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arXiv:2401.01308 · Physical Review Research

Hamiltonian learning in quantum field theories

R.J. Ott, Torsten V. Zache, Maximilian Prüfer, Sebastian Erne, Mohammadamin Tajik, Hannes Pichler, Jörg Schmiedmayer, P. Zoller

Published Dec 16, 2024• 8 citations
arXivHTMLPDF

Abstract

Quantum field theories (QFTs) as relevant for condensed-matter or high-energy physics are formulated in continuous space and time, and typically emerge as effective low-energy descriptions. In atomic physics, an example is given by tunnel-coupled superfluids, which realize the paradigmatic sine-Gordon model, and can act as quantum simulators of continuous QFTs. To quantitatively characterize QFT simulators, or to discover the Hamiltonian governing the dynamics of a continuous many-body quantum system, we discuss Hamiltonian learning as a method to systematically extract the operator content and the coupling constants of Hamiltonians from experimental data. In contrast to Hamiltonian learning for lattice models with a given lattice scale, we learn QFT Hamiltonians on a resolution scale set by the experiment. Varying the resolution scale gives access to QFTs at different energy scales, and allows to learn a flow of Hamiltonians reminiscent of the renormalization group. Applying these techniques to available experimental data from a tunnel-coupled quantum gas experiment allows a definite distinction between a free quadratic theory from an interacting sine-Gordon model, as the underlying QFT description of the system. Published by the American Physical Society 2024

DOI
10.1103/physrevresearch.6.043284

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Quantum field theories (QFTs) as relevant for condensed-matter or high-energy physics are formulated in continuous space and time, and typically emerge as effective low-energy descriptions. In atomic physics, an example is given by tunnel-coupled superfluids, which realize the paradigmatic sine-Gordon model, and can act as quantum simulators of continuous QFTs. To quantitatively characterize QFT simulators, or to discover the Hamiltonian governing the dynamics of a continuous many-body quantum system, we discuss Hamiltonian learning as a method to systematically extract the operator content and the coupling constants of Hamiltonians from experimental data. In contrast to Hamiltonian learning for lattice models with a given lattice scale, we learn QFT Hamiltonians on a resolution scale set by the experiment. Varying the resolution scale gives access to QFTs at different energy scales, and allows to learn a flow of Hamiltonians reminiscent of the renormalization group. Applying these techniques to available experimental data from a tunnel-coupled quantum gas experiment allows a definite distinction between a free quadratic theory from an interacting sine-Gordon model, as the underlying QFT description of the system. Published by the American Physical Society 2024
We experimentally study a pair of tunnel-coupled one-dimensional atomic superfluids, which realize the quantum sine-Gordon/massive Thirring models relevant for a wide variety of disciplines from particle to condensed-matter physics. From measured interference patterns we extract phase correlation functions and analyze if, and under which conditions, the higher-order correlation functions factorize into lower ones. This allows us to characterize the essential features of the model solely from our experimental measurements, detecting the relevant quasiparticles, their interactions and the topologically distinct vacua. Our method provides comprehensive insights into a non-trivial quantum field theory and establishes a general method to analyze quantum many-body systems through experiments.
Abstract Introduced as a quantum extension of Maxwell's classical theory, quantum electrodynamic (QED) has been the first example of a quantum field theory (QFT). Eventually, QFT has become the framework for the discussion of all fundamental interactions at the microscopic scale except, possibly, gravity. More surprisingly, it has also provided a framework for the understanding of second order phase transitions in statistical mechanics. In fact, as hopefully this work illustrates, QFT is the natural framework for the discussion of most systems involving an infinite number of degrees of freedom with local couplings. These systems range from cold Bose gases at the condensation temperature (about ten nanokelvin) to conventional phase transitions (from a few degrees to several hundred) and high energy particle physics up to a TeV, altogether more than twenty orders of magnitude in the energy scale. Therefore, although excellent textbooks about QFT had already been published, I thought, many years ago, that it might not be completely worthless to present a work in which the strong formal relations between particle physics and the theory of critical phenomena are systematically emphasized. This option explains some of the choices made in the presentation. A formulation in terms of field integrals has been adopted to study the properties of QFT. The language of partition and correlation functions has been used throughout, even in applications of QFT to particle physics. Renormalization and renormalization group (RG) properties are systematically discussed. The notion of effective field theory (EFT) and the emergence of renormalizable theories are described. The consequences for fine-tuning and triviality issue are emphasized. This fifth edition has been updated and fully revised.
Quantum field theory is a powerful tool to describe the relevant physics governing complex quantum many-body systems.Here, we develop a general pathway to extract the irreducible building blocks of quantum field theoretical descriptions and its parameters purely from experimental data.This determination is accomplished by extracting the one-particle irreducible (1PI) correlation functions from which one can construct all physical observables.To match the capabilities of experimental techniques, our approach employs a formulation of quantum field theory based on equal-time correlation functions only.We illustrate the theoretical foundations of our procedure by applying it to the sine-Gordon model in thermal equilibrium and then demonstrate explicitly how to extract these quantities from an experiment where we quantum simulate the sine-Gordon model by two tunnel-coupled superfluids.We extract all 1PI correlation functions up to the 1PI four-point function (interaction vertex) and their variation with momentum, encoding the "running" of the couplings.The measured 1PI correlation functions are compared to the theoretical estimates, verifying our procedure.Our work opens new ways of addressing complex many-body questions emerging in a large variety of settings from fundamental science to practical quantum technology.
Learning the Hamiltonian of a many-body system from its dynamics is a central task in quantum science, yet the algorithms with the strongest provable guarantees assume some level of quantum control--fast, arbitrary single-qubit gates interleaved with time evolution, and measurements in arbitrary bases--that is beyond the capabilities of near-term analog quantum simulators. Motivated by analog atom- and ion-based platforms, we study Hamiltonian learning under minimal access models. Uniform state preparation and measurements: We first consider the setting where in every experiment, one can rotate each qubit to the same state, perform short-time evolution, and measure every qubit in the same basis. Surprisingly, we show that for generic 2-local Hamiltonians on any interaction graph, all of the parameters can be reconstructed from such experiments. Computational basis state preparation and measurements: We then consider a similarly constrained setting, but where state preparation and measurement are restricted to the computational basis. For nearest-neighbor Hamiltonians with only Pauli X/ZX/ZX/Z interactions, a class which captures contemporary Rydberg atom platforms, we show that over 1D and 2D rectangular lattices, all of the parameters can be reconstructed from such experiments up to unavoidable gauges. Our protocols introduce new techniques for solving structured polynomial systems over an extensive number of parameters. Taken together, our results suggest that one can learn a great deal from the dynamics of quantum many-body systems even under the most stringent experimental constraints.
There is a powerful mathematical method for simulating quantum dynamics that goes beyond finite qubit representations called infinite-dimensional Hamiltonian systems. These systems make it possible to describe quantum fields, open quantum systems, and hybrid designs more accurately by simulating unbounded operators, spectral decompositions, and continuous-variable interactions. Quantum algorithms for partial differential equations, many-body physics, and error-tolerant models of infinite Hilbert spaces are all areas where they are useful. Adding Hamiltonians with infinite dimensions to computer systems makes it possible to use both classical approximation and quantum hardware. This method makes it easier to scale up, improves accuracy when dynamics aren’t simple, and sets the stage for more advanced robust quantum computing models in chemistry, information sciences, and physics.