Correctly characterizing the symmetry-protected topological (SPT) phases in open quantum systems is one of the most challenging problems in correlated quantum matter. Measurement-based quantum computation (MBQC) leverages the nontrivial edge states of SPT phases as logical qubits, making its computational power inherently linked to the topological properties of these phases. In this paper, we propose to use the gate fidelity which is a measure of the computational power of MBQC to identify the SPT phases in mixed-state settings. Specifically, we investigate the robustness of the Haldane phase by considering 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 nonlocal string order parameters, plays an important role. Furthermore, we demonstrate that 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 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 usual string order parameters.

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• Open quantum systems & decoherence
• Quantum computation
• Symmetry protected topological states
• 1-dimensional spin chains
• Quantum spin models
• Discrete symmetries in condensed matter
• Matrix product states
• Tensor network methods