Source Data for "Relaxation Critical Dynamics in Measurement-induced Phase Transitions"

In this paper, we investigate the relaxation critical dynamics near the MIPT for different initial states in a one-dimensional quantum circuit. Specifically, when the initial state is in the volume-law phase with vanishing measurement probability, we find that the half-chain entanglement entropy $S$ decays as $S\propto t^{-1}$ with the coefficients proportional to the size of the system in the short-time stage; In contrast, when the initial state is the product state, $S$ increases with time as $S\propto \ln{t}$, consistent with previous studies. Despite these contrasting behaviors, we develop a unified scaling form to describe these scaling behaviors for different initial states where the off-critical-point effects can also be incorporated. This framework offers significant advantages for experimental MIPT detection. Our novel scheme, leveraging relaxation dynamical scaling, drastically reduces post-selection overhead, and can eliminate it completely with trackable classical simulation.本文针对一维量子电路中不同初始态下的测量诱导相变（Measurement-Induced Phase Transition，MIPT）附近的弛豫临界动力学展开研究。具体而言，当初始态处于测量概率趋于零的体积律相时，我们发现短时阶段内半链纠缠熵$S$以$Spropto t^{-1}$的形式衰减，其衰减系数与系统尺寸成正比；与之相对，当初始态为直积态时，$S$随时间以$Spropto ln{t}$的形式增长，该结果与已有研究结论一致。尽管上述两类行为存在显著差异，我们仍构建了统一的标度形式，用以描述不同初始态下的标度行为，该形式亦可纳入非临界点效应的影响。该框架为实验探测MIPT提供了显著优势。我们提出的全新方案借助弛豫动力学标度，大幅降低了后选择开销，且通过可追踪的经典模拟可完全消除该开销。