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Numerical study of computational cost of maintaining adiabaticity for long paths

Thomas D. Cohen, Hyunwoo Oh, Veronica Wang

TL;DR

The paper investigates how the cost of maintaining adiabaticity along a Hamiltonian path scales with path length when the final error is fixed. By numerically testing simple non-periodic, low-dimensional Hamiltonians and evaluating the dimensionless proxy $Q_D$ (and variants), the authors provide evidence that $Q_D$ grows roughly as $L\log L$, i.e., superlinearly in $L$ in the large-path limit. This supports the conjecture that path-length scaling captures adiabatic cost better than time and suggests that adiabatic state preparation may be competitive only for moderate path lengths, with linear-scaling or projection-based methods preferable for very long paths. The study highlights that the scaling is generically superlinear but permits special linear cases, guiding future work toward broader Hamiltonians and optimized traversal strategies.

Abstract

Recent work argued that the scaling of a dimensionless quantity $Q_D$ with path length is a better proxy for quantifying the scaling of the computational cost of maintaining adiabaticity than the timescale. It also conjectured that generically the scaling will be superlinear (although special cases exist in which it is linear). The quantity $Q_D$ depends only on the properties of ground states along the Hamiltonian path and the rate at which the path is followed. In this paper, we demonstrate that this conjecture holds for simple Hamiltonian systems that can be studied numerically. In particular, the systems studied exhibit the behavior that $Q_D$ grows approximately as $L \log L$ where $L$ is the path length when the threshold error is fixed.

Numerical study of computational cost of maintaining adiabaticity for long paths

TL;DR

The paper investigates how the cost of maintaining adiabaticity along a Hamiltonian path scales with path length when the final error is fixed. By numerically testing simple non-periodic, low-dimensional Hamiltonians and evaluating the dimensionless proxy (and variants), the authors provide evidence that grows roughly as , i.e., superlinearly in in the large-path limit. This supports the conjecture that path-length scaling captures adiabatic cost better than time and suggests that adiabatic state preparation may be competitive only for moderate path lengths, with linear-scaling or projection-based methods preferable for very long paths. The study highlights that the scaling is generically superlinear but permits special linear cases, guiding future work toward broader Hamiltonians and optimized traversal strategies.

Abstract

Recent work argued that the scaling of a dimensionless quantity with path length is a better proxy for quantifying the scaling of the computational cost of maintaining adiabaticity than the timescale. It also conjectured that generically the scaling will be superlinear (although special cases exist in which it is linear). The quantity depends only on the properties of ground states along the Hamiltonian path and the rate at which the path is followed. In this paper, we demonstrate that this conjecture holds for simple Hamiltonian systems that can be studied numerically. In particular, the systems studied exhibit the behavior that grows approximately as where is the path length when the threshold error is fixed.

Paper Structure

This paper contains 12 sections, 14 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Errors
  4. Path length
  5. A no-go theorem
  6. Proxies for the computational cost
  7. Quantities reflecting computational costs
  8. Conjecture
  9. Approach for numerical tests
  10. Results
  11. Random Hamiltonians
  12. Discussion

Figures (3)

  • Figure 1: Comparison of the path length with different dimensions of Hamiltonians at fixed end time $T_{\rm end}=10$. 100 Hamiltonians are randomly chosen for each dimension in order to estimate means and standard deviations. The form of Hamiltonians are given by Eq. (\ref{['Eq:RandomH']}).
  • Figure 2: Comparison of different $Q_D$'s suggested in Eq. (\ref{['Eq:Qds']}) for a Hamiltonian with Eq. (\ref{['Eq:RandomH']}).
  • Figure 3: $Q_D$ is plotted with respect to the path length using four different choices of random Hamiltonians with the form of Eq. (\ref{['Eq:RandomH']}). The threshold error $\epsilon_{\rm th}$ is chosen as 0.1. $R_i$ denotes the different random matrices for $H_1$ and $H_2$ in Eq. (\ref{['Eq:RandomH']}).