Part 1 of the series From quarks to quantum advantage
A hadron is not a star ball with a quark and an antiquark that remain quietly in place. It is a dynamic bound state. The matter and the field that holds the matter together can move internally, exchange energy, and form or destroy pairs of particles. It is difficult to calculate exactly these real-time dynamics. The paper Observation of Robust and Coherent Non-Abelian Hadron Dynamics on Noisy Quantum Processors investigates whether a large, still error-prone quantum processor can still provide access to that behavior.
The authors do not simulate a complete proton nor the complete theory of the strong nuclear force. They choose a simpler, but still non-Abelian model: an SU(2) lattice-gauge theory in one space dimension and one time dimension. They allow a mesonic state to evolve on a grid of 60 sites. For each site, two dynamic binary variables are set to qubits. The calculation therefore uses 120 logical qubits on the 156-qubit IBM Boston processor.
The result is a spacetime picture in which the meson expands, but its components do not fly away from each other indefinitely. Within that limited structure, an early internal oscillation also appears: a breathing mode. The hadron becomes alternately wider and narrower. The authors treat the frequency of that oscillation as a spectroscopic clue to the meson's lowest excitation energy.
That sounds like one quantum experiment, but the convincing power comes from four interconnected calculations. To understand why, we must first see why these dynamics are so classically difficult.
Why real-time strong interaction physics is difficult
The strong interaction is described in the standard model by quantum chromodynamics, or QCD. In that theory, quarks carry a color charge and react via non-Abelian gauge fields. Non-Abelian means, among other things, that the local symmetry operations are not simply interchangeable with each other. The order of operations can matter, and local Gauss laws strongly link matter and field configurations.
For static properties, Euclidean Monte Carlo is particularly powerful. However, real-time evolution causes a phase or sign problem, which causes that method to lose its usual probabilistic interpretation. Exact state simulation faces a second barrier: the Hilbert space grows exponentially with the number of local degrees of freedom.
Tensor networks can often postpone that problem in one dimension for a long time. They compress a quantum state by mainly preserving the relevant entanglement. However, after a quench or other non-equilibrium evolution, this entanglement can grow rapidly. Then the internal bond dimension of the tensor network must also grow to maintain the same accuracy. That is the so-called entanglement wall.
A quantum processor does not need to store the entire state as a classical list of amplitudes. The 120 qubits carry the state directly. That doesn't solve the other problems: the model must be coded efficiently, the time evolution must be decomposed into gates, and the hardware adds noise. The question is therefore not only whether the circuit can be implemented, but whether the measured bits still demonstrably say something about the intended hadron physics.
A simplified but non-trivial hadron model
The paper uses SU(2) instead of the SU(3) symmetry of full QCD and restricts the space to a line. This makes the model not realistic enough to directly predict the behavior of a proton or a collision experiment. It does retain two properties that are essential for the computational challenge: the gauge group is non-Abelian and matter is dynamically coupled to the gauge field.
On the staggered lattice, even sites represent quark-like matter and odd sites represent anti-quark-like matter. In this description, a meson consists of a bound quark-antiquark configuration with electric color flux in between. The local Gauss law does not allow for any arbitrary combination of matter and flux. Only gauge-invariant states are physical.
To efficiently handle that constraint, the paper uses the Loop-String-Hadron or LSH basis. Instead of first encoding redundant gauge fields and then filtering out the unphysical states, LSH describes the local state with gauge-invariant building blocks: through flux, incoming and outgoing string ends, and local hadron configurations. In the weak coupling regime used, the two dynamic string end variables per site can be stored directly in two qubits.
The result is an encoding inherent in physics: 60 sites become 120 qubits. In part 3 of this series we will build this mapping exactly. For now, the important point is that the qubits do not represent abstract optimization variables. Their occupancy is linked to local matter and flux information from the lattice.
Two experiments for each time point
The desired signal is small relative to the background and the hardware is noisy. That is why the authors carry out two evolutions for each Trotter time.
The first begins in the strong-coupling vacuum, abbreviated SCV. In the particle convention used, this state has no local fermion excitations. The second starts with the same vacuum, but with a meson at the two middle sites. Both states are evolved with the same dynamic circuit and then measured in the Z-base.
A local fermion density n_f(r,t) is reconstructed from the bit probabilities. The researchers then subtract the evolved vacuum signal from the evolved meson signal:
\Delta n_f(r,t) = n_f^{\mathrm{meson}}(r,t)
– n_f^{\mathrm{SCV}}(r,t).
\]
This differential measurement is a central part of the experiment. Noise that affects both circuits in approximately the same way, and dynamics already present in the vacuum background, are partially suppressed. What remains is better focused on the extra dynamics caused by the central meson.
flowchart LR
A["Strong-coupling vacuum"] --> C["Same time evolution U(t)"]
B["Vacuum plus central meson"] --> C
C --> D["Z measurements on 120 qubits"]
D --> E["Reconstruction of n_f(r,t)"]
E --> F["Meson minus SCV"]
F --> G["Spacetime profile and breathing mode"]It is not a general error correction. Errors that affect the two circuits differently do not automatically disappear, and a difference of two noisy measurements can also contain additional statistical uncertainty. The method is useful here because it answers the physical question: what changes when we place one meson on top of the same background?
What appears in the spacetime image?
The measured Delta n_f(r,t) is displayed as a heatmap, with the grid position on one axis and the time step on the other. From the center the signal expands to the left and right. The edges form a bounded area that resembles a light cone, but the quark-like components do not fly apart as free particles. The paper interprets this as confined meson propagation.
Within that area, positive and negative density differences alternate. An early oscillation occurs especially around the central sites. The meson configuration breathes, as it were. The frequency is extracted from the central density, the spatial distribution and the total density fluctuation in different ways. These methods provide mutually consistent frequencies within the time window examined.
The authors link this breathing frequency to the energy difference between the initial meson state and the lowest excited state that is addressed by evolution. This makes the time series a form of hadron spectroscopy: not by directly diagonalizing a static energy spectrum, but by measuring a characteristic oscillation in real time.
Why one heatmap is not physical evidence
A noisy quantum circuit can produce patterns that look convincing but do not arise from the intended Hamiltonian. Therefore, the paper uses a layered validation.
The first classical method is a tensor network calculation with MPS, MPO and 2-site TDVP. This route evolves the full LSH Hamiltonian, with a bosonic cutoff and a maximum bond dimension of 200. It does not include the weak coupling and Trotter approaches to the quantum circuit. The comparison with this route therefore asks: does the approximate algorithm still follow the physics of the full LSH model in the accessible time window?
The second method is Pauli propagation. This calculates in the Heisenberg picture how the measured Pauli observables evolve backward through the ideal circuit. It thus simulates the noise-free quantum circuit, not directly the full LSH Hamiltonian. The comparison between tensor network and Pauli propagation mainly checks the model approach and digital implementation. The comparison between Pauli propagation and QPU then shows how much the hardware deviates from the ideal circuit.
In addition, all methods follow the global charges. For the main instance, the total charge should remain in the sector Q=60 and the net lattice flux at q=0. Maintaining these quantities is a powerful sanity check: large deviations would mean that evolution leaves the intended physical sector. But here too there is a limit. A correct total charge does not prove that the entire spatial meson profile is correct. Many different local states can have the same global charge.
The four visible routes in the paper therefore each have a different role:
| Route | What is calculated? | What does it check? |
|---|---|---|
| TN/TDVP | full LSH Hamiltonian | physical reference dynamics |
| PP CPU | ideal circuit with term limit | digital circuit dynamics |
| PP GPU | ideal circuit with coefficient cutoff | independent PP control |
| QPU | actual circuit with hardware errors | experimental signal |
It is precisely the chain of comparisons that makes the interpretation stronger than a separate hardware measurement.
The scale of the experiment
The paper considers a grid of 60 sites in the weak coupling regime, with x=100 as the main instance. The Trotter step is delta tau = 0.0015. The circuit has two logical qubits per site and uses a local gate construction whose depth per Trotter step is constant with respect to the lattice length.
The two-qubit depth grows as 13t-1. For 20 steps that is 259; at 25 steps
- The entire 25-step circuit contains 17,660 two-qubit gates, according to the paper
and more than 90,000 single-qubit gates. 10,000 shots are used for each measured time.
The Quantum Advantage Tracker specifically captures the 20-step instance. This is important for later timing tables: the paper discusses dynamics up to 25 steps, while the tracker claim and the downloaded step-20 QASM provide a defined benchmark point. We will keep those two scopes apart throughout the series.
What has been demonstrated here?
Within the early time window examined, the paper shows a noise-resistant, differential signal consistent with bounded meson dynamics and an internal oscillation mode. The agreement between the full LSH tensor calculation, the ideal circuit simulation and the hardware supports three consecutive steps: the physical approach used, the circuit construction and the experimental realization.
The experiment also shows why physics-native encoding can be important. Because gauge-invariant building blocks are directly encoded, a large non-Abelian system with two qubits per site and a local circuit can be implemented.
What has not yet been proven?
This is not a complete simulation of SU(3)-QCD in three space dimensions. The weak-coupling approximation, finite lattice spacing, open boundaries, first-order Trotterization and limited early time all determine the range of validity.
Nor does a general quantum advantage claim follow from circuit size or hardware time alone. A fair speed claim must compare the same task, the same observable, the classical convergence and the chosen time definition. The paper shows that the available classical baselines quickly become more expensive or more difficult to control with greater time and weaker coupling. We will only discuss how strong that evidence is for practical quantum advantage in Part 7.
Our local reproduction will add a second layer later. In this project, the public QASM circuits and data were collected, the circuits were run via the standalone Fire Opal API on ibm_fez, and checks with Aer, ITensor, MPO and Majorana were prepared. Those results are not used to make the first explanation more beautiful than it is. They are only addressed when the observables and timing boundaries are fully defined.
From observation to understanding
We now have the final picture: a central meson develops a bounded spacetime profile and an internal oscillation, measured with 120 logical qubits and controlled with two different classical strategies. What is still missing is a precise meaning of the words quark, color, gauge field, confinement and meson within this model.
That is the subject of part 2. Only then will we build the Kogut-Susskind Hamiltonian, the LSH basis and finally the quantum circuit step by step.
Sources
- Fran Ilčić et al., Observation of Robust and Coherent Non-Abelian Hadron Dynamics on Noisy Quantum Processors, arXiv:2602.18080v3 (2026).
- Quantum Advantage Tracker, issue #149.
- LSH-IBM, public circuit implementation.
- lsh_data, public classic benchmark data.
- Hadron LSH Quantum Advantage Workbench, local reproduction and claim limits.