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

Jan F. Haase1,2, Luca Dellantonio1,2, Alessio Celi3,4, Danny Paulson1,2, Angus Kan1,2, Karl Jansen5, and Christine A. Muschik1,2,6

1Department of Physics & Astronomy, University of Waterloo, Waterloo, ON, Canada, N2L 3G1
2Institute for Quantum Computing, University of Waterloo, Waterloo, ON, Canada, N2L 3G1
3Departament de Física, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain
4Center for Quantum Physics, Faculty of Mathematics, Computer Science and Physics, University of Innsbruck, Innsbruck A-6020, Austria
5NIC, DESY, Platanenallee 6, D-15738 Zeuthen, Germany
6Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada, N2L 2Y5

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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.

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.

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[5] Angus Kan, Lena Funcke, Stefan Kühn, Luca Dellantonio, Jinglei Zhang, Jan F. Haase, Christine A. Muschik, and Karl Jansen, "Investigating a (3+1)D topological θ -term in the Hamiltonian formulation of lattice gauge theories for quantum and classical simulations", Physical Review D 104 3, 034504 (2021).

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