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Quantum Computing for High-Energy Physics: State of the Art and Challenges. Summary of the QC4HEP Working Group

Alberto Di Meglio, Karl Jansen, Ivano Tavernelli, Constantia Alexandrou, Srinivasan Arunachalam, Christian W. Bauer, Kerstin Borras, Stefano Carrazza, Arianna Crippa, Vincent Croft, Roland de Putter, Andrea Delgado, Vedran Dunjko, Daniel J. Egger, Elias Fernandez-Combarro, Elina Fuchs, Lena Funcke, Daniel Gonzalez-Cuadra, Michele Grossi, Jad C. Halimeh, Zoe Holmes, Stefan Kuhn, Denis Lacroix, Randy Lewis, Donatella Lucchesi, Miriam Lucio Martinez, Federico Meloni, Antonio Mezzacapo, Simone Montangero, Lento Nagano, Voica Radescu, Enrique Rico Ortega, Alessandro Roggero, Julian Schuhmacher, Joao Seixas, Pietro Silvi, Panagiotis Spentzouris, Francesco Tacchino, Kristan Temme, Koji Terashi, Jordi Tura, Cenk Tuysuz, Sofia Vallecorsa, Uwe-Jens Wiese, Shinjae Yoo, Jinglei Zhang

arXiv:2307.03236v1quant-phhep-exhep-lathep-th

TL;DR

The paper addresses how quantum computing could tackle HEP problems that are difficult or inaccessible to classical methods, including scattering, dense matter, and quantum-state analysis. It synthesizes theoretical and experimental use cases, candidate algorithms, resource estimates, and near-term hardware considerations. The supported outlook is promising but bounded by algorithmic limitations and the need for effective error mitigation on noisy devices.

  • Problem

    HEP includes real-time scattering, dense-matter, and phase-transition problems that classical methods cannot fully access, while experimental analyses involve complex quantum-correlated data.

  • Method

    The paper develops a QC4HEP roadmap by surveying theoretical and experimental applications, quantum algorithms, resource estimates, and error-mitigation requirements.

  • Results

    The paper identifies quantum dynamics, lower-dimensional lattice models, quantum-data learning, and experimental HEP applications as promising near-term targets for quantum computing.

  • Takeaways & Limitations

    Lower-dimensional quantum simulations can help develop methods for higher-dimensional HEP systems and may enable non-perturbative studies of scattering and unexplored QCD phases.

  • Takeaways & Limitations

    Variational and Trotter-based approaches face barren plateaus, accumulated gate errors, and an optimal finite discretization that limits achievable accuracy on noisy devices.

Abstract

from arXiv · show

Quantum computers offer an intriguing path for a paradigmatic change of computing in the natural sciences and beyond, with the potential for achieving a so-called quantum advantage, namely a significant (in some cases exponential) speed-up of numerical simulations. The rapid development of hardware devices with various realizations of qubits enables the execution of small scale but representative applications on quantum computers. In particular, the high-energy physics community plays a pivotal role in accessing the power of quantum computing, since the field is a driving source for challenging computational problems. This concerns, on the theoretical side, the exploration of models which are very hard or even impossible to address with classical techniques and, on the experimental side, the enormous data challenge of newly emerging experiments, such as the upgrade of the Large Hadron Collider. In this roadmap paper, led by CERN, DESY and IBM, we provide the status of high-energy physics quantum computations and give examples for theoretical and experimental target benchmark applications, which can be addressed in the near future. Having the IBM 100 x 100 challenge in mind, where possible, we also provide resource estimates for the examples given using error mitigated quantum computing.

I. INTRODUCTION

The paper maps near-term quantum-computing opportunities in theoretical modelling and experimental HEP, motivated by classical limits in real-time, dense-matter, and large-scale data problems. It emphasizes representative low-dimensional models, quantum algorithms, and connections between theoretical and experimental applications.

  • Roadmap scope: The QC4HEP Working Group develops a common roadmap for HEP problems where noisy, near-term quantum devices may provide significant impact.The roadmap targets physically relevant use cases and potential demonstrations associated with IBM’s 100 ⊗100 challenge.
  • Theoretical challenges: Classical lattice simulations remain unable to access out-of-equilibrium dynamics and high-fermion-density nuclear matter with standard Euclidean Monte Carlo techniques.Examples include particle collisions, thermalization, quenches, neutron stars, and the early universe.
  • Theoretical challenges: Hamiltonian formulations avoid the sign problem but require storing many-particle wave functions whose memory scales exponentially with lattice size.Tensor-network methods address this by compactly representing physically relevant low-energy subspaces.
  • Quantum approaches: Quantum computers encode lattice degrees of freedom in qubits, with qubit requirements growing linearly with lattice sites, and support polynomial-time real-time algorithms for selected theories.The Hamiltonian formulation also allows quantum computations to avoid the sign problem.
  • Theoretical roadmap: The theoretical program progresses from (1+1)D to (2+1)D and ultimately (3+1)D systems while preserving key features of higher-dimensional HEP models.Lower-dimensional examples such as (2+1)D QED retain phenomena including asymptotic freedom and confinement.
  • Experimental challenges: Experimental HEP workloads span detector operation, event reconstruction and identification, and simulation and inference over large, structured datasets.These workloads require efficient, robust, and unbiased processing while identifying signals, physics structures, and possible anomalies.
  • Experimental challenges: Quantum computing is relevant to HEP experiments through potential speed-ups, sensitivity to correlations, and increased expressivity for data-processing algorithms.These characteristics align with faster processing, correlation-rich reconstruction, and expressive analysis models.

II. IBM ROADMAP ON QUANTUM COMPUTING

The IBM roadmap frames near-term quantum computing as a progression toward larger, more reliable devices, while noise and error correction remain decisive constraints. The paper connects this hardware trajectory to shallow-circuit, error-mitigated HEP applications and the 100 ⊗100 challenge.

  • Hardware requirements: Useful HEP quantum computing requires hardware and software capable of running algorithms at scale with reliable components, long coherence times, and high-quality gate parameters.The IBM roadmap presents progressively improved devices as stepping stones toward this goal.
  • Noise and fault tolerance: Current fault-tolerant quantum computing remains impractical because error-correcting codes require system sizes several orders of magnitude beyond available hardware.Complex, high-depth algorithms and long-time quantum-dynamics simulations are expected to require error correction.
  • Hardware progress: An order-of-magnitude improvement in two-qubit gate fidelities accompanies the construction of larger quantum devices.The paper presents this as evidence of substantial hardware progress.
  • Near-term devices: A 65-qubit Hummingbird processor could run circuits with a few thousand gates without error correction when two-qubit gate fidelities reach 99.99%.Such circuits may no longer be exactly simulable on a classical computer.
  • Error mitigation: Noise and decoherence bias shallow-circuit expectation values, motivating error-mitigation methods such as zero-noise extrapolation.These methods modify circuits and combine measurement outcomes to estimate noise-free values, adding computational and sampling overhead.
  • 100 ⊗100 Challenge: IBM’s 100 ⊗100 Challenge targets unbiased observables from 100-qubit circuits with depth 100 within a reasonable runtime.The challenge invites algorithms for problems notoriously hard for classical computers.
  • HEP applications: HEP is positioned to contribute relevant early demonstrations because it supplies computationally challenging problems and problem-relevant heuristics.The roadmap’s selected applications include theoretical and experimental domains, with real-time simulations among the motivating use cases.

1. Simulations of Real-Time Phenomena

Real-time and scattering simulations remain difficult for classical methods, while Hamiltonian quantum simulations offer a route to gauge-theory dynamics and direct event distributions. Key obstacles include resource-efficient encoding, state preparation, and hardware-scale implementation.

  • Motivation: Classical Monte Carlo methods are hindered by the sign problem when modeling non-perturbative, out-of-equilibrium scattering dynamics.This limits direct predictions of scattering-event distributions, especially in the non-perturbative QCD sector.
  • Quantum simulation strategies: Hamiltonian quantum simulations can represent real-time gauge-theory evolution while avoiding the sign problem inherent in Euclidean path-integral methods.Both analog and digital strategies evolve qubits or qudits encoding the many-body quantum-field state.
  • Experimental connection: Direct non-perturbative simulations could produce product statistics that are compared immediately with collision-event statistics measured in high-energy experiments.This provides a direct connection between simulated scattering outcomes and observed laboratory data.
  • State preparation: Preparing localized composite-particle wave packets remains an additional conceptual challenge because their internal wave functions can be highly entangled and complex.The general optimal preparation strategy is still unclear and requires further investigation.
  • Experimental progress: A 71-site U(1) quantum-link simulator used linear gauge protection to keep gauge violations below 10% throughout Coleman’s phase-transition dynamics.The platform was subsequently used to study thermalization and quantum many-body scarring, with extensions proposed for higher dimensions and larger spin representations.
  • Encoding and resources: Resource-efficient mappings, discretizations, and truncation schemes are needed to encode bosonic gauge degrees of freedom on quantum computers.Multiple ansätze must be tested against one another to identify their advantages and shortcomings.

3. (2+1)D SU(2)

The (2+1)D SU(2) program targets non-Abelian gauge theories as a step toward quantum chromodynamics, requiring explicit treatment of both fermionic and gauge degrees of freedom. Existing formulations and hardware studies provide early guidance, while larger computations must follow hardware progress.

  • Motivation: SU(2) Yang–Mills theory is identified as a natural first step toward the long-term goal of studying quantum chromodynamics.The standard Kogut–Susskind Hamiltonian formulation provides the starting point for the lattice model.
  • Hamiltonian ingredients: The SU(2) Hamiltonian uses fundamental-representation indices α, β and adjoint indices b, with chromoelectric and chromomagnetic fields governed by fermion mass m and coupling g.The chromomagnetic field arises from the plaquette term in the Hamiltonian.
  • Complexity: Lattice gauge theories can be simulated with a polynomial number of gates in lattice size, bosonic-field truncation, and simulation time.This establishes a general scalability result for universal quantum-computer implementations.
  • Quantum-simulator implementations: Quantum simulators realize non-Abelian gauge models by embedding gauge invariance in underlying symmetries such as angular-momentum or nuclear-spin conservation.Quantum-link formulations have been proposed for Rydberg-based architectures and superconducting circuits.
  • Model formulation: Two-dimensional SU(2) studies require both fermion and gauge-field degrees of freedom, unlike the one-dimensional case where gauge fields can be rewritten as long-range fermion interactions.Several formulations have been proposed, and practical comparisons are needed to understand their advantages and disadvantages.
  • Outlook: Upcoming computations can expand in scale and scope alongside continuing deployment of quantum hardware in the noisy intermediate-scale quantum era.

4. Quantum Link Models and D-Theory

Quantum-link models replace continuous classical gauge fields with discrete quantum degrees of freedom while preserving exact gauge symmetry. Their finite-dimensional representations enable resource-efficient qubit encodings for Abelian and non-Abelian lattice gauge theories.

  • D-theory: D-theory replaces continuous classical fields with discrete quantum degrees of freedom and realizes lattice gauge theories through quantum-link models.The formulation includes dimensional reduction from an extra dimension of short extent.
  • Quantum-link formulation: Quantum links are generalized quantum spins on lattice links that carry an exact gauge symmetry.They provide the link variables of the quantum-link formulation.
  • Representations: Quantum-link representations are finite-dimensional, contrasting with the infinite-dimensional link representations used in standard Wilson-type lattice gauge theory.
  • Embedding algebras: U(N) and SU(N) quantum links use SU(2N) embedding algebras, while U(1) models use ordinary SU(2) quantum spins.SO(N) and Sp(N) models similarly use SO(2N) and Sp(2N) embedding algebras.
  • Abelian implementation: The U(1) quantum-link model can be embodied by individual qubits and mapped efficiently onto a triangular lattice aligned with IBM’s 127-qubit Eagle topology.
  • Non-Abelian implementation: The simplest non-Abelian SU(2) quantum-link model uses Sp(2) = SO(5), with a four-dimensional link representation encoded by two qubits or by dual Z(2)-valued height variables.The height-variable formulation can be embodied by individual qubits.

5. (1+1)D CP(N −1) Models from (2 + 1)D SU(N) Quantum Spin Ladders

The section presents D-theory SU(N) quantum spin ladders as a regularization of (1+1)D CP(N−1) models, enabling quantum-simulation studies of dynamics difficult for classical methods.

  • D-theory replaces classical CP(N−1) fields with discrete SU(N) quantum spins while preserving exact continuous symmetries, including gauge symmetry.
  • The spin ladder uses fundamental and anti-fundamental SU(N) representations on alternating sites of a two-dimensional square lattice.
  • The ladder regularizes (1+1)D CP(N−1), with even or odd short-dimension extent producing vacuum angles θ = 0 or θ = π.
  • For moderately large L′/a ≳4, dimensional reduction yields the (1+1)D CP(N−1) model because the correlation length exceeds the short lattice extent.
  • Analog SU(N) quantum-antiferromagnet simulators using ultracold alkaline-earth atoms could access real-time dynamics, including false-vacuum decay in the CP(2) model.

6. Collective Neutrino Oscillations

The section models collective neutrino flavor oscillations as interacting two-level systems, emphasizing realistic many-body settings and the resource limits of current quantum implementations.

  • Dense neutrino clouds form strongly coupled many-body systems because two-body neutrino interactions substantially affect flavor oscillations.
  • A two-flavor approximation represents each neutrino as an interacting two-level system governed by vacuum, electron-background, and neutrino-neutrino terms.
  • The neutrino-neutrino coupling µ = 2GF nν scales with local neutrino density, while angular factors encode propagation geometry and suppress collinear interactions.
  • Spatially varying environments can be modeled by allowing λe and µ to change with distance from emission or elapsed time.
  • Current digital implementations remain restricted to small neutrino numbers and simple initial states, while larger systems, correlations, and longer evolution remain challenges.A first-order product-formula step costs 3N(N−1)/2 CNOT operations, and a second-order step costs 3(N^2−3N/2+1).

B. Selected Applications for Experiments

The section surveys quantum and classical approaches for experimental HEP data processing, spanning data reduction, classification, anomaly detection, and uncertainty-aware analysis. Current quantum prototypes cover many workflow tasks at reduced scale, but realistic data sizes and trainability remain limiting factors.

  • Experimental HEP requires scalable algorithms for highly structured datasets, motivating quantum approaches alongside distributed computing, artificial intelligence, and new hardware.
  • 1. Rare Signal Extraction: QML evaluation must account for realistic event-sample sizes and complexity, while barren plateaus can make variational-model gradients vanish exponentially with qubit number.
  • Quantum machine-learning pipelines reduce input features, define training sets, and embed compressed classical data into quantum states before processing.
  • Current quantum algorithms have been designed and implemented for most typical data-processing tasks, but only at reduced scale; 100⊗100 hardware could support more realistic sizes.
  • 1. Rare Signal Extraction: For rare-signal classification, precision-recall curves are preferred over accuracy for imbalanced data because precision reflects signal rarity and recall measures recovered signal events.
  • 1. Rare Signal Extraction: Machine-learning models significantly outperform kinematic variables for measuring the small longitudinal fraction in same-sign WW production.The Standard Model predicts a longitudinal fraction of 0.07 at large dijet invariant mass.
  • 1. Rare Signal Extraction: Quantum anomaly-detection studies report faster convergence for a quantum autoencoder and high-accuracy identification of realistic BSM events with a quantum support-vector classifier.

2. Pattern Recognition Tasks: Reconstructing Particle Trajectories and Particle Jets

HEP pattern-recognition workloads include track reconstruction and jet identification, where rapidly growing candidate spaces motivate quantum and hybrid approaches. Current demonstrations remain small, while larger devices are needed to test scalability and practical advantage.

  • Track Reconstruction: Track reconstruction becomes difficult as detector occupancy rises because possible candidates scale quadratically or cubically with the number of hits.Tracks are built from detector space-points, and each track may contain a variable number of measurements.
  • Track Reconstruction: QUBO formulations encode doublets or triplets as binary track candidates and use quantum optimization algorithms to seek the Hamiltonian ground state.Proposed solvers include QAOA, VQE, and HHL, with triplets mapped to qubit operators in the Hamiltonian.
  • Track Reconstruction: Quantum-kernel triplet classification uses a 9-qubit circuit, but its advantage may require higher-dimensional inputs than triplets provide.The method encodes spatial coordinates of triplet hits into quantum states and may outperform classical kernels when more hits are considered.
  • Track Reconstruction: Hybrid QGNNs perform similarly to classical equivalents up to 16 qubits, while large-scale realization still requires more qubits.Classical GNNs show empirical linear scaling in input space-points, partly attributed to GPU parallelization.
  • Scaling and Hardware: A 100 ⊗100 device could study 100-site local Hamiltonians, test QUBO viability, and implement QGNNs comparable in size to classical models.Local track-reconstruction analyses also indicate lower quantum complexity for reproducing the same tracks with bounded-error probability, but input size remains a short-term constraint.
  • Jet Reconstruction and Identification: Several proposed digital quantum jet algorithms are unsuitable for noisy devices because they require QRAM-like parallel access to particle information.Quantum versions of k-means, affinity propagation, and kT clustering have also been investigated.
  • Jet Reconstruction and Identification: Jet tagging uses global jet characteristics and individual particle properties, requiring many features to distinguish jets relevant to Higgs coupling measurements.Jet clustering first combines observed particles to estimate the kinematics of the initiating particle.

3. Interpretable Models and Inference

The paper reviews quantum models as inference tools for extracting HEP dataset characteristics, focusing on PDFs and EFT Wilson coefficients. Early PDF results agree with classical fits, while broader quantum benefits and QML-based interpretation remain open research questions.

  • Scope: Quantum models are reviewed as inference tools for HEP datasets, with examples covering Parton Distribution Functions and Effective Field Theory Wilson coefficients.The stated aim is precision modelling and extracting quantum descriptors from models learned from data.
  • Parton Distribution Functions: PDF estimation matters because PDFs determine predicted collider-process rates and distributions while carrying experimental and theoretical uncertainties.PDFs describe parton momentum-fraction distributions that cannot be obtained from perturbative methods alone.
  • Parton Distribution Functions: Parameterized quantum circuits estimate PDF functional forms by optimizing circuit parameters against experimental data.Preliminary results show good agreement with existing PDF fits obtained through classical optimization.
  • Parton Distribution Functions: The reported PDF agreement is an initial step, but the estimated PDFs remain classical approximations and their possible advantages are currently computational.The paper states that substantial work is needed to exploit the quantum nature of the underlying problem.
  • Effective Field Theories: QML representations of EFT Hamiltonians may reduce the parameters needed to capture essential physics and identify correlations among observables and effective couplings.The approach models relationships between Wilson coefficients and measurements rather than the dynamics of EFT operators themselves.
  • Interpretation: Selected QML benchmarks emphasize new ways to interpret experimental HEP data rather than only removing classical computational bottlenecks.The paper notes that existing studies provide only hints toward a complete understanding of quantum information in particle-physics data.

4. Generative Models for Simulation

Quantum generative models are investigated as possible tools for detector simulation and event generation, a major computational burden in collider experiments. Current demonstrations are accurate only for very small systems, with resolution, connectivity, and portability limiting scale-up.

  • Motivation: Detector simulation and event generation are major computational burdens, consuming more than 50% of the LHC computing grid directly or through simulated-data reconstruction.The next generation of detectors is expected to intensify this challenge.
  • Classical Simulation: Fast classical simulation trades some accuracy for speed using parametric or deep generative models that learn multidimensional conditional distributions.These models can learn particle-feature correlations while incorporating experimental effects.
  • Limitations: Deep generative simulation outputs are inherently analysis specific and cannot be used outside the scope for which they were designed.This is identified as the main limitation of the described computationally lightweight and flexible approach.
  • Quantum Generative Models: Quantum and hybrid generative implementations include QGANs, quantum autoencoders, and Quantum Circuit Born Machines.Several studies investigate quantum architectures inspired by classical generative models.
  • Current Scale: Current quantum models generate accurate simulations only for very small 10-sized setups, often using one qubit per detector sensor.Autoencoders can compress simulations into latent representations that quantum models learn before classical decoding.
  • Limitations: Detector simulation resolution is limited by discretization because continuous detector features must be represented with qubits’ naturally discrete quantities.Accurate cross-particle correlations also require good connectivity and the ability to reproduce complex multiqubit entanglement.

1. Product Formulas

Product formulas approximate Hamiltonian evolution by decomposing it into simpler operations, with accuracy improved through finer time slicing or higher-order constructions. Variational approaches instead evolve a parametrized state through classical parameter updates, offering constant-depth circuits but introducing ansatz, measurement, and numerical challenges.

  • 1. Product Formulas: Product formulas use the general Trotter approximation to decompose Hamiltonian evolution into simpler operations.
  • 1. Product Formulas: O(M 2t2/n) decomposition error decreases with more time slices, while higher-order formulas achieve O((Mτ)2k+1/n2k) at order 2k.Higher accuracy increases gate count; randomization, adaptive methods, and multi-product formulas can further reduce Trotter errors.
  • 1. Product Formulas: Product formulas also support time-dependent Hamiltonians, extending their use beyond static evolution.
  • 2. Variational Approaches: Variational time evolution represents dynamics with a parametrized wavefunction ansatz and propagates its parameters by solving a classical equation of motion.
  • 2. Variational Approaches: Quantum measurements provide the matrix elements needed for variational evolution, while the resulting equations are integrated classically through varQTE or varQITE.
  • 2. Variational Approaches: Variational evolution is attractive when direct unitary decomposition becomes demanding for growing systems, including fermionic, bosonic, and gauge-field problems.
  • 2. Variational Approaches: Ansatz expressivity is difficult to relate systematically to accuracy, while matrix inversion sensitivity and measurement requirements limit practical variational evolution.
  • 3. Algorithmic Limitations: Trotter methods face both discretization and gate-infidelity errors, producing an optimal discretization n∗ that minimizes their combined effect.

4. Near-term Applications

Near-term quantum applications span quantum simulation and quantum-enhanced machine learning, with resource estimates framed around the 100 × 100 challenge. Their practicality depends on circuit resources, data encoding, trainability, and optimization landscapes.

  • Resource estimates target qualitative QED ground-state and dynamical studies within the 100 × 100 challenge, reporting qubits and layers rather than precision or runtime.
  • Table I estimates 20/100 and 30/150 qubits for (2+1)D QED configurations with gauge-field truncations l = 2, 3.
  • For two-flavor neutrino evolution, the largest 40-neutrino system requires 2340 CNOT gates at depth 120 under a first-order product formula.
  • Quantum-enhanced machine learning includes quantum acceleration of classical learning methods and genuinely quantum models based on parametrized quantum circuits.
  • Quantum models commonly predict through fθ(x) = Tr[ρ(x, θ)O(x, θ)], with ρ prepared by a parametrized circuit and data entering through the mapping x 7→ρ(x, θ).
  • Quantum learning advantages in these settings concern learning quality rather than the dimensionality or number of data points.
  • PQC-based methods face barren plateaus, where cost-function gradients vanish exponentially with the number of qubits and optimization may require exponentially many shots.
  • Barren plateaus can arise from expressive or entangling ansätze, global cost functions, random datasets, or quantum errors.

3. Near-term Applications

The paper identifies theoretically hard low-dimensional models and experimental HEP workflows as near-term quantum-computing targets. It emphasizes hybrid methods, quantum-data learning, and hardware-aware error and measurement mitigation while treating applications as examples rather than a complete catalog.

  • A noiseless 100-qubit system could in principle address high-dimensional learning problems if the PQC architecture is carefully tailored for trainability.
  • Quantum hierarchical classifiers have reproduced two-dimensional HEP detector images, while quantum convolutions achieved optimal image-analysis and image-generation results while mitigating barren plateaus.
  • Quantum graph methods and equivariant neural networks are proposed for HEP tasks including tracking, jet reconstruction, jet tagging, and event generation.
  • Generative models such as QGAN and QAE are identified as promising candidates for moving from hybrid to fully quantum architectures.
  • The selected theoretical and experimental applications are examples chosen partly for IBM’s 100 × 100 challenge, with resource estimates provided where possible.
  • Quantum dynamics is a principal target because classical costs grow exponentially while suitable quantum algorithms are available for scattering, string breaking, quenching, and phase transitions.
  • Quantum computing has an advantage over classical Markov Chain Monte Carlo methods in these cases, but its advantage over tensor networks near continuum limits or phase transitions remains unresolved.
  • Hybrid combinations of tensor networks and quantum circuits are proposed for strongly entangled systems and longer time scales or near phase transitions.

Appendix A: Resource Requirements for Quantum Simulation of lattice QED

The appendix analyzes qubit and Pauli-string resources for lattice QED encodings. Plaquette terms dominate Pauli-string counts, while logarithmic encoding minimizes scaling and qubit requirements at the cost of a degenerate ground state for perfectly representable spin dimensions.

  • The resource analysis evaluates quantum-link U(1) lattice gauge theories with dynamical Wilson fermions across arbitrary spatial dimensions.
  • The notation tracks lattice sites, edges, plaquettes, spinor components, spatial dimension, spin-system dimension, matrix sparsity, and Pauli-string counts.
  • Operator expansions distinguish Pauli strings with real, imaginary, or mixed coefficients, and Table A summarizes the resulting analysis.
  • Table II reports scaling relations for Pauli terms under different truncated-gauge-operator encodings, independent of the fermionic mapping choice.
  • The plaquette Hamiltonian term requires the most Pauli strings because of its strong scaling with dS.
  • Logarithmic encoding gives the best overall scaling and lowest Pauli-term count while also reducing the required number of qubits.
  • Perfectly representable spin dimensions omit Sz = 0 when dS is a power of two, producing a degenerate ground state.
  • The appendix situates these encodings within a broader overview of classical and quantum algorithms, including VQE and methods for excited states.

2. Tensor Networks

Quantum methods discussed for HEP include variational optimization, quantum machine learning, generative modelling, reinforcement learning, and topological data analysis. Their potential is balanced by limitations involving entanglement, circuit depth, data embeddings, and classical comparators.

  • 2. Tensor Networks: Tensor Networks efficiently parametrize moderately entangled states but cease to work for highly entangled out-of-equilibrium dynamics.They can compute ground states, low-lying excitations, thermal states, and some real-time dynamics.
  • Quantum optimization: QAOA encodes combinatorial optimization in an Ising-type Hamiltonian and alternates problem-Hamiltonian evolution with RX(β_i) mixing gates.In the infinitely layered limit, it can be interpreted as adiabatic evolution from an X-operator eigenstate to a problem-Hamiltonian eigenstate.
  • Quantum optimization: QAOA performance depends on various factors, does not necessarily outperform classical algorithms, and can produce circuits too deep for noisy hardware.Warm-starts and counteradiabatic driving are cited as possible ways to alleviate some issues.
  • Quantum machine learning: Quantum kernels classify or regress by embedding classical inputs into quantum states, with speed-up depending on feature maps that recognize classically intractable structure.Quantum generative models learn target distributions and may encode distributions that cannot be modeled efficiently classically.
  • Other quantum methods: HEP applications also include reinforcement learning for sequential decision and control tasks and quantum algorithms for extracting persistent Betti numbers in topological data analysis.These approaches extend quantum methods toward optimization, continuous state-action environments, and robust topological features from complex datasets.
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