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Observation of glueball excitations and string breaking in a $2+1$D $\mathbb{Z}_2$ lattice gauge theory on a trapped-ion quantum computer
Kaidi Xu, Umberto Borla, Kevin Hemery, Rohan Joshi, Henrik Dreyer, Enrico Rinaldi, Jad C. Halimeh
TL;DR
The paper evaluates how Trotterization and hardware noise affect simulated lattice-gauge dynamics, with primary attention to string observables. It uses a short-depth commuting-term decomposition and classical convergence checks, finding controlled Trotter errors alongside noise effects at later times.
Problem
Determining acceptable Trotter and hardware errors for string observables is necessary because fidelity alone does not prescribe observable-level accuracy, while smaller time steps increase circuit cost and noise exposure.
Source: TROTTER ERROR ANALYSIS AND SCALING · p. 14Method
The study decomposes the Hamiltonian into two commuting sub-lattices for short parallel circuits, compares Trotter and exact evolution, and validates classical simulations across bond dimensions and time steps.
Source: TROTTER ERROR ANALYSIS AND SCALING · pp. 13–14, CONVERGENCE TEST OF CLASSICAL SIMULATION · p. 17Results
At dt = 0.1, total infidelity reaches 0.024 after four applications, while tested classical trajectories overlap and noise becomes noticeable after t = 0.5.
Source: TROTTER ERROR ANALYSIS AND SCALING · p. 14, CONVERGENCE TEST OF CLASSICAL SIMULATION · p. 17, THE NOISE AND NOISELESS EMULATION OF… · p. 18Takeaways & Limitations
The chosen circuit and simulation parameters remain within a regime that agrees acceptably with continuous-time evolution, although plaquette-flip transitions can appear earlier under Trotterization.
Source: TROTTER ERROR ANALYSIS AND SCALING · p. 14Smaller Trotter steps are not automatically preferable because their higher circuit cost can increase hardware-noise effects, so the acceptable regime is device-dependent.
Source: TROTTER ERROR ANALYSIS AND SCALING · p. 14
Abstract
from arXiv · showhide
A major goal of the quantum simulation of high-energy physics (HEP) is to probe real-time nonperturbative far-from-equilibrium quantum processes underlying phenomena such as hadronization in quantum chromodynamics (QCD). The quantum simulation of the dynamics of confining strings and glueballs, both essential aspects of quark confinement, in a controllable first-principles way is an important step towards this goal. Here, we realize a $\mathbb{Z}_2$ lattice gauge theory in $2+1$D with a tunable plaquette term on a \texttt{Quantinuum System Model H2} trapped-ion quantum computer. We implement a shallow depth-6 Trotter circuit on a $6 \times 5$ matter-site square lattice utilizing all $56$ available qubits to execute over $1000$ entangling gates. We prepare far-from-equilibrium initial string configurations that we quench across a range of parameters to observe rich dynamical phenomena, such as the formation of gauge-invariant closed-loop excitations reminiscent of glueballs in QCD and multi-order string breaking accompanied by spontaneous matter creation. We further demonstrate experimentally that the system displays genuine $2+1$D dynamics, as evidenced by string snapshots over time that cannot be trivially mapped to $1+1$D physics. Our results demonstrate digital quantum simulations of nonequilibrium dynamics in a higher-dimensional lattice gauge theory and provide an experimentally accessible setting for phenomena related to confinement physics.
Supplemental Online Material for “Observation of glueball excitations and string breaking in a 2 + 1D Z2 lattice gauge
The supplemental material describes the model derivation and numerical techniques used in the study.
- The supplement details the model derivation underlying the study.Source: Supplemental Online Material for… · p. 13
- It also describes the numerical techniques employed in the analysis.Source: Supplemental Online Material for… · p. 13
GAUGE THEORY FORMULATION
The model is formulated as a Z2 lattice gauge theory coupled to Ising matter, with matter fields integrated out through Gauss-law resolution and a unitary-gauge choice.
- The model maps exactly to a Z2 lattice gauge theory coupled to Ising matter.Source: GAUGE THEORY FORMULATION · p. 13
- Pauli τ matrices represent Ising matter fields on sites, while σ matrices represent Z2 gauge fields on links.Source: GAUGE THEORY FORMULATION · p. 13
- The Ising matter fields are integrated out by resolving Gauss law and fixing the unitary gauge τ_z = +1.Source: GAUGE THEORY FORMULATION · p. 13
- Background charges satisfy Q_s = +1 everywhere except at the endpoints of the initial string states.Source: GAUGE THEORY FORMULATION · p. 13
COST
Experiments repeat Trotter steps to reach selected stroboscopic times, with deeper circuits requiring leakage detection that substantially increases gate costs.
- The circuit for each experiment is iterated to reach different stroboscopic times t_n, with at most 8 Trotter steps.Source: COST · p. 13
- Measurements are performed only at even times t_n with n = 2, 4, 6, 8.Source: COST · p. 13
- For t_2 and t_4, no leakage detection is needed and the two-qubit gate depth is 6n.Source: COST · p. 13
- Deeper circuits reaching t_6 and t_8 use leakage detection, which increases the gate count significantly.Source: COST · p. 13
TROTTER ERROR ANALYSIS AND SCALING
The study analyzes Trotterization errors from noncommuting Hamiltonian terms, finding timestep-dependent corrections and experimentally acceptable qualitative agreement despite a slight time-scale shift.
- Noncommuting electric-field–plaquette and matter-fluctuation–star terms generate Trotterization errors analyzed using the Baker–Campbell–Hausdorff formula.Source: TROTTER ERROR ANALYSIS AND SCALING · pp. 13–14
- The effective unitary contains corrections quadratic in the timestep and linear in model coefficients.Source: TROTTER ERROR ANALYSIS AND SCALING · p. 14
- In the confining regime hE ≫ hM, Jp, Js, the Jp hE contribution is the most relevant error source and induces plaquette-flip transitions.Source: TROTTER ERROR ANALYSIS AND SCALING · p. 14
- For dt = 0.1, total infidelity reaches 0.024 after 4 applications, while operator 2-norm error and infidelity show little system-size sensitivity.Source: TROTTER ERROR ANALYSIS AND SCALING · p. 14
- The chosen Trotter step gives good qualitative hardware agreement but introduces a slight time-scale shift.Source: TROTTER ERROR ANALYSIS AND SCALING · p. 14
- Figure S1 compares error scaling with timestep for Lx = 2, Ly = 1 and accumulated error versus Lx at fixed Ly = 1.Source: Figure S1 · p. 15
OBSERVABLES FROM PROJECTIVE MEASUREMENTS
String-configuration probabilities are estimated from projective measurements by locally projecting each link qubit in the target configuration onto |1⟩ and averaging over measurement shots.
- For a string configuration γ, a local projector acts on every link qubit belonging to γ.Source: OBSERVABLES FROM PROJECTIVE MEASUREMENTS · p. 15
- Each link projector is P_l^(1) = |1⟩⟨1|_l = (1 − Z_l)/2.Source: OBSERVABLES FROM PROJECTIVE MEASUREMENTS · p. 15
- The resulting expectation value measures the probability that all links in γ are simultaneously in state |1⟩, regardless of other qubits.Source: OBSERVABLES FROM PROJECTIVE MEASUREMENTS · p. 15
- All simulations use N_shots = 200 measurement shots.Source: OBSERVABLES FROM PROJECTIVE MEASUREMENTS · p. 15
DETAILS OF HARDWARE EXPERIMENTS
The hardware experiments use a 56-qubit trapped-ion processor with parallel two-qubit operations, while leakage detection, dynamical decoupling, and convergence checks support the experimental protocol.
- The Quantinuum System Model H2 provides 56 physical qubits with any-to-any connectivity and supports up to four parallel two-qubit operations.Source: DETAILS OF HARDWARE EXPERIMENTS · p. 15
- Its native gate set includes single-qubit rotations and a parameterized-angle ZZ gate.Source: DETAILS OF HARDWARE EXPERIMENTS · p. 16
- Benchmark infidelities are 2.8×10^-5 for single-qubit gates, 8.4×10^-4 for two-qubit gates, and 6.7×10^-4 for preparation and measurement in |0⟩.Source: DETAILS OF HARDWARE EXPERIMENTS · p. 16
- Leakage detection couples each data qubit to an ancilla with two exp(−iπ/4 ZZ) gates, exploiting the effective deletion of gates acting on leaked qubits.Source: DETAILS OF HARDWARE EXPERIMENTS · p. 16, Figure S2 · p. 16
- Dynamical-decoupling X pulses are inserted automatically when idle windows exceed 0.03 seconds.Source: DETAILS OF HARDWARE EXPERIMENTS · p. 16
CONVERGENCE TEST OF CLASSICAL SIMULATION
Classical tensor-network simulations are tested across bond dimensions and time steps, with the selected baseline lying in a converged regime.
- The dynamics show perfect overlap across all tested bond-dimension and time-step combinations.Source: CONVERGENCE TEST OF CLASSICAL SIMULATION · p. 17
- The baseline χ = 256 and dt = 0.025 is therefore within the converged regime without finite-entanglement or time-discretization errors.Source: CONVERGENCE TEST OF CLASSICAL SIMULATION · p. 17
THE NOISE AND NOISELESS EMULATION OF SMALL SYSTEM SIZE
Small-system emulations compare continuous tensor-network dynamics with noiseless and noisy Trotter circuits, while parity checks detect possible bit-flip errors during hardware-style execution.
- The comparison includes state fidelity, minimal-string probabilities, and charge creation for continuous, noiseless, and noisy simulations.Source: Figure S4 · p. 17
- Noiseless Trotter emulation on 24 gauge links generally matches continuous tensor-network simulations, apart from a constant Trotter-error shift.Source: THE NOISE AND NOISELESS EMULATION OF… · p. 17, Figure S4 · p. 17
- The parity check uses four CNOT gates per vertex and terminates a shot when the measured vertex-link parity is odd.Source: Figure S5 · p. 18
- The noise model produces noticeable effects after t = 0.5, including increased charge creation and degraded string probabilities.Source: Figure S4 · p. 17, THE NOISE AND NOISELESS EMULATION OF… · p. 18
A SIMPLE QUANTUM-CLASSICAL HYBRID ERROR DETECTION CIRCUIT
The protocol uses ancilla-assisted parity checks after each Trotter step to verify that evolution remains in the zero-charge subspace. On a 4 × 4 lattice, error detection is evaluated during noisy Trotterized evolution up to t = 1.0.
- Ancilla-assisted parity checks are performed after each Trotter step to verify that the state remains in the zero-charge subspace.The checks rely on the star operator acting as a Z stabilizer under a sufficiently strong confining potential.Source: A SIMPLE QUANTUM-CLASSICAL HYBRID… · p. 18
- The error-detection protocol monitors transitions away from the initial string configuration and the average charge density during time evolution.The ideal noiseless Trotter evolution provides the reference trajectory for both observables.Source: Figure S6 · p. 19
- The noisy dynamics are simulated on a 4 × 4 lattice up to t = 1.0 with Trotter step dt = 0.05.Source: Figure S6 · p. 19