DOI: 10.5281/zenodo.18001896
title: Data for "Learning transitions of topological surface codes"

description: # Zenodo repository for the paper "Learning transitions of topological surface codes" #


### Abstract: ###


For the surface code, topological quantum order allows one to encode logical quantum information in a ro-

bust, long-range entangled many-body quantum state. However, if an observer probes this quantum state by

performing measurements on the underlying qubits, thereby collecting an ensemble of highly correlated clas-

sical snapshots, two closely related questions arise: (i) do measurements decohere the topological order of the

quantum state; and (ii) how much of the logical information can one learn from the snapshots? Here we address

these questions for measurements in a uniform basis on all qubits. We find that for generic measurement an-

gles, sufficiently far away from the Clifford X, Y, and Z directions (such as the X+Y +Z basis) the logical

information is never lost in one of the following two ways: (i) for weak measurement, the topological order

is absolutely robust; (ii) for projective measurement, the quantum state inevitably collapses, but the logical

quantum information is faithfully transferred from the quantum system to the observer in the form of a tomo-

graphically complete classical shadow. At these generic measurement angles and in the projective-measurement

limit, the measurement ensemble enforced by Born probabilities can be represented by a 2D tensor network

that can be fermionized into a disordered, free-fermion network model in symmetry class DIII, which gives

rise to a Majorana “metal” phase. When the measurement angle is biased towards the X or Z limits, a critical

angle indicates the threshold of a learning transition beyond which the classical shadow no longer reveals full

tomographic information (but reduces to a measurement of the logical X or Z state). This learning transition

can be described in the language of the network model as a “metal to insulator” transition. For the anisotropic

“insulating” angles, there is also a finite learnability threshold in terms of the measurement strength, above

which the surface code breaks down. In terms of universality, this latter transition is generically a Nishimori

transition. Our results can be generalized from qubit measurement to syndrome measurement, demonstrating

that – in the presence of coherent errors – even syndrome measurements can enable the observer to learn the

logical information and collapse the topological order across a learnability threshold.


## Repository structure ##


This repository is structured into three subfolders.


All numerical data is provided in the folder "Data".


In "Code" there notebooks in the programming language Julia, with which the numerical plots of the paper can be replicated.


Additionally, some interactive visualizations are included. (Phase diagram and reference qubit distribution)


"Exact_paper_figures" simply contains the exact versions of the paper figures in the paper.


### Using the code ###


The plots of this paper were produced with Julia version 1.12.1 and largely based on the package Makie.


To make sure the plots are reproduced the same way, use Julia version 1.12.1 and the package versions that are saved in the "Project.toml" and "Manifest.toml" files also provided in the "Code" folder.


To do that the project should be instantiated. This can be done the following steps:


Open a terminal and move into the folder "Code".


Start julia in the folder with the command


julia --project


Press "]" to open the package manager


Run the command


instantiate


With this all needed package versions should be installed.


When running the notebooks, now just make sure that these packages are installed / the right environment is activated.


All notebooks should work simply by executing them top to bottom. Where multiple plots are produced, different sections are marked.


Particularly worth of note is that the notebook "Fig12.ipynb" for figure 12 opens the phase diagram as an interactive plot (usually in a seperate window), where you can zoom and drag however you want.


The same is true for the plots for the distributions of the reference qubit/ the reference qubit samples scattered over the sphere,


which is the last part of "Fig5.ipynb" for figure 5.


### Data explanation ###


The data is structured in eight subfolders. Generally the names of the subfolder should give an impression of what the folder is about.


For more detailed explanations of the data make sure to read the file beginning with "Readme..." inside each folder. For every folder the data and its structure are laid out in its Readme-file.


To make sure it is easy to match the data to its corresponding parts in the paper, here an overview of the connections of the different data folders with the corresponding figures:


Phasediagram_data - Figure 1, Figure 2, Figure 12

Weak_measurement_transition_data - Figure 4

Perfect_distribution_Kulback_Leibler_divergence_data - Figure 5

Entanglement_arcs_data - Figure 7

Entanglement_structure_data - Figure 8

Postselection_data - Figure 9

Decoding_figure_data - Figure 10

Perfect_distribution_Kulback_Leibler_divergence_data - Figure 13