Improved Ising Meson Spectroscopy Simulation on a Noisy Digital Quantum Device

TL;DR

This work addresses extracting $E_8$-related meson spectra from real-time dynamics of the 1D transverse-field Ising model on a noisy quantum device. It compares two error-resilient strategies: direct first-order Trotter evolution with native gates and tensor-network–based Riemannian circuit compression to fixed-depth circuits, both coupled with error mitigation. Using the IBM Quantum device ibm_torino, the central-spin magnetization dynamics are Fourier-transformed to reveal peaks corresponding to $E_8$ mass ratios, validated against classical benchmarks and tensor-network results. The results show that, with circuit compression and mitigation, essential $E_8$ spectral features can be observed on NISQ hardware, highlighting a path to probing nontrivial quantum field theory phenomena with limited circuit depth. The work also outlines future extensions to quench dynamics in related models such as XXZ and the massive Thirring model.

Abstract

The transverse-field Ising model serves as a paradigm for studying confinement and excitation spectra, particularly the emergence of $E_8$ symmetry near criticality. However, experimentally resolving the Ising meson spectroscopy required to verify these symmetries is challenging on near-term quantum hardware due to the depth of circuits required for real-time evolution. Here, we demonstrate improved spectroscopy of confined excitations using two distinct error-resilient circuit construction techniques on the IBM Torino device: first-order Trotter decomposition utilizing native fractional gates, and a tensor-network-based circuit compression via Riemannian optimization. By analyzing the Fourier spectrum of error-mitigated time-series data, we successfully identify key signatures of $E_8$ symmetry despite hardware noise. These results validate the viability of both circuit compression and hardware-efficient compilation for probing complex topological phenomena on NISQ devices.

Figures (11)

• Figure 1: Simulation results obtained via exact diagonalization on a classical computer. The system evolves from the initial state $|\uparrow \uparrow \downarrow \downarrow \downarrow \downarrow \downarrow\downarrow\downarrow\uparrow \uparrow\rangle$ using the parameters $h_x = 1$ and $h_z = 3$. The observable is the central spin expectation value $\langle \sigma^z_{\text{cen}}(t) \rangle$. (a) Time-domain dynamics of the observable. (b) Frequency-domain spectrum plotted on a semi-log scale. Colored dashed lines indicate several expected peak positions predicted by the $E_8$ mass ratios. We identify the value of $m_1$, and the positions of all other expected peaks are computed by applying the known $E_8$ mass ratios. The peak values obtained from ED are denoted directly on the plot.
• Figure 2: Quantum circuit construction of the real-time evolution operator for the TFIM using first-order Trotter decomposition, as described in Eq. \ref{['eq:trotter']}. The circuit consists of sequential layers implementing the longitudinal field (RZ gates), transverse field (RX gates), and Ising interaction (RZZ gates) terms.
• Figure 3: Real-time evolution results on the IBM Quantum device ibm_torino for an 8-site transverse-field Ising model with parameters $h_x = 1$, $h_z = 3$, initial state $|\uparrow \uparrow \downarrow \downarrow \downarrow \downarrow \uparrow \uparrow\rangle$, and 8192 shots per circuit. The observable $\langle \sigma^z(t) \rangle$ is measured at site 4 (the central site). (a) Comparison of two reference circuit strategies for error mitigation: $\bullet$ markers correspond to the raw measured signal, $\square$ markers correspond to setting all RX and RZ angles to zero while leaving RZZ gates unchanged, and $\triangle$ markers correspond to setting all gate parameters to zero. A small discontinuity is visible around $t = 10$, which may arise from splitting the job into two separate submissions. (b) Post-processed results: $\bullet$ markers show the original raw data, $\square$ markers are the refined reference circuit results, and $\blacktriangle$ markers represent the final corrected signal obtained by dividing the raw data by the refined reference.
• Figure 4: TN representation of the overlap between the target unitary operator and the quantum gates. The blue tensors MPO form of the real-time evolution operator, while the pink tensors correspond to the quantum circuit with a fixed number of layers, initialized using second-order Trotter decomposition. This overlap corresponds to the trace term in the cost function defined in Eq. \ref{['eq:compress_cost']}. The goal of the Riemannian optimization is to optimize the quantum gates such that this overlap is maximized, i.e., the cost function is minimized.
• Figure 5: Simulation results using Riemannian optimization to construct time-evolution circuits with different numbers of layers, executed on a classical simulator. The initial state is $|\uparrow \uparrow \downarrow \downarrow \downarrow \downarrow \uparrow \uparrow\rangle$, and $\langle \sigma^z(t) \rangle$ is measured at site 4. The model parameters are $h_x = 1$, $h_z = 3$, and each simulation uses 8192 shots. (a) Results using 9 circuit layers; (b) results using 41 layers. The black dashed line represents the exact evolution, the blue line with marker shows the simulator results with the optimized quantum gates, and the red solid line indicates the cost function defined in Eq. \ref{['eq:compress_cost']}. In both cases, the fidelity of the simulation decreases over time, and longer-time evolution requires more circuit layers to maintain accuracy.