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Role of spectral structure in adiabatic ground-state preparation of the XXZ model

Francisco Albarrán-Arriagada, Juan Carlos Retamal

Abstract

Adiabatic ground-state preparation is fundamentally limited by the spectral structure of the time-dependent Hamiltonian, particularly by gap reductions and degeneracies that induce nonadiabatic transitions. We examine this dependence in the anisotropic Heisenberg (XXZ) model on an eight-site ring by comparing three strategies: optimization of the initial Hamiltonian, addition of auxiliary terms, and considering approximate counterdiabatic driving. Owing to anisotropy-dependent level crossings among low-energy states, the XXZ model provides a stringent benchmark. We find that performance is mainly constrained by spectral degeneracies between the ground and excited states. Simple strategies such as initial-Hamiltonian optimization or site-dependent Zeeman fields, suppresses critical crossings and drastically enhance ground-state preparation. In contrast, counterdiabatic terms alone do not improve the protocol when the spectral structure remains level-crossings, becoming effective only after degeneracies are removed. These results identify spectral engineering as a prerequisite for efficient adiabatic ground-state preparation in interacting spin systems.

Role of spectral structure in adiabatic ground-state preparation of the XXZ model

Abstract

Adiabatic ground-state preparation is fundamentally limited by the spectral structure of the time-dependent Hamiltonian, particularly by gap reductions and degeneracies that induce nonadiabatic transitions. We examine this dependence in the anisotropic Heisenberg (XXZ) model on an eight-site ring by comparing three strategies: optimization of the initial Hamiltonian, addition of auxiliary terms, and considering approximate counterdiabatic driving. Owing to anisotropy-dependent level crossings among low-energy states, the XXZ model provides a stringent benchmark. We find that performance is mainly constrained by spectral degeneracies between the ground and excited states. Simple strategies such as initial-Hamiltonian optimization or site-dependent Zeeman fields, suppresses critical crossings and drastically enhance ground-state preparation. In contrast, counterdiabatic terms alone do not improve the protocol when the spectral structure remains level-crossings, becoming effective only after degeneracies are removed. These results identify spectral engineering as a prerequisite for efficient adiabatic ground-state preparation in interacting spin systems.
Paper Structure (16 sections, 15 equations, 5 figures, 3 tables)

This paper contains 16 sections, 15 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Model and Adiabatic Protocol
  3. Adiabatic interpolation
  4. Standard adiabatic evolution
  5. Auxiliary Hamiltonian
  6. Initial Hamiltonian optimization
  7. Counterdiabatic driving
  8. Performance metrics
  9. Results
  10. Spectral limitations of standard adiabatic evolution
  11. Effect of auxiliary fields
  12. Initial Hamiltonian optimization
  13. Interplay with counterdiabatic driving
  14. Quantitative comparison of protocols and dynamical behavior
  15. Conclusion
  16. ...and 1 more sections

Figures (5)

  • Figure 1: Energy spectrum as a function of normalized time for pure adiabatic evolution. Left panel $\Delta=0.5$. Central panel $\Delta=1$, Right panel $\Delta=1.5$
  • Figure 2: Standard adiabatic evolution performance to reach the ground state for the XXZ model considering total evolution time $T=300J^{-1}$, $\Delta=0.5$ (blue line), $\Delta=1.0$ (red line) and $\Delta=1.5$ (green line). Top panel: Fidelity between the state at time $t$ against the ground state of final Hamiltonian $H_f$. Bottom panel: Fidelity between the state at time $t$ and the instantaneous ground state of $H(t/T)$, see Eq. (\ref{['Eq02']}).
  • Figure 3: Energy spectrum as a function of normalized time for adiabatic evolution with auxiliary Hamiltonian. Left panel $\Delta=0.5$. Central panel $\Delta=1$, Right panel $\Delta=1.5$
  • Figure 4: Energy spectrum as a function of normalized time for initial Hamiltonian optimization. Left panel $\Delta=0.5$. Central panel $\Delta=1$, Right panel $\Delta=1.5$
  • Figure 5: Optimil initial Hamiltonian with counterdiabatic driving strategy performance considering total evolution time $T=10J^{-1}$, $\Delta=0.5$ (blue line), $\Delta=1.0$ (red line) and $\Delta=1.5$ (green line). Top panel: Fidelity between the state at time $t$ against the ground state of final Hamiltonian $H_f$. Bottom panel: Fidelity between the state at time $t$ and the instantaneous ground state of $H(t/T)$, see Eq. (\ref{['Eq02']}).