Pauli Propagation for Imaginary-Time Evolution

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.