Robust decoding in monitored dynamics of open quantum systems with Z₂ symmetry

We explore a class of "open" quantum circuit models with local decoherence ("noise") and local projective measurements, each respecting a global Z_2 symmetry. The model supports a spin glass phase where the Z_2 symmetry is spontaneously broken, a paramagnetic phase characterized by a divergent susceptibility, and an intermediate "trivial" phase. Within the spin glass phase the circuit dynamics can be interpreted as a quantum repetition code, with each stabilizer of the code measured stochastically at a finite rate, and the decoherences as effective bit-flip errors. Motivated by the geometry of the spin glass phase, we devise a novel decoding algorithm for recovering an arbitrary initial qubit state in the code space, assuming knowledge of the history of the measurement outcomes, and the ability of performing local Pauli measurements and gates on the final state. With this simple decoder, we find that the information of the initial encoded qubit state can be retained (and then recovered) for a time logarithmic in L for a 1d circuit, and for a time at least linear in L in 2d below a finite error threshold. We also outline a connection of the simple decoder to correlation functions in a random bond Ising model, which leads to an improved decoder that has a finite threshold in both 1d and 2d, both for T linear in L. The improved decoder has a time complexity O(L^{d+1} T), thus preferable as compared to existing ones based on a "perfect matching" of error defects.