Thermalization of SU(2) Lattice Gauge Fields on Quantum Computers

Abstract

We simulate the thermalization dynamics for minimally truncated SU(2) pure gauge theory on linear plaquette chains with up to 151 plaquettes using IBM quantum computers. We study the time dependence of the entanglement spectrum, Rényi-2 entropy and anti-flatness on small subsystems. The quantum hardware results obtained after error mitigation agree with extrapolated classical simulator results for chains consisting of up to 101 plaquettes. Our results demonstrate the feasibility of local thermalization studies for chaotic quantum systems, such as nonabelian lattice gauge theories, on current noisy quantum computing platforms.

Figures (9)

• Figure 1: Single- and two-qubit quantum gates for the implementation of the three-qubit unitary operator $e^{-i\theta Z_i X_{i+1} Z_{i+2}}$. The $H$ and $R_Z$ in the circuit indicate the standard Hadamard and $Z$-rotation gates and the two-qubit gate is the $\textrm{CNOT}$ gate with the filled circle as the control and the "$+$" as the target.
• Figure 2: Quantum circuit for the time evolution of an initial strong-coupling vacuum state $|\downarrow \downarrow \downarrow \downarrow \downarrow\rangle$ for a plaquette chain of length $N=5$. $q_0,q_1,q_2,q_3,$ and $q_4$ are the quantum registers and $c$ denotes the classical (measurement) registers. The single- and two-qubit gates $U_Z=R_Z(2\theta,i)$, and $U_X=R_X(2\theta,i)$, $U_{ZZ}=R_{ZZ}(2\theta,i,i+1)$, $U_{ZX}=R_{ZX}(2\theta,i,i+1)$, and $U_{XZ}=R_{ZX}(2\theta,i+1,i)$ correspond to standard Qiskit gates. The quantum circuit for the three-qubit gate $U_{ZXZ}=e^{-i\theta Z_i X_{i+1} Z_{i+2}}$ is shown in Fig. \ref{['fig:time_evol']}. At the end of the time evolution, qubit $q_1$, $q_2$, and $q_3$ representing a $N_A=3$ subsystem are measured in the $X$, $Y$ and $Z$ eigenbasis, respectively, as an example of subsystem state tomography. The first vertical gray barrier is inserted to minimize the two-qubit gate depth of the quantum circuit within a single Trotter step, while the second barrier separates two consecutive Trotter steps.
• Figure 3: Scaling of two-qubit gate counts with system size $N$ for a single Trotter step in the time evolution quantum circuit. The data points (black dots) are tabulated in Tab. \ref{['Tab:qubitdepth']} and exhibit a linear dependence for $N\le 129$, which is demonstrated by a linear fit $c_0+c_1N$ (dashed line) with $c_0=-139.147$ and $c_1=15.388$. The deviation from the linear dependence for $N>129$ is caused by the breakdown of the linear connectivity of the plaquette chain when mapped onto the qubit layout of the IBM quantum hardware ibm_aachen, which can be found in Fig. 21 of Ref. Mayo:2026xxf.
• Figure 4: Entanglement spectrum for $N=101$ and $N_A=3$ is shown at time $t= 0, \frac{1}{3}, \frac{2}{3}, 1$. The diamond markers denote classical simulator results for $N=101$ that are obtained by extrapolating results from shorter chains. The open circle markers with errorbars represent results obtained from the quantum hardware devices ibm_boston and ibm_torino by combining measurements of $4,000$ shots from each. The insert figure is a zoom-in of the eigenvalues for the $5^{th}- 8^{th}$ eigenstates. All data points are listed in Tab. \ref{['Tab:E_spectrum']} in Appendix \ref{['app:B_data']}. Fitted parameter values used in extrapolating the classical simulator results are listed in Tab. \ref{['Tab:ES_fit']} in Appendix \ref{['app:C']}.
• Figure 5: Rényi-2 entropy $S_A^{(2)}$ as a function of the system size $N$ at time $t = 0,\,1/3,\, 2/3,$ and 1 for subsystem sizes (a) $N_A=1$, (b) $N_A=2$, and (c) $N_A=3$. Red dots with errorbars denote results obtained from the classical simulator with Trotter and statistical errors, black diamonds with errorbars represent results from real quantum hardware with hardware noise, Trotter and statistical errors, and the blue squares represent exact results obtained from classical exact diagonalization. For clarity, the $t=1$ curves are shown with $S_A^{(2)}$ multiplied by a factor of 2. The light red bands represent the cubic polynomials of $1/N$ fitted from the classical simulator results and extrapolated towards large $N$s where classical simulator results cannot be directly obtained.