  We reveal the connection between two-dimensional subsystem symmetry-protected
topological (SSPT) states and two-dimensional topological orders via a
self-dual frustrated toric code model. This model, an enrichment of the toric
code (TC) with its dual interactions, can be mapped to a model defined on the
dual lattice with subsystem symmetries and subextensive ground state
degeneracy. The map connects exactly the frustrated TC to two copies of the
topological plaquette Ising model (TPIM), as a strong SSPT model with linear
subsystem symmetries. The membrane order parameter of the TPIM is exactly
mapped to dual TC stabilizers as the order parameter of the frustrated TC
model, SSPT gapless edge states of the TPIM are mapped to zero-energy dangling
operators under open boundaries, and the transition from the SSPT-ordered TPIM
to the trivial paramagnetic phase is mapped to the transition between two
distinct topological orders. We also demonstrate that this mapping can be used
to elucidate the structure of other SSPT models, reflecting the subtle linkage
between SSPT order and topological order in two dimensions.
