A Framework for Quantum Simulations of Energy-Loss and Hadronization in Non-Abelian Gauge Theories: SU(2) Lattice Gauge Theory in 1+1D

TL;DR

The work presents a comprehensive framework for real-time quantum simulations of energy loss and hadronization in non-Abelian lattice gauge theories, demonstrated on a 1+1D SU(2) lattice with heavy quarks mapped to qubits. It combines domain decomposition, SC-ADAPT-VQE ground-state preparation, and fermionic-SWAP heavy-quark motion to handle color entanglement and non-commuting dynamics, enabling ground-state and dynamical studies on modest lattices. Classical benchmarks guide operator pools and variational circuits, while quantum experiments on IBM hardware validate the approach, with error mitigation aligning quantum results with classical simulations. The methods generalize to SU(3) and higher dimensions, offering a pathway toward more realistic jet-hadronization physics in QCD and related non-Abelian theories.

Abstract

Simulations of energy loss and hadronization are essential for understanding a range of phenomena in non-equilibrium strongly-interacting matter. We establish a framework for performing such simulations on a quantum computer and apply it to a heavy quark moving across a modest-sized 1+1D SU(2) lattice of light quarks. Conceptual advances with regard to simulations of non-Abelian versus Abelian theories are developed, allowing for the evolution of the energy in light quarks, of their local non-Abelian charge densities, and of their multi-partite entanglement to be computed. The non-trivial action of non-Abelian charge operators on arbitrary states suggests mapping the heavy quarks to qubits alongside the light quarks, and limits the heavy-quark motion to discrete steps among spatial lattice sites. Further, the color entanglement among the heavy quarks and light quarks is implemented using hadronic operators, and Domain Decomposition is shown to be effective in quantum state preparation. Scalable quantum circuits that account for the heterogeneity of non-Abelian charge sectors across the lattice are used to prepare the interacting ground-state wavefunction in the presence of heavy quarks. The discrete motion of heavy quarks between adjacent spatial sites is implemented using fermionic SWAP operations. Quantum simulations of the dynamics of a system on $L=3$ spatial sites are performed using IBM's ${\tt ibm\_pittsburgh}$ quantum computer using 18 qubits, for which the circuits for state preparation, motion, and one second-order Trotter step of time evolution have a two-qubit depth of 398. A suite of error mitigation techniques are used to extract the observables from the simulations, providing results that are in good agreement with classical simulations. The framework presented here generalizes straightforwardly to other non-Abelian groups, including SU(3) for quantum chromodynamics.

Figures (35)

• Figure 1: The lattice layout for heavy-quark, light-quark and light anti-quark fields on a lattice of spatial extent $L$. The spatial sites are labeled by $x$, which takes values from $x=0$ to $x=L-1$, the staggered sites are labeled by $n$ which takes values $n=0$ to $n=3L-1$, and the fermion (or qubit) sites are labeled by $j$ which takes values $n=0$ to $n=6L-1$. The boundaries between quark and anti-quark sectors, denoted by $b$, correspond to the locations of chromo-electric field contributing to the Hamiltonian via the non-local charge operators.
• Figure 2: A schematic of one heavy quark moving toward and passing a second heavy quark, implemented by FSWAP gates. The circles denote spatial lattice sites, and the vertical colored bars denote heavy quarks. Panels (a) through (d) show the discrete change of spatial site. Panel (b) shows the action of the FSWAP gate when the heavy quarks are on adjacent site, in which the stationary heavy quark exchanges position with the moving heavy quark, displacing it by one lattice site.
• Figure 3: The $\langle \hat{Z}_j \rangle$ as a function of the fermion site $j$, starting from the SC vacuum, through the classically optimized layers, through to the exact values for $L=2$ with $m_q=0.1$ and $g=1.0$. The left panel shows the $\langle \hat{Z}_j \rangle$ in the $n_Q=0$ sector with up to five layers of optimization starting from $|SC; 2, 0\rangle$ toward $|I; 2, 0\rangle$. The right panel shows the analogous results in the $n_Q=1$ sector with up to six layers of optimization starting from $|SC; 2, 1\rangle$ toward $|I; 2, 1\rangle$. The numerical values of the results are given in Table \ref{['tab:L2Q0Z']} and Table \ref{['tab:L2Q1Z']}.
• Figure 4: A block diagram for the DDec of the variational wavefunction in the $n_Q=1$ sector for $L=3$. The $L=3$ wavefunction is initialized from the variationally optimized $n_Q=1, L=2$ (with the heavy quark located at $x=0$) and $n_Q=0, L=1$ wavefunctions, as indicated by the operators acting either side of the dashed red line. The re-optimized operators in each sector are implemented in parallel, followed by the application of boundary operators that entangle the two sectors, which are also optimized (with all of the operators globally optimized).
• Figure 5: The $\langle \hat{Z}_j \rangle$ as a function of fermion site $j$, from $|SC; 3, 1, x=0\rangle$, through the 8 classically-optimized layers (corresponding to the application of the 10 operators in Eq. \ref{['eq:L3U']}, as shown in Fig. \ref{['fig:DDecL3']}), through to the exact values from $|I; 3, 1, x=0\rangle$ for $m_q=0.1$ and $g=1.0$. The heavy quark is located at $x=0$ in sites $j=\{0,1\}$, and $j=\{6,7,12,13\}$ correspond to unoccupied heavy-quark sites. The numerical values of the results are given in Table \ref{['tab:L3Q1Z']}.