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First-principle predictions of fragmentation functions via quantum computing

This paper presents a quantum computing algorithm for calculating fragmentation functions from first-principles QCD, demonstrating its feasibility through a classical simulation of heavy-quark fragmentation into quarkonium that aligns with NRQCD benchmarks.

Original authors: Juan J. Gálvez-Viruet, Felipe J. Llanes-Estrada, Nicolas M. Arenaza, María Gómez-Rocha, T. J. Hobbs

Published 2026-09-01

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CC BY 4.0

Original authors: Juan J. Gálvez-Viruet, Felipe J. Llanes-Estrada, Nicolas M. Arenaza, María Gómez-Rocha, T. J. Hobbs

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Inside the heart of every atom, protons and neutrons are not solid, unchanging spheres but rather bustling cities of smaller particles called quarks and gluons. These particles are bound together by the strong force, the most powerful interaction in nature, which behaves in ways that are notoriously difficult to predict using standard computers. When high-energy particles collide, they often shoot out streams of these smaller particles, known as jets. As these jets travel outward, the individual quarks and gluons within them must transform into stable particles like protons or mesons to be detected by scientists. The mathematical map that describes this transformation, known as a fragmentation function, is a crucial key to understanding the structure of matter. However, calculating this map from the fundamental laws of physics has long been a stumbling block, as the complexity of the interactions grows too fast for traditional supercomputers to handle with the necessary precision.

A team of researchers has now taken a significant step toward solving this problem by using a new kind of computing power: quantum computers. Instead of trying to simulate the behavior of particles on a conventional machine, which struggles with the sheer number of possibilities, the team developed an algorithm designed to run on a quantum device. This approach treats the particles not as fixed objects but as dynamic entities that can appear and disappear, mimicking the actual rules of quantum mechanics. The researchers successfully demonstrated this method by simulating a specific scenario where a heavy quark breaks apart to form a particle called a quarkonium. They ran this simulation on a classical computer cluster that was programmed to act like a quantum machine, limited to about 29 qubits, which is the basic unit of information in a quantum computer.

The results of this simulation were compared against established theoretical calculations to see if the new method held up. The team found that their quantum-based approach produced results that agreed reasonably well with the known theories, despite the limitations of the small-scale simulation. This agreement suggests that the method is on the right track. The researchers noted that their current simulation was a proof of concept, constrained by the memory limits of classical computers which could not fully represent the vast complexity of the real quantum system. To perform a full, high-precision calculation that could rival the best existing methods, they estimate that future quantum computers would need to operate with around 1,000 qubits and perform roughly one billion entangling operations.

This work does not claim to have solved the problem of fragmentation functions entirely, but it opens a viable path forward. The team showed that by encoding the creation and destruction of particles directly into the memory registers of a quantum computer, they could bypass some of the bottlenecks that have hindered previous attempts. Their method is particularly efficient for systems with a small number of active particles, a common scenario in high-energy collisions. While the current demonstration was a small-scale simulation, it provides a concrete blueprint for how future quantum hardware, potentially running alongside major particle accelerators in the 2030s, could extract these complex particle maps from first principles. The study confirms that the theoretical framework is sound and that the primary hurdle is no longer the idea itself, but the hardware required to execute it at a useful scale.

Technical Summary: First-Principle Predictions of Fragmentation Functions via Quantum Computing

Problem Statement
The paper addresses the challenge of computing fragmentation functions (FFs) from first principles using Quantum Chromodynamics (QCD). While Lattice Gauge Theory (LGT) has been successful in many hadronic observables, it currently struggles with certain sectors, particularly fragmentation functions, which describe the probability of a parton fragmenting into a specific hadron. The authors propose utilizing quantum computing to simulate QCD in the Light-Front (LF) gauge ( $A^+ = 0$ ), a framework naturally suited for describing high-energy jet structures and longitudinal momentum distributions, which are difficult to access via Euclidean lattice methods.

Methodology
The authors develop a specific algorithm and encoding scheme to simulate QCD on a quantum computer:

Light-Front Hamiltonian Formulation: The approach begins with the QCD Lagrangian, transformed via a Legendre transform into the Light-Front Hamiltonian ( $P^-$ ). This Hamiltonian includes terms that change particle number (creation and annihilation), necessitating a second-quantized treatment.
Particle-Register Encoding: Unlike standard lattice gauge theory or Jordan-Wigner encodings which map fields to fixed lattice sites, the authors employ a "particle-register" encoding.
- Creation/Annihilation: Operators ( $a, a^\dagger$ ) are implemented as computer methods acting on memory registers.
- Qubit Structure: For a given particle type (e.g., gluons), a register consists of an "Absence/Presence" qubit followed by qubits encoding the particle's quantum numbers (momentum, color, spin).
- Symmetrization: The algorithm includes step (anti)symmetrizers to ensure the wavefunction maintains correct (anti)symmetry upon the exchange of identical bosons or fermions.
- Boundary Conditions: A boundary term is introduced in the commutation relations $[a_i, a^\dagger_j] = \delta_{ij} + \text{boundary}$ to handle cases where the memory register is full, preventing the creation of particles beyond the available qubit capacity.
Simulation Scale: The authors simulate this algorithm on a classical cluster, limited to approximately 29 qubits (specifically for $SU(3)$ color). This simulation is constrained by memory limits, restricting the Fock space to a small number of particles (up to four partons: two quarks, one antiquark, one gluon) and a coarse longitudinal momentum grid. Transverse momentum is currently excluded.

Key Contributions and Results
The primary contribution is a demonstration of the feasibility of extracting fragmentation functions using this quantum algorithm, specifically benchmarking a heavy-quark process:

Case Study: The authors simulate a charm-initiated jet fragmenting into a $J/\psi$ meson.
Wavefunction Ansatz: Since diagonalizing the full QCD Hamiltonian is computationally prohibitive, they utilize an ansatz for the $J/\psi$ wavefunction based on a longitudinal distribution $\chi_0(x)$ and physical parameters ( $m_q = 1.27$ GeV, $\kappa = 1.34$ GeV).
Benchmarking: The results are compared against Non-Relativistic QCD (NRQCD) computations at a scale $\mu = 3m_c$ $μ = 3 m_{c}$ .
- Findings: Despite large systematic uncertainties arising from Fock space truncation, Trotter discretization, scale choice, and the wavefunction ansatz, the quantum simulation shows "reasonable qualitative agreement" with established NRQCD results.
- Systematic Controls: The authors explicitly note that, unlike previous iterations, they did not discard small Hamiltonian matrix elements to speed up computation, thereby removing one source of truncation error.
Resource Analysis: The paper provides an estimate of the resources required for credible, competitive calculations. It suggests that while current chips (approx. 100 qubits) are insufficient, future machines with $\sim 10^3$ qubits and $\sim 10^9$ entangling gates (potentially available during the HL-LHC era) would be required to overcome current limitations and explore larger Fock spaces.

Significance and Claims
The paper positions this work as a proof-of-concept for a new pathway in hadron physics:

Complementarity to Lattice QCD: The authors argue that this Light-Front quantum approach offers a distinct advantage for few-body problems with variable particle numbers, where their encoding is more qubit-efficient ( $n \log_2 N_p$ ) compared to Jordan-Wigner encodings ( $O(N_p)$ ) when the system is far from saturation.
Access to New Observables: The method provides access to the reduced density matrix of the final-state hadron, allowing for the calculation of quantum information metrics (concurrence, discord, magic) within the jet, which are difficult to access via Euclidean lattice approaches.
Modest Outlook: The authors are modest about the current state of the technology. They acknowledge that current quantum hardware is not yet capable of "serious-scale" hadron calculations due to noise and connectivity issues. However, they assert that the algorithm is ready and that the necessary hardware improvements are plausible within the timeline of the HL-LHC and the Electron-Ion Collider (EIC) in the 2030s.
Future Potential: The ultimate goal is to extract arbitrary parton-hadron combinations from first principles, potentially running concurrently with high-energy collider experiments to provide theoretical predictions for fragmentation functions that currently resist conventional lattice efforts.

Technical Summary: First-Principle Predictions of Fragmentation Functions via Quantum Computing

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