Bridging measurement-induced phase transition and quantum error correction in monitored quantum circuits with many-body measurements

Previous studies of $(1+1)$-dimensional random Clifford circuits and Haar random circuits indicate that the quantum error correction threshold occurs concurrently with measurement-induced phase transitions. This implies the area-law phase cannot be used for quantum error correction in nonunitary quantum channels. The situation might be changed when the many-body measurements and feedback operations are introduced into circuits. The channel invariant subspace generated by the combination of many-body measurement and feedback operations can support the quantum error correction in the area-law phase. To show these properties, four quantum circuits consisting of noncommuting four-body measurements and feedback operations have been put forward as a prominent example. The study of these circuits also shows that the nondeterminism (determinism) of the final state after many-body measurements (single-body measurements) plays an important role in the quantum phase transition, which also leads to the quantum error correction in the area-law phase.