Clifford-dressed variational principles for precise Loschmidt echoes

We extend the recently introduced Clifford-dressed time-dependent variational principle (TDVP) to efficiently compute many-body wave-function amplitudes in the computational basis. This advancement enhances the study of Loschmidt echoes, which generally require accurate calculations of the overlap between the evolved state and the initial wave function. By incorporating Clifford-disentangling gates during TDVP evolution, our method effectively controls entanglement growth while keeping the computation of these amplitudes accessible. Specifically, it reduces the problem to evaluating the overlap between a matrix product state (MPS) and a stabilizer state, a task that remains computationally feasible within the proposed framework. To demonstrate the effectiveness of this approach, we first benchmark it on the one-dimensional transverse-field Ising model. We then apply it to more challenging scenarios, including a nonintegrable next-to-nearest-neighbor Ising chain and a two-dimensional Ising model. Our results highlight the versatility and efficiency of the Clifford-augmented MPS, showcasing its capability to go beyond the evaluation of simple expectation values. This makes it a powerful tool for exploring various aspects of many-body quantum dynamics.