramatically increasing computational resources at small values of the bare couplings, where magnetic field effects become important. Such limitation precludes one from `taking the continuous limit’ while working with finite resources. To overcome this limitation, we provide a resource-efficient protocol to simulate LGTs with continuous gauge groups in the Hamiltonian formulation. Our new method allows for calculations at arbitrary values of the bare coupling and lattice spacing. The approach consists of the combination of a Hilbert space truncation with a regularization of the gauge group, which permits an efficient description of the magnetically-dominated regime. We focus here on Abelian gauge theories and use $2+1$ dimensional quantum electrodynamics as a benchmark example to demonstrate this efficient framework to achieve the continuum limit in LGTs. This possibility is a key requirement to make quantitative predictions at the field theory level and offers the long-term perspective to utilise quantum simulations to compute physically meaningful quantities in regimes that are precluded to quantum Monte Carlo.

Popular summary

Gauge theories are at the basis of the standard model, which describes fundamental particle interactions and thus the universe surrounding us. Besides other things, these theories tell us why particles bind together in nuclei to eventually form atoms and how they interact with each other. To mention a famous example where gauge theories have played a key role is the discovery of the Higgs boson which is responsible for the mass of the fundamental particles.
While the standard model, which is built on gauge theories,has already pushed the boundary of our knowledge, many open questions remain since their simulation is notoriously difficult in certain parameter regimes which are, however, important to understand physical phenomena such as the matter antimatter asymmetry of CP violation. Standard methods encounter problems that cannot or are extremely hardly be solved by using classical computing approaches — though quantum algorithms and hence quantum computers qualify as excellent candidates to drive another wave of innovations in particle physics.

In this work, we show how the gauge fields of a lattice gauge theory can be efficiently simulated. Generally, an exact description of the system requires an infinite amount of degrees of freedom, which practically cannot be taken into account. Therefore, any numerical method needs to apply a cutoff to these degrees of freedom, which in turn reduces the accuracy of the simulation. We provide a novel protocol leading to a cutoff, that allows for an efficient implementation of the problem. Importantly, it is applicable in both, quantum algorithms as well as classical simulation techniques. Furthermore, we provide the tools to estimate the error of the simulation if compared to the exact result from the untruncated theory. Our protocol is flexible and can be optimized for the available resources, for example the number of qubits in a quantum computer or the available memory in a classical machine.

As an application, we consider quantum electrodynamics in two spatial and one temporal dimensions as a benchmarking example. For intermediate values of the coupling – the most difficult regime – we reduce the required number of quantum states by one order of magnitude.
Our method will play an important role in the simulation of lattice gauge theories, whether they are quantum or classical and hence opens a path towards simulations of underlying models of the standard model of high energy physics, providing the possibility to address so far unanswered fundamental questions.

► BibTeX data

@article{Haase2021resourceefficient, doi = {10.22331/q-2021-02-04-393}, url = {https://doi.org/10.22331/q-2021-02-04-393}, title = {A resource efficient approach for quantum and classical simulations of gauge theories in particle physics}, author = {Haase, Jan F. and Dellantonio, Luca and Celi, Alessio and Paulson, Danny and Kan, Angus and Jansen, Karl and Muschik, Christine A.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{“{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {5}, pages = {393}, month = feb, year = {2021}<br /> }

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Cited by

[1] David B. Kaplan and Jesse R. Stryker, “Gauss’s law, duality, and the Hamiltonian formulation of U(1) lattice gauge theory”, Physical Review D 102 9, 094515 (2020).

[2] Danny Paulson, Luca Dellantonio, Jan F. Haase, Alessio Celi, Angus Kan, Andrew Jena, Christian Kokail, Rick van Bijnen, Karl Jansen, Peter Zoller, and Christine A. Muschik, “Towards simulating 2D effects in lattice gauge theories on a quantum computer”, arXiv:2008.09252.

[3] Zohreh Davoudi, Indrakshi Raychowdhury, and Andrew Shaw, “Search for Efficient Formulations for Hamiltonian Simulation of non-Abelian Lattice Gauge Theories”, arXiv:2009.11802.

[4] Julian Bender and Erez Zohar, “A gauge redundancy-free formulation of compact QED with dynamical matter for quantum and classical computations”, arXiv:2008.01349.

[5] Wibe A. de Jong, Mekena Metcalf, James Mulligan, Mateusz Płoskoń, Felix Ringer, and Xiaojun Yao, “Quantum simulation of open quantum systems in heavy-ion collisions”, arXiv:2010.03571.

[6] Arata Yamamoto, “Real-time simulation of (2+1)-dimensional lattice gauge theory on qubits”, arXiv:2008.11395.

[7] Ryan R. Ferguson, Luca Dellantonio, Karl Jansen, Abdulrahim Al Balushi, Wolfgang Dür, and Christine A. Muschik, “A measurement-based variational quantum eigensolver”, arXiv:2010.13940.

[8] Giuseppe Magnifico, Timo Felser, Pietro Silvi, and Simone Montangero, “Lattice Quantum Electrodynamics in (3+1)-dimensions at finite density with Tensor Networks”, arXiv:2011.10658.

[9] Erik J. Gustafson and Henry Lamm, “Toward Quantum Simulations of $mathbb{Z}_2$ Gauge Theory Without State Preparation”, arXiv:2011.11677.

[10] Lena Funcke, Tobias Hartung, Karl Jansen, Stefan Kühn, and Paolo Stornati, “Dimensional Expressivity Analysis of Parametric Quantum Circuits”, arXiv:2011.03532.

[11] Pierpaolo Fontana, Joao C. Pinto Barros, and Andrea Trombettoni, “Reformulation of gauge theories in terms of gauge invariant fields”, arXiv:2008.12973.

[12] Jens Nyhegn, Chia-Min Chung, and Michele Burrello, “$mathbb{Z}_N$ lattice gauge theory in a ladder geometry”, arXiv:2011.06534.

[13] Robert Ott, Torsten V. Zache, Fred Jendrzejewski, and Jürgen Berges, “Scalable cold-atom quantum simulator for two-dimensional QED”, arXiv:2012.10432.

[14] Anthony Ciavarella, Natalie Klco, and Martin J. Savage, “A Trailhead for Quantum Simulation of SU(3) Yang-Mills Lattice Gauge Theory in the Local Multiplet Basis”, arXiv:2101.10227.

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A resource efficient approach for quantum and classical simulations of gauge theories in particle physics

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