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Fast convergence of Majorana Propagation for weakly interacting fermions

Giorgio Facelli, Hamza Fawzi, Omar Fawzi

TL;DR

This work addresses the classical simulability of real-time dynamics for local observables under quartic fermionic Hamiltonians by introducing Majorana Propagation (MP), a truncation-enabled Trotter method that maintains a low-degree representation of the time-evolved operator. The authors prove a rigorous Frobenius-norm error bound linking MP to the best degree-$\ell$ approximant and show efficient simulation for $\Delta$-sparse Hamiltonians; in the weakly interacting regime $H=H_0+uV$ with $u\le1$, a time horizon $t_{\max}(u)$ exists beyond which low-degree approximations persist, enabling overall runtime $N^{O(\log((1+t)/\varepsilon))}$ for prescribed accuracy $\varepsilon$. The key technical advances include degree-dependent Frobenius bounds for commutators and a system-size-independent Trotter error bound achieved via a $G\le4\Delta$ color partition of Hamiltonian terms, together with a constructive, efficient truncation framework. These results illuminate when classical simulation via MP is feasible and lay groundwork for benchmarking quantum dynamics against controlled classical approximations in fermionic systems.

Abstract

Simulating the time dynamics of an observable under Hamiltonian evolution is one of the most promising candidates for quantum advantage as we do not expect efficient classical algorithms for this problem except in restricted settings. Here, we introduce such a setting by showing that Majorana Propagation, a simple algorithm combining Trotter steps and truncations, efficiently finds a low-degree approximation of the time-evolved observable as soon as such an approximation exists. This provides the first provable guarantee about Majorana Propagation for Hamiltonian evolution. As an application of this result, we prove that Majorana Propagation can efficiently simulate the time dynamics of any sparse quartic Hamiltonian up to time $t_{\text{max}}(u)$ depending on the interaction strength $u$. For a time horizon $t \leq t_{\text{max}}(u)$, the runtime of the algorithm is $N^{O(\log(t/\varepsilon))}$ where $N$ is the number of Majorana modes and $\varepsilon$ is the error measured in the normalized Frobenius norm. Importantly, in the limit of small $u$, $t_{\text{max}}(u)$ goes to $+\infty$, formalizing the intuition that the algorithm is accurate at all times when the Hamiltonian is quadratic.

Fast convergence of Majorana Propagation for weakly interacting fermions

TL;DR

This work addresses the classical simulability of real-time dynamics for local observables under quartic fermionic Hamiltonians by introducing Majorana Propagation (MP), a truncation-enabled Trotter method that maintains a low-degree representation of the time-evolved operator. The authors prove a rigorous Frobenius-norm error bound linking MP to the best degree- approximant and show efficient simulation for -sparse Hamiltonians; in the weakly interacting regime with , a time horizon exists beyond which low-degree approximations persist, enabling overall runtime for prescribed accuracy . The key technical advances include degree-dependent Frobenius bounds for commutators and a system-size-independent Trotter error bound achieved via a color partition of Hamiltonian terms, together with a constructive, efficient truncation framework. These results illuminate when classical simulation via MP is feasible and lay groundwork for benchmarking quantum dynamics against controlled classical approximations in fermionic systems.

Abstract

Simulating the time dynamics of an observable under Hamiltonian evolution is one of the most promising candidates for quantum advantage as we do not expect efficient classical algorithms for this problem except in restricted settings. Here, we introduce such a setting by showing that Majorana Propagation, a simple algorithm combining Trotter steps and truncations, efficiently finds a low-degree approximation of the time-evolved observable as soon as such an approximation exists. This provides the first provable guarantee about Majorana Propagation for Hamiltonian evolution. As an application of this result, we prove that Majorana Propagation can efficiently simulate the time dynamics of any sparse quartic Hamiltonian up to time depending on the interaction strength . For a time horizon , the runtime of the algorithm is where is the number of Majorana modes and is the error measured in the normalized Frobenius norm. Importantly, in the limit of small , goes to , formalizing the intuition that the algorithm is accurate at all times when the Hamiltonian is quadratic.
Paper Structure (23 sections, 19 theorems, 93 equations, 2 figures, 1 algorithm)

This paper contains 23 sections, 19 theorems, 93 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. Majorana Propagation for real-time quantum dynamics
  4. Application to weakly interacting Hamiltonians
  5. Overview of techniques
  6. Preliminaries
  7. Majorana algebra
  8. Norms on $\mathcal{A}$
  9. Hamiltonian
  10. Majorana Propagation algorithm
  11. Description of the algorithm
  12. Trotter dynamics
  13. Degree truncation
  14. Majorana Propagation algorithm
  15. Ingredients
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $H$ be a $\Delta$-sparse quartic fermionic Hamiltonian in $N$ Majorana modes (see eq:vertexbounded for the definition of $\Delta$-sparse) and $A$ an arbitrary observable with $\left\lVert A\right\rVert_\textup{F}=1$. Let $t \geq 0$ be a time horizon. Then the Majorana Propagation algorithm with where $A(t)$ is the time-evolved observable eq:intro1timeevolution, and $\eta^\star_{\textup{F}}$ i

Figures (2)

  • Figure 1: Average error for 1D Fermi-Hubbard model with $6$ sites. We plot the distance in Frobenius norm between the MP algorithm output and a pure Trotter evolution, as a function of the maximal degree $\ell$. Different lines indicate different time horizons. In all cases, there is an exponential suppression of the Frobenius norm as the maximal degree is increased. The time-step chosen for discretizing the dynamics is $\delta t = 0.01$.
  • Figure 2: Expectation value of the hole density in central site for $L\times L$ Fermi Hubbard model with $L\in\{3,5,7\}$. We plot, for different values of the interaction strength $U$, the behaviour of the hole density in the central site as a function of time. We compare across multiple values of $\ell\in\{4,6,8,10\}$ to show in a qualitative way that higher $\ell$ converges to the exact result. The time-step is $\delta t =0.02$.

Theorems & Definitions (38)

  • Theorem 1.1: Analysis of Majorana Propagation for time dynamics -- Informal version of Thm. \ref{['thm:majorana-prop']}
  • Theorem 1.2: Low-degree approximability of $A(t)$ for weakly interacting interacting Hamiltonians
  • Corollary 1.3: Efficient simulation of weakly interacting Hamiltonians with Majorana Propagation
  • Proposition 2.1
  • Corollary 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 28 more