Quench Spectroscopy of Magnetic Excitations on a Superconducting Quantum Processor

Technical Summary: Quench Spectroscopy of Magnetic Excitations on a Superconducting Quantum Processor

Problem Statement
The elementary excitation spectrum of many-body quantum systems encodes critical properties regarding transport, thermalization, and ground state structure. In strongly correlated systems, these spectra are typically encoded in dynamical structure factors, which are experimentally demanding to measure and classically challenging to compute, particularly for large systems where exact solutions are intractable (BQP-hard). While quantum simulators offer a path to probe these spectra by realizing exact non-equilibrium evolution, existing spectral probes often require complex experimental setups, linear response assumptions, or the reconstruction of Green's functions from multiple observables. Furthermore, many protocols rely on the costly preparation of complex ground states, which is a significant bottleneck on near-term quantum hardware, especially in gapless regimes where entanglement scales logarithmically with system size.

Methodology
The authors employ local quench spectroscopy on a superconducting quantum processor (IBM's ibm_boston) to extract excitation spectra of spin chains with $L=101$ sites. The core technique involves the Quench Spectral Function (QSF), defined as the space-time Fourier transform of a local observable $G(r, t) = \langle O(r, t) \rangle$ following a local perturbation.

The experimental protocol consists of the following steps:

1. State Preparation:
   • Ferromagnetic (FM) & Gapless (XY) Phases: The system is initialized in simple product states (e.g., fully polarized or $x$-polarized). Notably, for the gapless XY regime, the authors demonstrate that a high-fidelity ground state preparation is not required; instead, they utilize a global quench from a product state followed by a local quench.
   • Antiferromagnetic (AFM) Phase: Due to the entangled nature of the ground state, the authors first compute a classical Matrix Product State (MPS) using Density Matrix Renormalization Group (DMRG). This MPS is then compiled into a shallow "brickwork" quantum circuit using Approximate Quantum Compilation (AQC) to prepare the initial state on hardware.
2. Local Quench: A local perturbation is applied, typically a $R_y(\pi/2)$ rotation on a single site or a pair of adjacent sites.
3. Time Evolution: The system evolves under the XXZ Hamiltonian using a Trotterized time evolution circuit.
4. Measurement: Local observables (e.g., $\sigma_y$ or two-site operators) are measured across the chain at multiple time steps.
5. Analysis: The measured space-time signal $G(r, t)$ is symmetrized around the quench site and subjected to a 2D Fast Fourier Transform to yield the QSF, $G(k, \omega)$.

Error Mitigation
To address hardware noise without relying on Zero Noise Extrapolation (ZNE), the authors employ lightweight, standard error mitigation techniques integrated into Qiskit:

• Pauli Twirling: Applied to gates and measurements to convert coherent errors into stochastic noise with a flat Fourier spectrum, effectively filtering out noise during the spectral analysis.
• Dynamical Decoupling (DD): Used during evolution to suppress dephasing.
• Twirled Readout Error Extinction (TREX): Applied to correct systematic measurement biases.
• Antisymmetric Quench Protocol: By performing experiments with both $R_y(+\theta)$ and $R_y(-\theta)$ and taking the difference, static backgrounds and symmetric noise components are canceled, isolating the dispersive spectral features.

Key Results
The study successfully resolves excitation spectra across three distinct phases of the spin-1/2 XXZ chain:

1. Ferromagnetic Phase ($\Delta > 1$):
   • Single Magnons: Using a single-site quench and $\langle \sigma_y \rangle$ measurement, the authors resolve the single-magnon dispersion relation $E(k) = J(\Delta - \cos k)$, matching exact Bethe ansatz solutions.
   • Two-Magnon Bound States: By employing a two-site quench and a specific two-site observable, they selectively isolate and resolve the dispersion of two-magnon bound states, demonstrating the method's ability to access multi-particle sectors.
2. Antiferromagnetic Phase ($\Delta < -1$):
   • Starting from a compiled DMRG ground state, the protocol resolves the two-spinon continuum. The resulting QSF displays a broad spectral weight confined within the exact Bethe ansatz boundaries, characteristic of fractionalized excitations, rather than a sharp quasiparticle branch.
3. Gapless XY Regime ($|\Delta| < 1$):
   • The authors demonstrate that detailed ground state preparation is unnecessary. By initializing the system in a simple product state (which acts as a global quench) and applying a local perturbation, they successfully extract the magnon-like excitation spectrum (string excitations). The results show that even when the initial state is far from the true ground state, provided it lies in the correct symmetry sector, the QSF reveals the stable, long-lived excitations of the system.

Significance and Claims
The paper establishes quench spectroscopy as a fast, flexible, and robust probe for many-body excitation spectra on digital quantum hardware. Its primary contributions are:

• Scalability and Efficiency: The method successfully extracts spectra for $L=101$ spins using only lightweight error mitigation, with sampling overhead independent of system size.
• No Ground State Requirement: A novel finding is that the protocol does not rely on costly ground state preparation. In the classically challenging gapless regime, spectra can be extracted directly from the dynamics of easily prepared product states, provided the symmetry sector is respected.
• Direct Spectral Access: Unlike methods requiring linear response theory or Green's function reconstruction, this approach obtains the spectrum directly from the evolution of a single local observable.
• Hardware Validation: The quantitative agreement with exact analytical solutions (Bethe ansatz) across all phases serves as a non-trivial benchmark, confirming that the quantum hardware faithfully encodes the microscopic model despite noise.

The authors conclude that this approach opens a scalable route to studying regimes where classical simulation is intractable, such as two-dimensional strongly correlated systems and models with long-range interactions, without the immediate need for complex state preparation routines.