Computational Characterization of Symmetry-Protected Topological Phases in Open Quantum Systems

TL;DR

This paper introduces a method using gate fidelity to identify symmetry-protected topological phases in open quantum systems, analyzing the robustness of the Haldane phase under noise and symmetry conditions.

Contribution

It proposes a novel fidelity-based criterion for characterizing SPT phases in mixed states, linking topological features to computational robustness in noisy environments.

Findings

Gate fidelity depends on noise symmetry conditions.

String order parameters relate to identity gate fidelity.

Stronger symmetry conditions are needed for other gates.

Abstract

It is a challenging problem to correctly characterize the symmetry-protected topological (SPT) phases in open quantum systems. As the measurement-based quantum computation (MBQC) utilizes non-trivial edge states of the SPT phases as the logical qubit, its computational power is closely tied to the non-trivial topological nature of the phases. In this paper, we propose to use the gate fidelity which is a measure of the computational power of the MBQC to identify the SPT phases in mixed-state settings. Specifically, we investigate the robustness of the Haldane phase by considering the MBQC on the Affleck-Kennedy-Lieb-Tasaki state subject to different types of noises. To illustrate how our criterion works, we analytically and numerically calculated the gate fidelity to find that its behavior depends crucially on whether the noises satisfy a certain symmetry condition with respect to the on-site $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry. In particular, the fidelity for the identity gate, which is given by the sum of the non-local string order parameters, plays an important role. Furthermore, we demonstrate that a stronger symmetry conditions are required to be able to perform other (e.g., the $Z$-rotation gate) gates with high fidelity. By examining which unitary gates can be implemented with the MBQC on the decohered states, we can gain a useful insight into the richer structure of noisy SPT states that cannot be captured solely by the string order parameters.