Noise-induced coding and entanglement phase transition in Clifford circuits

In addition to the traditional measurement-induced phase transition, the effects of noise have attracted intense research interest, which is unavoidable in real physical systems. Recently, the boundary noise-induced coding transition and noise-induced phase transition in random circuit sampling have been studied. Here, we investigate the entanglement and coding transitions induced by quantum noises with various system-size-dependent strength and time correlations. Based on the numerous and convincing numerical calculations from Clifford simulations, we have demonstrated that the mutual information within the system obeys a $q^{-1/3}$ scaling, where $q$ is the noise strength, and this scaling is independent of the type of time correlation in quantum noise. In terms of the effective statistical model, the entanglement corresponds to the free energy of the directed polymer. Although time-dependent and time-independent noises have very different impacts on fluctuations of directed polymers and the wandering exponents are distinct, the subleading term of free energy always follows the Kardar-Parisi-Zhang (KPZ) prediction $(q^{-1/3})$ with an effective length scale $q^{-1}$. From the perspective of protecting encoded information, via varying rescaled noise strength $q sim L^{-1}$, there is a noise-induced coding transition, where the critical point and behaviors are consistent with the entanglement phase transition between the power-law and volume-law phases. We have shown that consistency is natural and it is a first-order transition with the help of the statistical model.