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Practical limitations of the switching theorem for adiabatic state preparation

Thomas D. Cohen, Andrew Li, Hyunwoo Oh, Maneesha Sushama Pradeep

Abstract

The viability of adiabatic quantum computation depends on the slow evolution of the Hamiltonian. The adiabatic switching theorem provides an asymptotic series for error estimates in $1/T$, based on the lowest non-zero derivative of the Hamiltonian and its eigenvalues at the endpoints. Modifications at the endpoints in practical implementations can modify this scaling behavior, suggesting opportunities for error reduction by altering endpoint behavior while keeping intermediate evolution largely unchanged. Such modifications can significantly reduce errors for long evolution times, but they may also require exceedingly long timescales to reach the hyperadiabatic regime, limiting their practicality. This paper explores the transition between the adiabatic and hyperadiabatic regimes in simple low-dimensional Hamiltonians, highlighting the impact of modifications of the endpoints on approaching the asymptotic behavior described by the switching theorem.

Practical limitations of the switching theorem for adiabatic state preparation

Abstract

The viability of adiabatic quantum computation depends on the slow evolution of the Hamiltonian. The adiabatic switching theorem provides an asymptotic series for error estimates in , based on the lowest non-zero derivative of the Hamiltonian and its eigenvalues at the endpoints. Modifications at the endpoints in practical implementations can modify this scaling behavior, suggesting opportunities for error reduction by altering endpoint behavior while keeping intermediate evolution largely unchanged. Such modifications can significantly reduce errors for long evolution times, but they may also require exceedingly long timescales to reach the hyperadiabatic regime, limiting their practicality. This paper explores the transition between the adiabatic and hyperadiabatic regimes in simple low-dimensional Hamiltonians, highlighting the impact of modifications of the endpoints on approaching the asymptotic behavior described by the switching theorem.

Paper Structure

This paper contains 10 sections, 18 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Switching theorem
  3. Switching theorem in practice
  4. Switching on of the switching theorem
  5. A subtlety
  6. Examples
  7. Two-level system
  8. Three-level system
  9. Extreme Scenario: Essential Singularity at the endpoints
  10. Discussion

Figures (10)

  • Figure 1: $f(t;k)$ with various $k$. $k=0$ corresponds to $f_0(t)$, whose first-order derivatives are not zero at the endpoints.
  • Figure 2: Comparison of errors, $\bar{\epsilon}_T$, $\bar{\epsilon}_1$, and $\bar{\epsilon}_2$, with $k=10^{-3}$ and $(E_0, E_1) = (1, 1)$. The left plot shows the ratio of switching errors and the averaged true error with the solid line being $\bar{\epsilon}_1$ and the dotted line being $\bar{\epsilon}_2$. The right panel displays the three average errors with respect to the timescale. Note that the y-axis for the right plot is $\bar{\epsilon}$ times $T^2$, not just the error $\bar{\epsilon}$, for visualizing the scaling behavior.
  • Figure 3: Comparison of average errors, $\bar{\epsilon}_T$, $\bar{\epsilon}_1$, and $\bar{\epsilon}_2$, with two $k$, $10^{-4}$ and $10^{-3}$, and $(E_0, E_1) = (1, 1)$. Note that switching errors can be larger than 1 since they are estimation, but true errors, Eq. (\ref{['Eq:errordef']}), are always less than 1.
  • Figure 4: Comparison of average errors, $\bar{\epsilon}_1$, and $\bar{\epsilon}_2$, with respect to $\bar{\epsilon}_T$, for the three-level system with identical off-diagonal elements at a fixed $k_1 =k_2 =k_3 = 10^{-4}$, while varying $T$.
  • Figure 5: Comparison of the error ratio, $\bar{\epsilon}_2$/$\bar{\epsilon}_T$, by changing $k$ for the three-level system with identical off-diagonal elements, Eq. (\ref{['Eq:H_3levelex1']}).
  • ...and 5 more figures