Pauli Propagation for Imaginary Time Evolution

TL;DR

The paper addresses computing thermal and ground-state properties of quantum many-body systems by extending Pauli Propagation to imaginary time. It derives explicit imaginary-time Pauli update rules and builds the imaginary-time Pauli Propagation (ITPP) algorithm, combining Pauli-basis evolution with a Trotter approximation and truncation schemes. Through a TFIM benchmark, the work demonstrates controllable accuracy-cost trade-offs and highlights operator-growth challenges in non-unitary evolution, while suggesting a path toward open-system dynamics by unifying imaginary- and real-time propagation. Overall, ITPP provides a flexible operator-based framework for accessing thermal states and ground-state properties in small-to-intermediate systems, setting a baseline for future improvements in truncation strategies and open-system simulations.

Abstract

We extend the Pauli Propagation framework to simulate imaginary time evolution. By deriving explicit update rules for the propagation of Pauli operators under imaginary time evolution generated by Pauli strings, we introduce an imaginary time Pauli Propagation (ITPP) algorithm for approximating imaginary time dynamics directly in the Pauli basis. This approach enables the computation of thermal and ground-state properties while retaining the key computational advantages of Pauli Propagation. Benchmarking ITPP on the one-dimensional transverse-field Ising model demonstrates that truncation provides a controlled trade-off between accuracy and computational cost, while also revealing challenges associated with operator growth under imaginary time evolution. Finally, combining imaginary time and real-time Pauli Propagation naturally suggests a pathway toward simulating open quantum system dynamics within a unified framework.

Figures (3)

• Figure 1: Comparison of Pauli Propagation imaginary time evolution (ITPP) with Trotterized imaginary time evolution and exact imaginary time evolution for the one-dimensional transverse-field Ising model (TFIM) with $N = 10$ spins, $J = 1$, and $h = 0.5$, using a Trotter step size $\Delta t = 0.04$. The left panel shows the evolution of the energy as a function of imaginary time, while the right panel displays the corresponding relative error with respect to the exact imaginary time evolution, shown on a logarithmic scale. The comparison illustrates the accuracy of ITPP relative to standard Trotterized evolution and the exact result over the course of imaginary time propagation.
• Figure 2: Performance of ITPP with coefficient-threshold truncation for various thresholds in the ordered-phase TFIM ($J = 1$, $h = 0.5$) with $N = 12$ spins and Trotter step $\Delta t = 0.04$. The three panels show, from left to right, the energy during imaginary time evolution, the relative error with respect to the exact ground-state energy (displayed on a logarithmic scale), and the number of Pauli terms retained under different truncation thresholds. In the left-most panel, the solid black line indicates the exact ground-state energy computed using the Bogoliubov–de Gennes solution. The results illustrate the trade-off between accuracy and computational cost controlled by the truncation threshold.
• Figure 3: Scalability of imaginary time Pauli Propagation with fixed-$K$ truncation in the ordered phase of the one-dimensional TFIM ($J = 1$, $h = 0.5$). The left panel shows the converged ground-state energy obtained with ITPP as a function of system size $N$, while the right panel displays the corresponding relative error (on a logarithmic scale) with respect to the exact ground-state energy computed using the Bogoliubov--de Gennes solution. The truncation parameter is fixed to $K = 2{,}704{,}156$, equal to the number of Pauli terms generated for $N = 12$ in the absence of truncation. The results indicate the range of system sizes for which this fixed Pauli basis size is sufficient to accurately capture the underlying operator growth under imaginary time evolution, as well as the onset of a breakdown when the fixed computational budget becomes inadequate.