  The exploration of topologically-ordered states of matter is a long-standing
goal at the interface of several subfields of the physical sciences. Such
states feature intriguing physical properties such as long-range entanglement,
emergent gauge fields and non-local correlations, and can aid in realization of
scalable fault-tolerant quantum computation. However, these same features also
make creation, detection, and characterization of topologically-ordered states
particularly challenging. Motivated by recent experimental demonstrations, we
introduce a new paradigm for quantifying topological states -- locally
error-corrected decoration (LED) -- by combining methods of error correction
with ideas of renormalization-group flow. Our approach allows for efficient and
robust identification of topological order, and is applicable in the presence
of incoherent noise sources, making it particularly suitable for realistic
experiments. We demonstrate the power of LED using numerical simulations of the
toric code under a variety of perturbations. We subsequently apply it to an
experimental realization, providing new insights into a quantum spin liquid
created on a Rydberg-atom simulator. Finally, we extend LED to generic
topological phases, including those with non-abelian order.
