Error-correction properties of an interacting topological insulator

Abstract

We analyze the phase diagram of a topological insulator model including antiferromagnetic interactions in the form of an extended Su-Schrieffer Heeger model. To this end, we employ a recently introduced operational definition of topological order based on the ability of a system to perform topological error correction. We show that the necessary error correction statistics can be obtained efficiently using a Monte-Carlo sampling of a matrix product state representation of the ground state wave function. Specifically, we identify two distinct symmetry-protected topological phases corresponding to two different fully dimerized reference states. Finally, we extend the notion of error correction to classify thermodynamic phases to those exhibiting local order parameters, finding a topologically trivial antiferromagnetic phase for sufficiently strong interactions.

Learning of error statistics for the detection of quantum phases

“…The antiferromagnetic order computed using the DNN in Fig. 2(c) is in strong agreement with the phase diagram established using other methods 17,22 . Hence, we have demonstrated that a DNN trained with a certain perturbation of the reference state is capable of detecting the gapped phase in the presence of different perturbations.…”

“…In the following, we introduce our method to detect gapped quantum phases, which is inspired by the operational definition for topological order 16 . We therefore briefly review this approach and extensions to phases exhibiting local order 17 . Essentially, this operational definition interprets topological order as the intrinsic ability of a system to perform topological error correction and classifies topological phases in terms of reference states without any errors.…”

“…The operational definition can be generalized towards phases exhibiting symmetry-protected topological order and phases undergoing spontaneous symmetry breaking exhibiting local order parameters, by imposing symmetry constraints on the error correction circuits 17 . In the following, we will also discuss how these generalizations can be implemented within our machine learning approach.…”

“…For this, we weakly perturb the reference state as well as a disordered paramagnetic state, for which we know which phase the resulting states are in. Using a Monte-Carlo sampling 17 , we obtain simulated measurement data representing snapshots of the errors in the perturbed ground state. This training set is then fed into a deep neural network (DNN), see.…”

“…These states are said to have symmetry-protected topological (SPT) order. In the context of the recently introduced operational definition 17 , a state is said to be SPT ordered if it FIG. 3.…”