Data for "Learning Potts Models and Z3 Toric Codes: Higher and Ordinary Nishimori Criticality"

Zenodo repository containing data and analysis code for the paper'Learning Potts Models and Z3 Toric Codes: Higher and Ordinary Nishimori Criticality' Abstract:Motivated by the existence of a higher Nishimori line in the Bayesian inference/learning phase diagram ofthe classical Ising model under bond-energy measurements, we identify an analogous line in the correspondingphase diagram of the two-dimensional (2D) q-state Potts model (2 < q <= 4). Tuning the Potts model to itscritical temperature beta_c, this higher Nishimori line meets the critical line (at beta_c) in a distinct higherNishimori critical point - a tricritical point at finite inference strength that separates a paramagnetic, aferromagnetic and a 'spin-glass' phase. With analytical tools, we discuss the general structure of the rich phasediagram, which contains two unstable and three stable fixed points, and obtain a number of exact results foruniversal quantities, including the decay exponent of the Edwards-Anderson correlator, using a Gaussianmeasurement protocol which allows for exact calculations. Using extensive numerical tools, we confirm thesestatements for a generic, discrete q-state measurement protocol and determine precise numerical estimates forthe location of higher and ordinary Nishimori critical points as well as RG flows between the various fixedpoints. We also discuss the Casimir effective central charges of the critical points in the learning phase diagram,and their monotonic decrease along measurement-induced RG flows, as established non-perturbatively by thec-effective theorem and its extensions, and contrast it to the monotonic increase along the corresponding RGflows in the random-bond Potts model. Finally, we discuss a simple general argument based on Elitzur's theoremthat establishes the ordinary Nishimori critical point(s) as stable universality class(es) in monitored as well asoptimal decoding problems. Our results can be recast in the language of Born measurements on a deformed Zqtoric code where the tricritical higher Nishimori point is an 'information' critical point that separates phaseswith strong, weak, and broken Zq symmetry, which correspond to stable quantum, classical, and no memoryphases, respectively. ##### Repository structure ##### The repository is organized into three top-level folders: code/ Julia/Jupyter analysis and plotting notebooks, together with the Julia environment data/ Raw simulation batches, averaged observables, and finite-size-scaling results in JLD2 format figures/ PDF figures generated by the notebooks ### Code: The numerical analysis and plotting code is written in Julia. To reproduce the same environment, use Juliaversion 1.12.6 and instantiate the project recorded in `code/Project.toml` and `code/Manifest.toml`. From therepository root, run: cd code julia --project=. -e 'using Pkg; Pkg.instantiate()' Open and run the notebooks from the `code` directory so that their relative paths to `../data` and `../figures`resolve correctly. The notebooks are organized by paper figure: code/fig01.ipynb Generates Fig. 1, the schematic illustration of noisy domain-wall measurements, reconstruction, and the corresponding three-state Potts configuration. This notebook does not read numerical data. code/fig03.ipynb Generates Fig. 3, the numerical learning phase diagram. It combines the coherent-information scans, finite-size-scaling estimates of the phase boundaries, the regular phase-diagram grid, and the higher Nishimori-line estimates stored in `data/Potts_sisj_betacuts/2/γN2.jld2`. code/fig04_fig08.ipynb Generates Fig. 4 (coherent information and scaling collapses at the higher and ordinary Nishimori points) and Fig. 8 (the correlation-length exponent along the phase boundary). The notebooks finite-size-scaling analysis writes `data/Potts_Ic_horizontal/1/FSS.jld2`. code/fig05_fig07.ipynb Generates Fig. 5 (Edwards-Anderson correlator at the higher Nishimori and clean Potts critical points) and Fig. 7 (the fitted Edwards-Anderson exponent 2X2 along beta = beta_c). code/fig06.ipynb Generates Fig. 6 by fitting the Edwards-Anderson correlation length on high-temperature cuts and comparing it with the clean Potts correlation length. This determines points on the emergent higher Nishimori line. code/fig10.ipynb Generates Fig. 10, the correlation-length exponent obtained from vertical finite-size-scaling cuts along beta = beta_c. The analysis writes `data/Potts_Ic_vertical/4/FSS.jld2`. ### Data: All numerical data are stored as Julia JLD2 files. The principal quantities are: Ics coherent information sisj measurement-averaged spin-spin correlator sisj_sq Edwards-Anderson correlator (the second moment of the spin-spin correlator) ds linear system sizes L betas inverse temperatures (stored under the Unicode key `βs`) gammas measurement strengths (stored under the Unicode key `γs`) Most simulation folders use the following file convention: results.jld2 Simulation parameters and metadata, including system sizes, parameter grids, sample count, and tensor- network cutoff. results1.jld2, results2.jld2, ... Raw batched simulation output. These files contain the task descriptions and the corresponding results. averaged_results.jld2 Processed observables averaged over the raw samples. Statistical uncertainties are stored using Julia's `Measurements.jl` mean-plus-standard-error values. FSS.jld2 Finite-size-scaling outputs, where present, including fitted critical points and critical exponents. The data folders are: data/Potts_Ic_horizontal/1/ Coherent-information scans along horizontal cuts (varying measurement strength gamma at fixed beta) across 40 beta values for L = 8, 16, 32, and 64. The folder also contains the derived FSS results used for Figs. 3 and 8. data/Potts_Ic_horizontal/3/ Higher-resolution horizontal scans at beta = 0 and beta = beta_c for L = 8, 16, 32, 64, and 128. These data are used for the critical-point estimates and scaling collapses in Fig. 4. data/Potts_Ic_vertical/2/ Coherent-information scans along vertical cuts (varying beta at fixed gamma) across 40 measurement strengths for L = 8, 16, 32, and 64. These data contribute to the phase boundary in Fig. 3. data/Potts_Ic_vertical/4/ Higher-resolution vertical scans at 30 measurement strengths for L = 8, 16, 32, 64, and 128. The folder contains the FSS results used for the beta-direction critical exponent in Fig. 10 and the phase boundary in Fig. 3. data/Potts_Ic_phasediagram/4/ Coherent information on a regular 40 x 40 (beta, gamma) grid for L = 8, 16, 32, and 64. These data provide the background interpolation of the numerical phase diagram in Fig. 3. data/Potts_sisj_betac/3/ Spin-spin and Edwards-Anderson correlation data along beta = beta_c for a 256 x 256 system and 21 measurement strengths. These data are analyzed in Figs. 5 and 7. data/Potts_sisj_betacuts/2/ Spin-spin and Edwards-Anderson correlation data for a 256 x 256 system along six high-temperature beta cuts. The file `γN2.jld2` stores the (beta, gamma) points obtained from correlation-length matching; these are shown as the emergent higher Nishimori line in Fig. 3. The underlying fits are shown in Fig. 6. data/Potts_sisj_clean/ Clean (unmeasured) Potts spin-spin correlations for L = 16, 32, 64, 128, 256, and 512 at seven beta values. `sisj_clean.jld2` contains the clean beta = beta_c reference for L = 256 used in Fig. 5, while `256.jld2` supplies the clean correlation-length reference used in Fig. 6. ### Figures: The `figures` directory contains the generated PDF files `fig01.pdf`, `fig03.pdf`, `fig04.pdf`, `fig05.pdf`,`fig06.pdf`, `fig07.pdf`, `fig08.pdf`, and `fig10.pdf`. Running the corresponding notebooks saves updated PDFsto this directory.