Part 2 of the series From quarks to quantum advantage
In the first part we saw the end result: a meson-like object expands over a lattice, but does not break up into two free-flying parts. An internal oscillation also appears in the middle. To understand why the authors call it a hadron, we have to go back to the physical building blocks: quarks, color charge, gauge fields and confinement.
The words seem familiar from particle physics, but in this paper they are given a precise and limited meaning. The experiment does not simulate a full proton and does not simulate a three-dimensional SU(3)-QCD. It uses a SU(2) lattice gauge theory with one space dimension. That model is simpler, but retains the non-Abelian character and the local coupling between matter and gauge field that make the calculation interesting.
Quarks are not loose beads
In the quark model, mesons consist of a quark and an antiquark, while baryons in ordinary QCD consist of three quarks. That image is useful, as long as we remember that a hadron is more than the sum of a few point particles. The quantum state also contains the strong field, virtual excitation, and superpositions of different matter and field configurations.
In addition to electric charge, a quark also carries color charge. Red, green and blue are names for them, not visible colors. In QCD, the gauge group SU(3) forms the mathematical symmetry of that color charge. Physical hadrons are color neutral: their overall state is a singlet under local gauge transformations.
The paper replaces SU(3) with SU(2). The fundamental representation then has two color components instead of three. The singlet structure also differs: SU(2) is not literally a model in which every detail of the QCD meson or baryon is preserved. The authors use meson, baryon, string and hadron for the corresponding gauge-invariant LSH building blocks of this SU(2) model. That terminology describes the role in model theory, not a direct prediction for a physical pion or proton.
Why the field is indispensable
We can describe two electrically charged particles in simple cases with a potential between their positions. In a gauge theory, the field itself is a dynamic part of the quantum state. On a spatial grid, the matter resides on the sites and the gauge field lives on the links between neighboring sites.
For SU(2), the local charge density and the electric fields have three algebra components, denoted \(a=1,2,3\). A local Gauss law applies to each site. In compact operator form is the physical condition
\[G^a(r)|\mathrm{phys}\rangle = 0,
\qquad a=1,2,3.
\]
The operator \(G^a(r)\) counts not only the matter charge at site \(r\), but also the electric flux entering and leaving via the adjacent links. A random bit pattern for matter and flux is therefore usually unphysical. When color charge appears on a site, the change in the surrounding field must match it exactly.
This is the local nature of gauge invariance. It is more stringent than just maintaining one total charge for the entire grid. A state can be globally neutral and yet locally violate all Gauss laws. This is an important risk for a quantum algorithm: noise or clumsy encoding can leak amplitudes into a large unphysical subspace.
flowchart LR
L["incoming color flux"] --> S["site with quark charge"]
S --> R["outgoing color flux"]
G["Gauss law at the site"] --> L
G --> S
G --> RFrom color flux to a string
Suppose we create a quark and antiquark in different places. The local Gauss law then requires a matching flux configuration between the two ends. In one dimension of space, that flux cannot bend around an object or spread across a transverse plane. It follows the lattice line.
We call this continuous color flux a string. It is not a mechanical rope, but a gauge-invariant description of how the electric flux connects the matter charges. The energy of the configuration depends on the flux and the length over which it is present. This makes it energetically unfavorable to separate the ends without a boundary.
That is the intuition behind confinement: the relevant excitation is not a free quark, but a bound, color-neutral combination of matter and field. Given sufficient energy, a string in a dynamical matter theory can break by forming an extra pair. The new quarks shield the original ends and again produce gauge-invariant objects. The individual colored constituent still does not appear as a free end product.
The paper mentions processes such as pair creation, pair annihilation, string breaking and rejoining in the measured dynamics. On the heatmap these are not individual classical events that we follow per shot. The measured density profile is an average over many shots of a quantum state containing a superposition of allowed configurations.
Staggered fermions in a line
Placing a continuous fermion field theory directly on a finite lattice can produce unwanted additional fermion species. The Kogut-Susskind or staggered fermion formulation therefore distributes the components across alternating sites. In the convention of the paper, even sites carry the quark interpretation and odd sites carry the antiquark interpretation.
The local LSH variables \(n_i(r)\) and \(n_o(r)\) indicate whether a site has an incoming or outgoing string end. Their sum is not necessarily the physical fermion density, because the filled odd sites form the reference vacuum. The paper uses
\[n_f(r)=
\begin{cases}
n_i(r)+n_o(r), & r\ \text{even},\\
2-[n_i(r)+n_o(r)], & r\ \text{odd}.
\end{cases}
\]
For the strong-coupling vacuum condition, both string ends are absent at even sites and both are present at odd sites. Due to the parity-dependent definition, \(n_f(r)=0\) is on every site. A local bit pattern that looks like filled on odd sites actually means vacuum in particle interpretation.
This is a source of much confusion when reading the QASM. Without site parity, the total particle density would be incorrectly reconstructed, even if all measured bits are technically correct.
What is the central meson?
The second initial state differs from the vacuum at the two middle sites. There \(n_f=1\) is placed on an even-odd pair. In the LSH description these are two string ends that together form a central meson. The rest of the lattice remains in the same background sector as the SCV.
During evolution, the meson profile can be spread over more sites. This does not mean that a classic object becomes slightly larger in every shot. The unitary evolution builds up a superposition in which different allowed matter and flux configurations have different amplitudes. The measured \(n_f(r,t)\) is the expectation of the local occupancy in that condition.
Subtracting the SCV evolution asks which density change is specifically caused by the central meson. The resulting
\[\Delta n_f(r,t)=n_f^{\mathrm{meson}}(r,t)-n_f^{\mathrm{SCV}}(r,t)
\]
can be both positive and negative. A negative value is not a negative probability or negative number of particles. It means that the local density in the meson run is lower than in the separately evolved vacuum run.
Confinement in the measured profile
For free particles, a localized wave packet would disperse according to their dispersion relation. In the paper, the relevant fluctuations remain within a bounded, curved space-time region. The authors interpret the shape of that edge together with the classical references as confined propagation: the constituents move, but remain part of a bound meson profile.
Within the bounded region, alternating red and blue bands appear in the difference heatmap. That structure is used to fit a breathing frequency. Physically, this is comparable to a bound object that periodically exchanges energy between different internal configurations. The frequency is a proxy for the difference \(E_1-E_0\) between the addressed low-lying excitation and the initial meson state.
The word proxy is important. The quantum processor does not diagonalize the full Hamiltonian and does not directly provide an exact energy spectrum. The energy information is extracted indirectly from a limited, damped time series. The reliability therefore depends on the time window, the fit model, the noise and the agreement between the different methods.
What has been demonstrated here?
The SU(2) theory used contains local non-Abelian Gauss laws, dynamic fermionic matter and flux strings. The central object is a gauge-invariant, bound meson configuration within that theory. The measured spacetime dynamics are consistent with bounded propagation and an internal oscillation mode in the early time window.
What has not yet been proven?
The simulation does not reproduce all colors, dimensions and interactions of physical QCD. The visual boundaries of one heatmap in itself is not evidence of confinement. The interpretation is based on the Hamiltonian, the gauge-invariant encoding and the comparison with the classical references.
In part 3 we make this underlying calculation concrete. We start with the Kogut-Susskind Hamiltonian and show how the LSH basis converts the local non-Abelian Gauss laws into directly usable physical building blocks.
Sources
- Fran Ilčić et al., paper v3, 2026.
- I. Raychowdhury and J. R. Stryker, Loop, string, and hadron dynamics in SU(2) Hamiltonian lattice gauge theories, Phys. Rev. D 101, 114502 (2020).
- C. W. Bauer et al., Quantum simulation of fundamental particles and forces, Nature Reviews Physics 5, 420-432 (2023).
- J. Kogut and L. Susskind, Hamiltonian Formulation of Wilson's Lattice Gauge Theories, Phys. Rev. D 11, 395 (1975).