Fig. 5: LED for generic string-net models.
From: Enhancing detection of topological order by local error correction
a An ancilla qudit is used to measure the topological charge within each local region \({{{{{{{\mathcal{R}}}}}}}}\): we initialize the ancilla in \(\left\vert 0\right\rangle\), apply a local unitary \(U=\mathop{\sum }\nolimits_{i,j=0}^{N-1}\left\vert (i+j)\,{{{{{{{\rm{mod}}}}}}}}\,N\right\rangle {\left\langle j\right\vert }_{{{{{{{{\rm{anc}}}}}}}}}\otimes {P}_{i}\), where Pi projects \({{{{{{{\mathcal{R}}}}}}}}\) onto the subspace with topological charge αi, and finally measure the ancilla’s state. b Local error correction is performed by inputting the fusion rules of \({{{{{{{\mathcal{C}}}}}}}}\) into a maximum-likelihood patch-based decoder. Given any l × l patch, one identifies possible groupings of anyons (including groupings to the boundary) that can remove all nontrivial topological charges within the patch. The decoder performs the grouping of highest probability by fusing anyons or dragging them to the boundary of the patch59. If \({{{{{{{\mathcal{C}}}}}}}}\) is non-abelian, the vacuum topological charge may only be attained probabilistically with probability 1 − ∑αpα, or a nontrivial topological charge α remains with some probability pα. c The system is then coarse-grained by applying a quantum circuit corresponding to a multiscale entanglement renormalization ansatz (MERA) representation of the fixed-point state43. d At the final layer, S- and T-matrix elements can be measured by introducing an ancilla qubit in the \(\left\vert+\right\rangle\) state and applying controlled-anyon-braiding operations. More details on implementing Steps (c) and (d) can be found in Methods.
