DOI: 10.5281/zenodo.15530141
title: Data for "Flow to Nishimori universality in weakly monitored quantum circuits with qubit loss"

description: Zenodo repository containing data and analysis code for the paper'Flow to Nishimori universality in weakly monitored quantum circuits with qubit loss'

Abstract:In circuit-based quantum state preparation, qubit loss and coherent errors are circuit imperfections that im-peril the formation of long-range entanglement beyond a certain threshold. The critical theory at the thresholdis a continuous entanglement transition known to be described by a (2+0)-dimensional non-unitary conformalfield theory which, for the two types of imperfections of certain circuits, is described by either percolation orNishimori criticality, respectively. Here we study the threshold behavior when the two types of errors simultane-ously occur and show that, when moving away from the Clifford-regime of projective stabilizer measurements,the percolation critical point becomes unstable and the critical theory flows to Nishimori universality. Wetrack this critical renormalization group (RG) crossover flow by mapping out the entanglement phase diagrams,parametrized by the probability and strength of random weak measurements, of two dual protocols preparingsurface code or GHZ-class cat states from a parent cluster state via constant-depth circuits. Extensive numericalsimulations, using hybrid Gaussian fermion and tensor network / Monte Carlo sampling techniques on systemswith more than a million qubits, demonstrate that an infinitesimal deviation from the Clifford regime leads toa sudden, strongly non-monotonic entanglement growth at the incipient non-unitary RG flow. We argue thatthe spectra of scaling dimensions of both the percolation and Nishimori fixed points exhibit multifractality. Forpercolation, we provide the exact (non-quadratic) multifractal spectrum of exponents, while for the Nishimorifixed point we show high-precision numerical results for five leading exponents characterizing multifractality.

##### Repository structure ######## Code:In order to reproduce the same results, make sure to run Julia version 1.11.3 and instantiate the julia projectsaved in the Project.toml and Manifest.toml. You can do this by navigating to the `code` folder in your terminaland starting Julia with the following command:    julia --projectThen switch to the package manager mode by pressing `]` and run the command:    instantiateThis should install the same package versions that were used to generate and analyse the data in this repository.

The jupyter notebooks `code/fig*.ipynb` contain the code to generate the figures in the paper. The other code isloaded within the jupyter notebooks to e.g. calculate the cumulants and to define a plotting theme.

### Data:There is data from 8 different simulations in this repository:- data/sweep_pd.jld2 contains the coherent informations for radial cuts through the phase diagram (used to do    finite size scaling analysis).- data/nishimori_line.jld2 contains the coherent informations for an additional simulation along the Nishimori    line with larger system sizes.- data/entanglement_criticalline/* contain entanglement data from 3 different simulations on various points along    the critical line between percolation and Nishimori point. This data is averaged, since the raw data was over     50 GB.- data/moments_percolation_obc.jld2 and data/moments_percolation_pbc.jld2 contain the averaged entanglement data    (for open and periodic boundary conditions respectively), as well as the moments at the percolation point.    The raw data is around 220GB and is thus too large for this repository.- data/moments_Nishimori.jld2 contains the averaged entanglement data, as well as the moments at the Nishimori    point. The raw data is here around 2TB large, as we have around 18 million samples and 1023 (every possible    subsystemsize) x 13 (number of different renyi entropies we calculated) numbers per sample.