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Standard Model Physics and the Digital Quantum Revolution: Thoughts about the Interface

Natalie Klco, Alessandro Roggero, Martin J. Savage

arXiv:2107.04769v1quant-phhep-lathep-phnucl-th

TL;DR

The paper addresses how entanglement, complexity, and quantum simulation can advance nuclear and high-energy physics despite limited classical tractability and immature quantum hardware. It synthesizes entanglement-based diagnostics, organizational principles, and simulation strategies, concluding that these ideas can guide Standard Model studies and near-term quantum-simulation goals.

  • Problem

    Standard Model quantum many-body and field-theory problems involve non-local correlations and computational demands that challenge classical methods, while entanglement is difficult to characterize with a single criterion.

  • Method

    The article synthesizes perspectives on entanglement measures, quantum algorithms, state preparation, real-time probes, and mappings for simulating Standard Model systems.

  • Results

    The article identifies entanglement as a diagnostic and organizational tool that informs quantum simulation design, Standard Model structure, and near-term scientific objectives.

  • Takeaways & Limitations

    Understanding entanglement can guide theoretical frameworks, computational representations, and architectural codesign for quantum simulations of subatomic systems.

Abstract

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Advances in isolating, controlling and entangling quantum systems are transforming what was once a curious feature of quantum mechanics into a vehicle for disruptive scientific and technological progress. Pursuing the vision articulated by Feynman, a concerted effort across many areas of research and development is introducing prototypical digital quantum devices into the computing ecosystem available to domain scientists. Through interactions with these early quantum devices, the abstract vision of exploring classically-intractable quantum systems is evolving toward becoming a tangible reality. Beyond catalyzing these technological advances, entanglement is enabling parallel progress as a diagnostic for quantum correlations and as an organizational tool, both guiding improved understanding of quantum many-body systems and quantum field theories defining and emerging from the Standard Model. From the perspective of three domain science theorists, this article compiles thoughts about the interface on entanglement, complexity, and quantum simulation in an effort to contextualize recent NISQ-era progress with the scientific objectives of nuclear and high-energy physics.

I. INTRODUCTION

The article places quantum simulation of Standard Model systems within the broader quantum-information revolution, emphasizing entanglement as both a computational resource and an organizing concept. Early devices make these ideas experimentally tangible, while current hardware and complexity limits constrain near-term scientific applications.

  • Feynman’s vision motivated quantum simulation because many complex quantum systems exceed classical computational capabilities.
  • Current universal quantum devices operate at tens of qubits, lack intrinsic error correction and fault tolerance, and are not expected to deliver high-precision calculations at experimental scale during the NISQ era.Near-term motivations instead include real-time evolution, finite-density systems, and entangled observables that scale poorly classically.
  • For n qubits, the Hilbert-space dimension is d = 2^n, illustrating the exponential state growth relevant to many-body systems.For n = 299, the dimension is approximately 10^90.
  • Early digital quantum devices have enabled a first QFT simulation and calculations including the deuteron binding energy and light nuclei.
  • Quantum simulation targets Standard Model quantities while also connecting to communication, sensing, error correction, and general-purpose quantum computation.
  • Entanglement is treated as a diagnostic and organizational tool whose basis-dependent structure can guide quantum descriptions, simulation design, and architectural codesign.
  • Entanglement measures range from entropy and separability criteria to operational resource measures, but no single criterion fully characterizes complex quantum correlations.

B. The Role of Entanglement

The paper presents entanglement as a broad source of conceptual connections across quantum many-body systems, quantum field theories, and quantum simulation. Its structure can provide novel perspectives despite the difficulty of quantifying it.

  • Entanglement connects quantum many-body and quantum field theories with emerging approaches to quantum simulation.

1. ...as an organizational principle:

Entanglement organizes physical insight by linking symmetries, phases, operator hierarchies, and dynamical behavior across Standard Model systems. Examples range from scattering and nuclear symmetries to chiral symmetry breaking and tensor-network descriptions.

  • 1. ...as an organizational principle:: Tensor-network methods mitigate exponential demands by truncating states according to entanglement across spatial bipartitions.
  • 1. ...as an organizational principle:: Nuclear shell-model success partly reflects the near-product structure of valence-space and inert-core wavefunctions.
  • 1. ...as an organizational principle:: Maximally entangled helicity states in QED scattering motivated an entanglement-extremization perspective on Standard Model interactions.
  • 1. ...as an organizational principle:: In low-energy nuclear scattering, enhanced spin-flavor symmetries coincide with suppressed S-matrix entanglement power and may inform emergent-symmetry and EFT-operator hierarchies.
  • 1. ...as an organizational principle:: Entanglement measures provide non-local probes complementary to local observables for identifying phase transitions, symmetries, confinement, and topological states.
  • 2. ...as an order parameter for symmetry breaking:: Nucleon entanglement entropy tracks valence–parton-sea interactions, and entanglement functions as an order parameter for chiral symmetry breaking.
  • 2. ...as an order parameter for symmetry breaking:: Tensor networks have enabled early quantum simulations of collective neutrino oscillations by efficiently compressing states with low bipartite entanglement.

3. ...as insight into the structure of hadrons:

The paper connects entanglement to hadronic structure, gauge-theory correlations, statistical behavior, geometry, and quantum-field simulation design. These connections motivate both non-local physical descriptions and hardware-aware representations.

  • Effective descriptions in nuclear and high-energy physics often expand around bases that treat coherent quantum effects as perturbations to an effectively classical structure.
  • Color entanglement in non-Abelian gauge theories reflects coherent exchange of color degrees of freedom between gluons.
  • Spatial entanglement in protons can be studied across regions selected by deep-inelastic-scattering probes, with maximal entropy predicted at small Bjorken x.
  • Entanglement generation is linked to the local emergence of statistical mechanics even when the global quantum state remains pure.
  • Connections among entanglement, geometry, area laws, circuit complexity, and holographic ideas provide perspectives on quantum correlations and spacetime structure.
  • The S-matrix formulation uses entanglement power as a distance from non-entangling subspaces and as curvature of scattering trajectories, including non-local inelastic effects.
  • Momentum-space field representations can trivialize free-field vacuum entanglement, whereas position-space mappings expose local correlations in quantum hardware.
  • Classical tensor networks and reorganized circuits offer lower-cost ways to incorporate area-law, symmetry-embedded, or perturbative entanglement.

7. ...as a harbinger of computational complexity:

Entanglement does not by itself determine computational complexity: highly entangled stabilizer states can remain classically simulatable, while departures from stabilizer structure and sufficiently rapid entanglement growth matter for quantum computational complexity.

  • Stabilizer states provide a counterexample to equating entanglement directly with computational complexity.The formalism includes highly entangled states while retaining efficient classical simulation for suitable circuits.
  • Clifford circuits acting on stabilizer states remain efficiently classically simulatable, including GHZ-state preparation and quantum teleportation.The relevant gate set is {H, S, CNOT, X, Y, Z}.
  • T-gate count can characterize simulation complexity because non-Clifford operations mark departure from a classically efficient stabilizer strategy.
  • Tensor-network perspectives suggest entanglement must grow more rapidly than logarithmically with system size to support exponential computation.
  • QFT practitioners face a limited toolkit for systematically improvable calculations involving non-local quantities, motivating new intuition about non-locality and entanglement.The established toolkit is strongly based on locality and the operator product expansion.

III. THE VISION OF QUANTUM SIMULATION

The paper develops the vision of universal quantum simulation from ideal scaling arguments toward practical algorithms, resource estimates, hardware constraints, and codesigned implementations for Standard Model systems.

  • A. “Gedanken” Scaling of Quantum Resources: Universal quantum simulation is connected to intersimulatability: a universal device can efficiently simulate compatible systems within the same computational class.Feynman proposed quantum simulators built from quantum-mechanical degrees of freedom to simulate other compatible quantum systems.
  • A. “Gedanken” Scaling of Quantum Resources: For locally interacting systems with n degrees of freedom evolved for time T, the expected resource requirement is O(nT) operations.The corresponding device uses O(n) degrees of freedom and O(T) time steps.
  • A. “Gedanken” Scaling of Quantum Resources: Efficient simulation strategies have been formulated for quantum chemistry, fermion-lattice models, pionless EFT, and some relativistic QFTs.
  • B. Quantum Algorithm Development: “Gedanken” to Reality: Lie-Trotter, Suzuki, LCU, Taylor, Qubitization, and QSP methods trade approximation error, operation count, and qubit overhead in real-time simulation.The passage reports improved error scaling for LCU and Taylor methods and optimal scaling for Qubitization and QSP under stated conditions.
  • B. Quantum Algorithm Development: “Gedanken” to Reality: Finite gate fidelities create an inverse relationship between system size and feasible Trotter steps within the gate-fidelity coherence time.Quantum volume is introduced as a metric combining circuit size and reliable execution rather than using qubit count or error rate alone.
  • B. Quantum Algorithm Development: “Gedanken” to Reality: Resource-estimation efforts have reduced some naive requirements from billions of device-years to a handful of device-days.
  • B. Quantum Algorithm Development: “Gedanken” to Reality: Error-correction mappings can let one general-purpose device execute problems with different logical resource requirements, while NISQ applications generally use little or no error correction.Codesigned special-purpose architectures are presented as a complementary route to early quantum advantage.
  • B. Quantum Algorithm Development: “Gedanken” to Reality: Systematic uncertainty quantification requires ensembles of simulations spanning strategically chosen parameters for extrapolations, interpolations, observables, and uncertainties.

A. Asymptopia: The Utopia of Infinite Resources

The section frames quantum simulation through formal complexity classes and emphasizes that experimentally relevant Standard Model observables require comprehensive uncertainty quantification and careful measurement design.

  • Complexity classes characterize how computational resources scale with system size, with P containing polynomial-time classical problems and BQP bounded-error polynomial-time quantum problems.
  • NP and QMA contain important problems expected to require beyond-polynomial resources, despite polynomial resources being sufficient to verify solutions in NP.
  • Complexity-class membership describes worst-case instances and need not represent average-case or best-case performance.The passage uses Minesweeper to illustrate this distinction.
  • Connecting quantum simulations of Standard Model observables to experiment requires complete uncertainty quantification across classical and quantum sources.
  • Quantum simulations add digitization, field-truncation, state-preparation, measurement, Trotterization, and device-noise uncertainties to established theory and discretization uncertainties.The error decomposition explicitly includes these additional quantum-simulation sources.
  • Because wavefunction tomography scales exponentially with system size, efficient quantum simulation requires optimized observable estimators and attention to measurement-stage cancellations.

C. Reality: Hacking Bounded-Error Quantum Simulations

The paper argues that bounded-error quantum simulations can address formally difficult Standard Model problems by combining efficient quantum subproblems with perturbative approximations and lower-complexity theoretical inputs.

  • Reality: Hacking Bounded-Error Quantum Simulations: Complexity class membership does not by itself determine whether Standard Model observables can yield useful physical insight or motivate experiments.Some observables in P may still be impractical at required precision, while modest NP or QMA instances may remain tractable.
  • Reality: Hacking Bounded-Error Quantum Simulations: Bounded-error simulations may use a perturbatively close input theory when its induced error remains within the target total tolerance ϵ(t).This supports using effective field theories or phenomenological models for only the observables and precision required.
  • Reality: Hacking Bounded-Error Quantum Simulations: Lower-complexity leading-order interactions have enabled classical progress on problems formally beyond classical computing, motivating analogous strategies for quantum simulation.Examples include DFT, symmetry-assisted nuclear Monte Carlo, and other perturbative expansions.
  • Reality: Hacking Bounded-Error Quantum Simulations: Nuclear and lattice-QCD calculations span practical regimes from demonstrated smaller-nucleus computations to problems likely in QMA or beyond in PSPACE.Smaller-nucleus results can support bounded-error extraction of nuclear-EFT counterterms for somewhat larger nuclei.
  • Reality: Hacking Bounded-Error Quantum Simulations: Heavy-quark symmetry and factorization provide additional structured settings where leading-order descriptions organize higher-order corrections or make classical computation effective.For heavy-quark systems, 1/m_Q corrections can be included perturbatively; for high-momentum-transfer QCD, higher-order matrix elements are classically computable with required precision.
  • Reality: Hacking Bounded-Error Quantum Simulations: Perturbative problem decomposition places a large component inside BQP for efficient quantum execution while isolating terms outside BQP in a separately treated part.Soft-Collinear EFT is presented as a possible route for precision fragmentation simulations, though this remains to be demonstrated.

D. Bottom Line: Enhancing Leading Order Complexity

The paper presents entanglement and approximate symmetry as guides for reducing effective complexity, while emphasizing that near-term quantum reach depends on more than asymptotic scaling.

  • Bottom Line: Enhancing Leading Order Complexity: Approximate symmetries separate Standard Model Hamiltonians into large symmetry-preserving terms and small symmetry-breaking terms that can be treated perturbatively.The paper connects this separation to reduced entanglement and perturbatively close tensor-product states, while also noting leading-order entangled states.
  • Bottom Line: Enhancing Leading Order Complexity: Asymptotic resource scaling does not conclusively determine the reach of present-day or near-term quantum simulations.Formal scaling remains important, but it is not the sole criterion for selecting near-term Standard Model simulation goals.
  • Bottom Line: Enhancing Leading Order Complexity: The mapping from physical systems to quantum degrees of freedom affects state preparation, time evolution, and final measurement costs.Different representations can therefore support codesign between theoretical formulations, algorithms, and hardware.
  • Bottom Line: Enhancing Leading Order Complexity: Where to partition computational responsibility between quantum devices and classical resources is a central design question for quantum simulations.State preparation is one example where quantum devices may optimize interpolating operators for classical calculations.
  • Bottom Line: Enhancing Leading Order Complexity: Jordan-Wigner mapping assigns one qubit per fermionic mode but typically induces O(n)-qubit interactions for locally interacting systems in dimensions above one.Bravyi-Kitaev reduces this induced nonlocality to O(log(n)), while alternative mappings trade locality against other benefits.

2. Scalar Fields (The BQP-Complete “Gold Standard”)

Scalar-field mappings encode continuum fields into finite quantum registers while targeting recovery of continuum observables as discretization and digitization vanish.

  • 2. Scalar Fields (The BQP-Complete “Gold Standard”): Scalar-field mappings represent digitized field arguments in computational-basis states and wavefunction values in their amplitudes.The approach turns continuous field degrees of freedom into finite-dimensional quantum registers.
  • 2. Scalar Fields (The BQP-Complete “Gold Standard”): Continuum observables are recovered by taking the spacetime and field discretization errors toward zero.The paper emphasizes that the continuum limit is generally dependent on the chosen mapping.
  • 2. Scalar Fields (The BQP-Complete “Gold Standard”): n_Q ∼ log log(1/ϵ) qubits per site can yield exponentially improving precision in the digitized representation as the number of states increases.Spatial discretization errors remain and require n ∼ log(1/ϵ).
  • 2. Scalar Fields (The BQP-Complete “Gold Standard”): Position-space scalar mappings use on-site and nearest-neighbor interactions and require only a small number of unitary layers for each Trotterized time-evolution step.Alternative bases include digitized Hermite-polynomial states and truncated harmonic-oscillator eigenstates.
  • 2. Scalar Fields (The BQP-Complete “Gold Standard”): Gauge-theory simulation requires representing local symmetries and link couplings, with multiple bases and strategies developed for hardware-software codesign.Action-angle variables, discrete gauge-group subgroups, and irreducible-representation bases provide distinct mapping choices.
  • 2. Scalar Fields (The BQP-Complete “Gold Standard”): Integrating out gauge fields in one spatial dimension reduces qubit requirements by trading gauge-field registers for nonlocal fermionic interactions.This strategy was used in the first quantum simulation of the Schwinger model.
  • 2. Scalar Fields (The BQP-Complete “Gold Standard”): Field representations alter the gauge-invariant-to-gauge-variant Hilbert-space ratio and therefore the calculation’s sensitivity to device errors and noise.Representations can also differ in effective code distances and available error-mitigation strategies.

B. Quantum Fields for Quantum Information

Quantum fields and related lattice systems can organize quantum information through entanglement, topology, confinement, and protected logical degrees of freedom.

  • B. Quantum Fields for Quantum Information: Quantum simulation of fields is motivated by the expectation that atomic-scale and subatomic systems can exhibit commensurate quantum complexity.The paper connects this possibility to emergent properties whose resource scaling need not remain exponential in volume.
  • B. Quantum Fields for Quantum Information: Spin liquids connect QFTs, spin models, and the distribution and protection of quantum information.They provide a condensed-matter setting relevant to logical qubits and quantum memories.
  • B. Quantum Fields for Quantum Information: The Toric Code has a highly entangled, topologically ordered ground state supporting two logical qubits through Wilson-loop operators.Local errors below a threshold can be corrected using stabilizer measurements and decoders.
  • B. Quantum Fields for Quantum Information: Higher-dimensional SU(N) lattice gauge theories may localize qubit errors within hadrons through confinement, offering a mechanism distinct from the Toric Code.The paper contrasts this with unconfined electric charges and magnetic vortices in the Toric Code.
  • B. Quantum Fields for Quantum Information: Honeycomb spin systems realize gapped and gapless phases with Majorana-fermion descriptions and can develop chiral edge states under an applied magnetic field.These edge states are connected to topological structure and are used to simulate chiral fermions in lattice-gauge-theory calculations.

C. Preparing Wavefunctions: Ground States and Finite-Density

Ground-state preparation combines adiabatic, energy-filtering, and variational strategies, while real-time simulation supports static-property calculations and dynamical observables. Variational methods reduce quantum-operation demands and tolerate noise, but introduce difficult non-convex optimization and measurement costs.

  • Ground-state preparation: Real-time dynamics can support static-property studies, including ground-state energy density, through adiabatic preparation, energy filters, or variational approaches.These form the three main classes of state-preparation strategies discussed.
  • Ground-state preparation: QPE estimates Hamiltonian eigenvalues by simulating U(t) = exp(−itH), with energy-filter resolution δH = O(1/T).Ground-state measurement probability depends on the trial state’s overlap with the ground state.
  • Variational approaches: Variational algorithms construct parameterized circuits and optimize a variational principle to approximate the ground state.Their advantages include relatively few operations for simple parametrizations and partial robustness to device noise.
  • Variational approaches: VQA optimization is generally non-convex and NP-hard, so globally optimal parameters cannot be expected for all instances.This heuristic limitation accompanies the difficulty of classically simulating measurement statistics for sufficiently complex circuits.
  • Variational approaches: VQA can serve as a stepping stone for preparing high-accuracy trial states used by more sophisticated quantum algorithms.The paper compares this role with Variational Monte Carlo preceding more accurate, expensive GFMC calculations of nuclei.
  • Real-time probes: Reliable real-time information is difficult because sign problems affect QFT and quantum-many-body path integrals, especially for scattering-dominated Standard Model experiments.Classical Euclidean-to-real-time inversion is ill-posed because small correlator errors can produce large inversion errors.
  • Real-time probes: Frequency-domain projectors can avoid UV and IR Fourier-transform approximations while revealing response strength and semi-exclusive final-state information.The approach may support more complete scattering characterization, although resource estimates indicate further improvements are needed for near-term devices.

VI. STANDARD MODEL APPLICATIONS: SELECT IMPLEMENTATIONS ON DIGITAL QUANTUM HARDWARE

Digital quantum simulations have begun implementing Standard Model-motivated lattice gauge dynamics on small devices. Demonstrations include Schwinger-model pair production and early SU(2) and SU(3) Yang–Mills studies, while higher-dimensional gauge-field simulations remain unrealized on quantum hardware.

  • Scope and focus: The section focuses on energy scales between chemistry and roughly the TeV scale, excluding molecular, quantum-gravity, holography, baryogenesis, and other beyond-Standard-Model results.It emphasizes hardware implementations explicitly targeting the Standard Model.
  • Lattice gauge dynamics: A four-trapped-ion digital simulation represented fermionic degrees of freedom at two spatial sites of lattice 1+1-dimensional QED and showed dynamical e+e− pair generation.The experiment provided an early concrete demonstration of digital quantum simulation for microscopic physics.
  • Lattice gauge dynamics: The same small-lattice program included Schwinger-model vector-current and pair-production studies plus SU(2) and SU(3) Yang–Mills electric-energy and vacuum-persistence measurements.Figure 2 combines classical simulations, trapped-ion systems, superconducting quantum devices, and quantum annealing systems.
  • Lattice gauge dynamics: The trapped-ion experiment maintained greater than 70% survival in the physical zero-charge subspace through four Trotter steps despite approximately 50 gates per time-evolution step.This enabled error-mitigating post-selection while retaining sufficient statistics.
  • Model relevance and limitations: The Schwinger model shares features with QCD, including charge screening, a fermion condensate, composite bound states, and non-trivial θ-term topology.Its gauge field is nevertheless non-dynamical, limiting implications for Standard Model simulations.
  • Model relevance and limitations: Formal higher-dimensional QED simulation formalisms exist, but they had not yet been executed on a quantum device.Studies also address Gauss’s-law violations and evolution into gauge-variant hardware-Hilbert-space sectors.
  • Non-Abelian theories: Non-Abelian gauge-theory hardware implementations build on the Kogut–Susskind Hamiltonian and earlier proposals for implementing SU(N) Yang–Mills theories.Multiple formulations are available for hardware implementation.

B. Structure and Reactions of Nuclei

Quantum simulations have progressed from deuteron ground-state calculations to larger light-nucleus energies and prototype reaction observables. Frequency-space methods additionally target final-state information, but current device depth, encoding, and resource requirements constrain near-term nuclear scattering studies.

  • Nuclear structure: The first nuclear ground-state calculation used VQE for the deuteron on IBM and Rigetti superconducting quantum devices.Follow-on implementations used trapped ions, Gray-code encodings, and larger systems with A ≤4.
  • Nuclear structure: A quantum frequency processor enabled calculations for nuclei with A ≤4 in Hilbert spaces containing up to 68 states.The device represented each calculation with a single d-dimensional qudit manipulated using quantum-optics elements.
  • Nuclear structure: The quantum-frequency-processor encoding requires resources scaling exponentially with the number of orbitals, while photon loss limits achievable circuit depth.These constraints qualify the larger-system demonstrations.
  • Reaction dynamics: Frequency-space methods can expose possible nuclear final states, supporting more complete characterization of semi-exclusive scattering cross sections.This capability is connected to information requirements for long-baseline neutrino experiments such as DUNE.
  • Reaction dynamics: Figure 4 collects prototype observables including heavy-scalar decay Green’s functions, np ↔dγ transition and success probabilities, mitigated scattering probabilities, and neutron-neutron spectral density.The panels cover multiple reaction-dynamics calculations using quantum devices.
  • Limitations: Resource estimates could be reduced below O(10^6), but realistic nuclear-scattering studies remained out of reach for NISQ devices.The estimate concerns implementations using one- and two-qubit operations and improved integral kernels and real-time-evolution representations.
  • Reaction dynamics: Frequency-space reaction methods prepare a ground state, apply a generally non-unitary Hermitian vertex operator, and extract response information through frequency filtering.The vertex-operator step requires maintaining a normalized state proportional to Ô|Ψ0⟩.

C. Collective Neutrino Oscillations

Collective neutrino oscillations are difficult to solve exactly, motivating mean-field, MPS, and quantum-device approaches. These studies connect entanglement growth with flavor-evolution instabilities and enable simulations beyond small systems.

  • Physical setting: Dense neutrino systems can substantially modify flavor evolution through weak interactions with a background, especially in astrophysical settings.The two-flavor approximation maps the dynamics to an all-to-all Heisenberg model with interaction strength proportional to neutrino density.
  • Computational challenge: Exact real-time dynamics are generally out of reach for direct methods, so integrable limits and mean-field approximations are used for restricted or general conditions.Highly symmetric homogeneous cases admit Bethe-ansatz solutions, whereas more general conditions typically rely on mean-field simulations.
  • Representative studies: Recent demonstrations span MPS calculations for Nν = 96, quantum-device simulations for Nν = 4, and entanglement-entropy evolution for systems with Nν = 2−9.The highlighted works also extract flavor survival probabilities from Nν = 4 simulations.
  • Quantum-device implementation: A swap-network construction implements all-to-all interactions efficiently on devices with linear nearest-neighbor connectivity.The method addresses the limited connectivity of quantum devices when realizing pairwise neutrino interactions.
  • MPS simulations: For highly symmetric conditions, bipartite entanglement grows approximately as log(Nν), enabling controllable MPS simulations of more than a hundred neutrino amplitudes.The calculations begin from product states and use the slow entanglement growth to control the many-body approximation.
  • MPS simulations: MPS studies link instabilities in flavor evolution to a dynamical phase transition in the underlying spin model.This connection may provide a framework for identifying conditions that produce collective oscillations without relying on mean-field approximations.
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