Probing the 2D Error-Mitigation Threshold with Matrix Product States

Error-mitigated dynamics of random circuits are known to exhibit a sharp threshold: as the disorder strength is tuned, mitigated estimates switch abruptly from a phase that reliably recovers ideal observables to one in which mitigation fails. Whether this threshold exists on two-dimensional lattices has remained unresolved. Here, we introduce a numerical scheme based on constant-bond-dimension matrix product states to simulate error-mitigated random-circuit dynamics across diverse lattice geometries. Our approach reproduces the established mitigation thresholds for all-to-all and one-dimensional circuits. Applied to two-dimensional random circuits on both square and heavy-hex lattices, it provides the first quantitative evidence for a 2D mitigation threshold and yields an estimate of its critical disorder strength. Beyond these results, our method furnishes a computationally efficient framework for probing phase transitions in higher-dimensional error-mitigated noisy quantum circuits.