  We investigate the influence of local decoherence on a symmetry-protected
topological (SPT) phase of the two-dimensional (2D) cluster state. Mapping the
2D cluster state under decoherence to a classical spin model, we show a
topological phase transition of a
$\mathbb{Z}_2^{(0)}\times\mathbb{Z}_{2}^{(1)}$ SPT phase into the trivial phase
occurring at a finite decoherence strength. To characterize the phase
transition, we employ three distinct diagnostic methods, namely, the relative
entropy between two decohered SPT states with different topological edge
states, the strange correlation function of $\mathbb{Z}_2^{(1)}$ charge, and
the multipartite negativity of the mixed state on a disk. All the diagnostics
can be obtained as certain thermodynamic quantities in the corresponding
classical model, and the results of three diagnostic tests are consistent with
each other. Given that the 2D cluster state possesses universal computational
capabilities in the context of measurement-based quantum computation, the
topological phase transition found here can also be interpreted as a transition
in the computational power.
