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The Constant Speed Schedule for Adiabatic State Preparation: Towards Quadratic Speedup without Prior Spectral Knowledge

Mancheon Han, Hyowon Park, Sangkook Choi

TL;DR

The paper tackles the efficiency of adiabatic state preparation by linking evolution time to the spectral gap and introducing a constant-speed schedule that traverses the adiabatic path at uniform geometric speed. By reframing the problem with path length and curvature, the authors derive an $O(Δ^{-1})$ scaling bound, one order better than typical $O(Δ^{-2})$ results, and implement a practical, overlaps-based segmentation guided by Quantum Zeno Monte Carlo projections to realize the schedule without prior spectral knowledge. Across adiabatic Grover search and quantum-chemistry benchmarks (N$_2$ and [2Fe-2S]), the method achieves the optimal $Δ^{-1}$ scaling and delivers substantial speedups over linear schedules, while also improving robustness to initial-state variations. The work provides a broadly applicable, spectra-uninformed tool for quantum simulation with significant practical impact for achieving faster, more reliable adiabatic state preparation.

Abstract

The efficiency of adiabatic quantum evolution is governed by the adiabatic evolution time, \(T\), which depends on the minimum energy gap, \(Δ\). For a generic schedule, \(T\) typically scales as \(Δ^{-2}\), whereas the rigorous lower bound is \(\mathcal{O}(Δ^{-1})\). This indicates the potential for a quadratic speedup through the adiabatic schedule construction. Here, we introduce the constant speed schedule, which traverses the adiabatic path of the eigenstate at a uniform rate. We first show that this approach reduces the scaling of the upper bound of the required evolution time by one order in \(1/Δ\). We then provide a segmented constant speed schedule protocol, in which path segment lengths are computed from eigenstate overlaps along the adiabatic evolution. By relying on the overlaps on the fly, our method eliminates the need for prior spectral knowledge. We test our algorithm numerically on the adiabatic unstructured search, the N$_2$ molecule, and the [2Fe-2S] cluster. In our numerical experiments, the method achieves the optimal \(1/Δ\) scaling in a small gap region, thereby demonstrating a quadratic speedup over the standard linear schedule.

The Constant Speed Schedule for Adiabatic State Preparation: Towards Quadratic Speedup without Prior Spectral Knowledge

TL;DR

The paper tackles the efficiency of adiabatic state preparation by linking evolution time to the spectral gap and introducing a constant-speed schedule that traverses the adiabatic path at uniform geometric speed. By reframing the problem with path length and curvature, the authors derive an scaling bound, one order better than typical results, and implement a practical, overlaps-based segmentation guided by Quantum Zeno Monte Carlo projections to realize the schedule without prior spectral knowledge. Across adiabatic Grover search and quantum-chemistry benchmarks (N and [2Fe-2S]), the method achieves the optimal scaling and delivers substantial speedups over linear schedules, while also improving robustness to initial-state variations. The work provides a broadly applicable, spectra-uninformed tool for quantum simulation with significant practical impact for achieving faster, more reliable adiabatic state preparation.

Abstract

The efficiency of adiabatic quantum evolution is governed by the adiabatic evolution time, , which depends on the minimum energy gap, . For a generic schedule, typically scales as , whereas the rigorous lower bound is \(\mathcal{O}(Δ^{-1})\). This indicates the potential for a quadratic speedup through the adiabatic schedule construction. Here, we introduce the constant speed schedule, which traverses the adiabatic path of the eigenstate at a uniform rate. We first show that this approach reduces the scaling of the upper bound of the required evolution time by one order in . We then provide a segmented constant speed schedule protocol, in which path segment lengths are computed from eigenstate overlaps along the adiabatic evolution. By relying on the overlaps on the fly, our method eliminates the need for prior spectral knowledge. We test our algorithm numerically on the adiabatic unstructured search, the N molecule, and the [2Fe-2S] cluster. In our numerical experiments, the method achieves the optimal scaling in a small gap region, thereby demonstrating a quadratic speedup over the standard linear schedule.

Paper Structure

This paper contains 11 sections, 3 theorems, 41 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Methods
  3. Adiabatic Evolution Error
  4. The Constant Speed Schedule
  5. The implementation of the Constant Speed Schedule
  6. Applications
  7. Adiabatic Grover Search
  8. Quantum Chemistry
  9. Nitrogen Molecule (N2)
  10. [2Fe-2S] cluster
  11. Conclusion

Key Result

Lemma 1

Let $\ket{\Phi(s)}$ be a normalized and differentiable family of eigenstates of a parameter-dependent Hamiltonian $H(s)$, describing an adiabatic path for $s \in [0,1]$. Then, to the leading order in $\Delta s$, the overlap between nearby eigenstates satisfies where $\Delta l$ is defined in Eq. eq:delta_l.

Figures (4)

  • Figure 1: Schematic of the constant speed schedule construction. The procedure, implemented in Algorithm \ref{['alg:css_qzmc']}, computes the adiabatic path segment length $\Delta l$ from the overlap $|\langle\Phi(s)|\Phi(s+\Delta s)\rangle|^2$. The evolution time for each segment, $\Delta t$, is then adjusted to maintain a constant geometric speed, $\Delta l / \Delta t \approx \text{const}$.
  • Figure 2: Application of the constant speed schedule to the adiabatic Grover search. Panels (a--c) show results for $N = 2^{14}$. (a) Energy spectrum of the adiabatic Grover Hamiltonian $H(s)$, aligned with panel (b). (b) Comparison of the segmented constant speed schedule ($\hat{s}_c$), the exact optimal schedule ($s_c$), and the linear schedule ($s_l$); circles mark calculated points and the red line is the interpolation. (c) Fidelity $\mathcal{F}$ as a function of the total evolution time $T$, comparing $\hat{s}_c$ and $s_l$. (d) Total evolution time $T$ required to reach $\mathcal{F}=0.75$ versus the minimum gap $\Delta$. Circles show numerical results and dashed lines indicate fitted curves.
  • Figure 3: Application of the constant speed schedule to the nitrogen molecule. (a) Low-lying energy spectra from Density Functional Theory (DFT, blue) and Full Configuration Interaction (FCI, red) calculations. (b) Comparison of the segmented constant speed schedule ($\hat{s}_c$, red) with the linear schedule ($s_l$, black) at bond length $R = 3.5\,\text{\AA}$. (c) Fidelity $\mathcal{F}$ as a function of the evolution time $T$. (d) Evolution time $T$ required to reach 75% fidelity versus the minimum gap $\Delta$, showing the optimal $\Delta^{-1}$ scaling of $\hat{s}_c$ (red) compared with the $\Delta^{-2}$ scaling of $s_l$ (black). The total time including projection overhead (blue), also follows the optimal scaling.
  • Figure 4: Application of the constant speed schedule to the [2Fe-2S] cluster. (a) Molecular structure. (b) Estimated adiabatic evolution time $T_\textrm{ASP}^\textrm{est}$ versus the minimum energy gap $\Delta$, showing the optimal $\Delta^{-1}$ scaling for $\hat{s}_c$ (red) compared with the $\Delta^{-2}$ scaling of $s_l$ (black). (c,d) Results for the initial state chosen as the DFT ground state, corresponding to the crosses in (b). (c) Comparison of schedules. (d) Fidelity $\mathcal{F}$ as a function of evolution time $T$, showing a speedup of more than two orders of magnitude with $\hat{s}_c$.

Theorems & Definitions (3)

  • Lemma 1
  • Theorem 1
  • Theorem 2