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Improved gap dependence in adiabatic state preparation by adaptive schedule

Xi Guo, Dong An

TL;DR

Adiabatic quantum computing typically requires runtime scaling as ${\mathcal O}(1/\Delta_*^2)$ due to the spectral gap, motivating adaptive scheduling. The authors propose a nonlinear power-law schedule with $u'(s) \propto \Delta^p(u(s))$ ($p\in(1,2)$) under a gap-measure condition $\mu(\{\Delta\le x\})=\mathcal{O}(x)$, achieving an adiabatic error scaling of ${\mathcal O}(\Delta_*^{-1})$ and a runtime $T \sim {\mathcal O}(1/\Delta_*)$, i.e., a quadratic improvement in gap dependence. Through a variational analysis, they show that for linear gaps the power-law schedule with $p=3/2$ is optimal, and it remains practically favorable for general gaps, while the linear schedule is not optimal when the gap varies. The framework covers important applications such as adiabatic Grover search and adiabatic quantum linear system algorithms, offering a principled, gap-aware approach to faster adiabatic state preparation under known spectral structure.

Abstract

Adiabatic quantum computing is a powerful framework for state preparation, while its evolution time often scales quadratically in the inverse Hamiltonian spectral gap, leading to sub-optimal computational complexity. In this work, we introduce a nonlinear adaptive strategy for finding the time scheduling function, and show that the gap dependence can be quadratically improved to be inverse linear for a wide range of systems under a mild gap measure condition. Through variational analysis, we further demonstrate the optimality of our schedule for systems with linear gap and the partial optimality for general systems, while we also rigorously show that the commonly used linear schedule is never optimal.

Improved gap dependence in adiabatic state preparation by adaptive schedule

TL;DR

Adiabatic quantum computing typically requires runtime scaling as due to the spectral gap, motivating adaptive scheduling. The authors propose a nonlinear power-law schedule with () under a gap-measure condition , achieving an adiabatic error scaling of and a runtime , i.e., a quadratic improvement in gap dependence. Through a variational analysis, they show that for linear gaps the power-law schedule with is optimal, and it remains practically favorable for general gaps, while the linear schedule is not optimal when the gap varies. The framework covers important applications such as adiabatic Grover search and adiabatic quantum linear system algorithms, offering a principled, gap-aware approach to faster adiabatic state preparation under known spectral structure.

Abstract

Adiabatic quantum computing is a powerful framework for state preparation, while its evolution time often scales quadratically in the inverse Hamiltonian spectral gap, leading to sub-optimal computational complexity. In this work, we introduce a nonlinear adaptive strategy for finding the time scheduling function, and show that the gap dependence can be quadratically improved to be inverse linear for a wide range of systems under a mild gap measure condition. Through variational analysis, we further demonstrate the optimality of our schedule for systems with linear gap and the partial optimality for general systems, while we also rigorously show that the commonly used linear schedule is never optimal.

Paper Structure

This paper contains 18 sections, 10 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Quadratically improved error bound
  4. Variational optimality analysis
  5. Related works
  6. Organization
  7. Preliminary
  8. Set up
  9. Quantum Adiabatic Theorems (QATs)
  10. The improvement of error bound
  11. Measure condition for spectral gap
  12. Power-law scheduling function
  13. Error bound enhancement via schedule optimization
  14. Optimality analysis
  15. Variational method
  16. ...and 3 more sections

Key Result

Lemma 1

The adiabatic approximation error satisfies where Here $H^{(k)}(s) = \frac{\mathrm{d}^k}{\mathrm{d}s^k}H(u(s))=\left(H_1-H_0\right)\frac{\mathrm{d}^k}{\mathrm{d}s^k}u(s)$ denotes the $k$-th derivative with respect to $s$ for $k = 1, 2$, and $C$ is a constant independent of $s$, $\Delta$, and $T$.

Theorems & Definitions (13)

  • Lemma 1: QAT JansenRuskaiSeiler2007
  • Corollary 2: QATs for linear scheduling function
  • Definition 1: Spectral gap measure condition
  • Definition 2: Power-law scheduling condition
  • Theorem 3
  • Lemma 4: Spectral gap integral bound
  • Theorem 5
  • Theorem 6: First-order optimality condition
  • Theorem 7: Non-constant gap superiority
  • Theorem 8: Second-order derivative component optimality
  • ...and 3 more