Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Pauli Propagation: Simulating Quantum Spin Dynamics via Operator Complexity

Yuguo Shao, Song Cheng, Zhengwei Liu

TL;DR

The paper tackles the challenge of simulating real-time quantum spin dynamics without exponential state growth by introducing Pauli propagation, an observable-centric approach that evolves local operators in the Heisenberg picture via back-propagation of Pauli terms under a Trotter sequence. Truncation is controlled with a Top-$K$ strategy, and a priori error bounds are derived using the Operator Stabilizer Rényi entropy (OSE) $ ext{S}^ extalpha(O)$, yielding explicit prescriptions for the retained Pauli terms $K$ to meet a target accuracy. A key theoretical result shows that, for the 1D $J_z=0$ Heisenberg/XY model, the number of nonzero Pauli coefficients in evolved operators grows only as $ ext{O}(s^2)$, demonstrating operator compressibility; numerically, the method achieves high accuracy with modest $K$ in free regimes and competes with tensor-network methods in interacting regimes. Overall, the work provides a scalable, entropy-guided alternative to state-based methods for non-equilibrium dynamics, with potential applicability to transport and information-scrambling diagnostics where operator complexity, not entanglement, governs computational cost.

Abstract

Simulating real-time quantum dynamics in interacting spin systems is a fundamental challenge, where exact diagonalization suffers from exponential Hilbert-space growth and tensor-network methods face entanglement barriers. In this work, we introduce a scalable Pauli propagation approach that evolves local observables directly in the Heisenberg picture. Theoretically, we derive a priori error bounds governed by the Operator Stabilizer Rényi entropy (OSE) $\mathcal{S}^α(O)$, which explicitly links the truncation accuracy to operator complexity and prescribes a suitable Top-$K$ truncation strategy. For the 1D Heisenberg model with $J_z = 0$, we prove the number of non-zero Pauli coefficients scales quadratically in Trotter steps, establishing the compressibility of Heisenberg-evolved operators. Numerically, we validate the framework on XXZ Heisenberg chain benchmarks, showing high accuracy with small $K$ in free regimes ($J_z = 0$) and competitive performance against tensor-network methods (e.g., TDVP) in interacting cases ($J_z = 0.5$). These results establish an observable-centric simulator whose cost is governed by operator complexity rather than entanglement, offering a practical alternative for studying non-equilibrium dynamics in quantum many-body systems.

Pauli Propagation: Simulating Quantum Spin Dynamics via Operator Complexity

TL;DR

The paper tackles the challenge of simulating real-time quantum spin dynamics without exponential state growth by introducing Pauli propagation, an observable-centric approach that evolves local operators in the Heisenberg picture via back-propagation of Pauli terms under a Trotter sequence. Truncation is controlled with a Top- strategy, and a priori error bounds are derived using the Operator Stabilizer Rényi entropy (OSE) , yielding explicit prescriptions for the retained Pauli terms to meet a target accuracy. A key theoretical result shows that, for the 1D Heisenberg/XY model, the number of nonzero Pauli coefficients in evolved operators grows only as , demonstrating operator compressibility; numerically, the method achieves high accuracy with modest in free regimes and competes with tensor-network methods in interacting regimes. Overall, the work provides a scalable, entropy-guided alternative to state-based methods for non-equilibrium dynamics, with potential applicability to transport and information-scrambling diagnostics where operator complexity, not entanglement, governs computational cost.

Abstract

Simulating real-time quantum dynamics in interacting spin systems is a fundamental challenge, where exact diagonalization suffers from exponential Hilbert-space growth and tensor-network methods face entanglement barriers. In this work, we introduce a scalable Pauli propagation approach that evolves local observables directly in the Heisenberg picture. Theoretically, we derive a priori error bounds governed by the Operator Stabilizer Rényi entropy (OSE) , which explicitly links the truncation accuracy to operator complexity and prescribes a suitable Top- truncation strategy. For the 1D Heisenberg model with , we prove the number of non-zero Pauli coefficients scales quadratically in Trotter steps, establishing the compressibility of Heisenberg-evolved operators. Numerically, we validate the framework on XXZ Heisenberg chain benchmarks, showing high accuracy with small in free regimes () and competitive performance against tensor-network methods (e.g., TDVP) in interacting cases (). These results establish an observable-centric simulator whose cost is governed by operator complexity rather than entanglement, offering a practical alternative for studying non-equilibrium dynamics in quantum many-body systems.
Paper Structure (16 sections, 5 theorems, 49 equations, 5 figures, 1 algorithm)

This paper contains 16 sections, 5 theorems, 49 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Pauli Propagation Method
  3. Analytical Bounds
  4. 1D Heisenberg system
  5. Conclusions and Discussions
  6. Computational Cost
  7. Pauli conjugation
  8. Per Hamiltonian term within one time step.
  9. Per time step and total evolution.
  10. Final rescaling and expectation evaluation.
  11. Proof of Lemma \ref{['lemma:prop_entropy']}
  12. Proof of Lemma \ref{['lemma:error_bound']}
  13. Proof of Thm. \ref{['thm:sim_error_entropy']}
  14. Proof of Thm. \ref{['thm:1d_XY_trotter']}
  15. Weight Truncation
  16. ...and 1 more sections

Key Result

lemma 1

The OSE $\mathcal{S}^\alpha(O)$ satisfies the following properties:

Figures (5)

  • Figure 1: Time-evolution results for the one-dimensional Heisenberg model with length $L=50$, obtained using the Pauli propagation and TDVP methods, respectively. The orange lines denote TDVP results for different maximum bond dimensions $D$, and the blue lines denote Pauli propagation results for different $K$ values. For one representative curve, we also compare the absolute deviations from the exact results. (a) For $J_x=J_y=1$ and $J_z=0$, Pauli propagation attains accurate results at tiny cost, whereas for TDVP the error accumulates rapidly once the entanglement entropy exceeds the MPS capacity. (b) For $J_z \neq 0$, TDVP behaves similarly, while the computational cost of Pauli propagation is strongly affected by $J_z$ and becomes comparable to TDVP.
  • Figure 2: $\alpha = 1/2$ and $\alpha \rightarrow 1$(Shannon) OSE over time. Where the blue line denotes the entropy of the Pauli Z operator with time evolution on the open boundary, and the orange line denotes the entropy corresponding to the Pauli Z operator with time evolution at the center of the one-dimensional chain.
  • Figure 3: Growth behavior of Pauli words over time for different $J_z$ values.
  • Figure 4: Distribution of the squared Pauli coefficients as a function of time for different values of $J_z$.
  • Figure 5: Pauli weight distribution of the Pauli words coefficients as a function of time for different values of $J_z$.

Theorems & Definitions (11)

  • Definition 1: Operator Stabilizer Rényi entropy dowling2025magic
  • lemma 1
  • lemma 2
  • Theorem 1
  • Theorem 2
  • proof
  • proof
  • proof
  • proof
  • Theorem 3
  • ...and 1 more