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Time-Dependent Hamiltonian Simulation in the Low-Energy Subspace

Shuo Zhou, Zhaokai Pan, Weiyuan Gong, Tongyang Li

TL;DR

This work addresses time-dependent Hamiltonian simulation under a low-energy subspace constraint, showing that product-formula simulations can be significantly more efficient when the initial state is confined to a fixed number of low-energy eigenstates. By integrating adiabatic perturbation theory with projection lemmas and Floquet-based residual analysis, the authors derive commutator-scaling error bounds and explicit Trotter-number bounds that reduce dependence on system size, while quantifying leakage across the evolving spectral subspace. The results yield concrete r bounds for generalized and standard product formulas, with an explicit adiabatic regime where the reduction is exponential in the system size, and they provide a matching lower bound for generic time-dependent simulations. The paper also demonstrates practical impact through applications to non-equilibrium quantum dynamics and adiabatic state preparation, highlighting that low-energy structure can substantially improve quantum simulation performance.

Abstract

Hamiltonian simulations are key subroutines in adiabatic quantum computation, quantum control, and quantum many-body physics, where quantum dynamics often happen in the low-energy sector. In contrast to time-independent Hamiltonian simulations, a comprehensive understanding of quantum simulation algorithms for time-dependent Hamiltonians under the low-energy assumption remains limited hitherto. In this paper, we investigate how much we can improve upon the standard performance guarantee assuming the initial state is supported on a low-energy subspace. In particular, we compute the Trotter number of digital quantum simulation based on product formulas for time-dependent spin Hamiltonians under the low-energy assumption that the initial state is supported on a small number of low-energy eigenstates, and show improvements over the standard cost for simulating full unitary simulations. Technically, we derive the low-energy simulation error with commutator scaling for product formulas by leveraging adiabatic perturbation theory to analyze the time-variant energy spectrum of the underlying Hamiltonian. We further discuss the applications to simulations of non-equilibrium quantum many-body dynamics and adiabatic state preparation. Finally, we prove a lower bound of query complexity for generic time-dependent Hamiltonian simulations.

Time-Dependent Hamiltonian Simulation in the Low-Energy Subspace

TL;DR

This work addresses time-dependent Hamiltonian simulation under a low-energy subspace constraint, showing that product-formula simulations can be significantly more efficient when the initial state is confined to a fixed number of low-energy eigenstates. By integrating adiabatic perturbation theory with projection lemmas and Floquet-based residual analysis, the authors derive commutator-scaling error bounds and explicit Trotter-number bounds that reduce dependence on system size, while quantifying leakage across the evolving spectral subspace. The results yield concrete r bounds for generalized and standard product formulas, with an explicit adiabatic regime where the reduction is exponential in the system size, and they provide a matching lower bound for generic time-dependent simulations. The paper also demonstrates practical impact through applications to non-equilibrium quantum dynamics and adiabatic state preparation, highlighting that low-energy structure can substantially improve quantum simulation performance.

Abstract

Hamiltonian simulations are key subroutines in adiabatic quantum computation, quantum control, and quantum many-body physics, where quantum dynamics often happen in the low-energy sector. In contrast to time-independent Hamiltonian simulations, a comprehensive understanding of quantum simulation algorithms for time-dependent Hamiltonians under the low-energy assumption remains limited hitherto. In this paper, we investigate how much we can improve upon the standard performance guarantee assuming the initial state is supported on a low-energy subspace. In particular, we compute the Trotter number of digital quantum simulation based on product formulas for time-dependent spin Hamiltonians under the low-energy assumption that the initial state is supported on a small number of low-energy eigenstates, and show improvements over the standard cost for simulating full unitary simulations. Technically, we derive the low-energy simulation error with commutator scaling for product formulas by leveraging adiabatic perturbation theory to analyze the time-variant energy spectrum of the underlying Hamiltonian. We further discuss the applications to simulations of non-equilibrium quantum many-body dynamics and adiabatic state preparation. Finally, we prove a lower bound of query complexity for generic time-dependent Hamiltonian simulations.
Paper Structure (22 sections, 12 theorems, 132 equations, 1 figure, 2 tables)

This paper contains 22 sections, 12 theorems, 132 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Techniques
  4. Open questions
  5. Preliminaries
  6. Notations
  7. Model and setup
  8. Projection lemmas
  9. Product formula for time-dependent Hamiltonian simulation
  10. Error Analysis of Time-Dependent Low-Energy Simulation
  11. Nested commutator of $k$-local multi-linear Hamiltonian
  12. Short-time simulation error
  13. Long-time simulation error
  14. Applications
  15. Non-equilibrium quantum many-body dynamics
  16. ...and 7 more sections

Key Result

Theorem 1

When the initial state is only supported on the low-energy subspace, the Trotter number of both $p$-th order generalized and standard time-dependent product formula can be reduced to In the adiabatic regime where $T = \Omega(\frac{gN}{\Delta}\cdot(\sigma\frac{\|{\dot{H}}\|}{\gamma^2}+\sigma\sqrt{\sigma} \frac{\|{\dot{H}}\|^2}{\gamma^3}+\sigma\frac{\|{\ddot{H}}\|}{\gamma^2}))$, it suffices to choo

Figures (1)

  • Figure 1: Spectral Flow of Time-Dependent Hamiltonians. We define the spectral projector onto the subspace spanned by the three low-energy eigenstates in blue, which has no overlap with the remaining eigenstates in red. In this illustrated case $\sigma=3$, $\Pi_{3}(t)=\ket{\psi_1(t)}\bra{\psi_1(t)}+\ket{\psi_2(t)}\bra{\psi_2(t)}+\ket{\psi_3(t)}\bra{\psi_3(t)}$.

Theorems & Definitions (18)

  • Theorem 1: Informal
  • Definition 1
  • Lemma 1: Theorem 3 of jansen_bounds_2007
  • Lemma 2: Time-dependent version of Theorem 2.1 of arad2016connecting
  • proof
  • Lemma 3: Theorem S1 and Corollary S2 of Mizuta_2025
  • Lemma 4
  • Lemma 5
  • Theorem 2: Trotter number for simulating non-equilibrium quantum many-body dynamics
  • Theorem 3: Trotter number for simulating adiabatic state preparation
  • ...and 8 more