Generative modeling assisted simulation of measurement-altered quantum criticality

TL;DR

This paper tackles the challenge of simulating measurement-induced quantum criticality, where post-selection makes direct sampling of measurement outcomes exponentially hard. It proposes a physics-preserving conditional diffusion approach that learns the observation-indexed distribution of local reduced density matrices $D(\rho_{s_{[i]}})$ conditioned on truncated measurement labels $s_{[i]}$, ensured by Structure-Preserving Diffusion Models (SPDM) that enforce $\rho \succeq 0$, $\rho=\rho^{\dagger}$, and $\mathrm{Tr}(\rho)=1$. By exploiting locality, the method reduces global labeling to local strings and enables extrapolation to unseen labels, providing scalable generation of ensembles of local quantum states. This ML-assisted framework aims to mitigate post-selection costs and enable efficient quantum simulations of measurement-altered protocols in quantum many-body systems.

Abstract

In quantum many-body systems, measurements can induce qualitative new features, but their simulation is hindered by the exponential complexity involved in sampling the measurement results. We propose to use machine learning to assist the simulation of measurement-induced quantum phenomena. In particular, we focus on the measurement-altered quantum criticality protocol and generate local reduced density matrices of the critical chain given random measurement results. Such generation is enabled by a physics-preserving conditional diffusion generative model, which learns an observation-indexed probability distribution of an ensemble of quantum states, and then samples from that distribution given an observation.

Figures (3)

• Figure 1: This figure is taken from Fig. 1 of Ref.Murciano2023 which illustrates the protocol of measurement-altered criticality.
• Figure 2: Scattered plots of the imaginary vs the real part of the off-diagonal entry of the one-body reduced density matrix on site $i$ labeled by the measurement result on site $i-2$ to site $i+2$. The difference in distribution decreases as the difference in the measurement result moves away from site $i$.
• Figure 3: Average variance of one and two-body RDM with respect to the change in measurement results at each site. The blue and red lines represent the diagonal and off-diagonal entries respectively of the one-body RDM at site $0$. The yellow and purple lines represent the $ZZ$ and $XX$ correlators of the two-body RDM at site $-2$ and $3$. We have rescaled the $ZZ$ and $XX$ correlators by a factor of $3$ and $2000$ respectively.