Fig. 3: Application to mixed states.

a Without error correction, generic points in the topological and trivial, disordered phase (g_Z = 0.12, g_X = 0.18, p_flip = 0.0 and g_Z = 0.06, g_X = 0.0, p_flip = 0.11 resp. shown in the plot) appear very similar qualitatively, as closed loops decay exponentially with loop perimeter in both cases, while open strings remain close to zero (see Methods). In contrast, in the trivial, paramagnet phase (g_Z = 0.32, g_X = 0.2, p_flip = 0), open strings decay with the same perimeter-law as closed loops. b g_Z = 0.14 slice of mixed state phase diagram, containing topological, disordered, and X-paramagnetic phases. These phases are associated with fixed-point states g_x7D2 = 7D2g_z7D2 = 7D2p_flip7D2 = 7D20, g_z7D2 → 7D2∞, g_x7D2 → 7D2∞, and p_flip7D2 → 7D20.5, respectively. The flow of the closed-loop decay exponent α under LED provides a sharp divider between two kinds of perimeter-law decay, observed in different regimes of the mixed-state phase diagram. c In the uncorrectable regime (i), the local decoder of LED pairs anyons incorrectly, resulting in perimeter-law decay with large α in disordered and paramagnetic phases. Moreover, the probability of such an incorrect pairing can increase with the number n of LED iterations. Here, the black pairings are made by LED at or before one specific value of n, and gray pairings are made upon performing one additional LED iteration. In the correctable (topological) regime (ii), increasing n can reduce α to zero, as fluctuations of higher characteristic length ξ can be reliably corrected using only local information. In the conceptual framework where an LED operator is embedded in a surface code on an annulus (Fig. 2a), incorrect pairings corresponds to logical errors (e.g. X_L). d Expectation values of LED loop observables upon increasing n (d, L ∝ 2ⁿ), in thermal states of varying temperatures (between 0 and 0.35, with darker colors indicating higher temperatures) and p_flip = 0.02.