ParaToric 1.0-beta: Continuous-time quantum Monte Carlo for the toric code in a parallel field

TL;DR

ParaToric delivers a versatile, sign-problem-free CT-QMC solver for the extended toric code in a parallel field, supporting square, triangular, honeycomb, and cubic lattices with open or periodic boundaries at finite temperature. By extending the Wu, Deng, and Prokof'ev algorithm, it enables robust thermalization, production sampling, and hysteresis studies while providing a rich set of diagonal observables, Fredenhagen-Marcu loops, percolation metrics, and snapshot extraction in both X- and Z-bases. The package offers seamless integrations via C++, C, Python interfaces, and a CLI, with outputs in HDF5 and GraphML, and includes extensive diagnostics for autocorrelation and error estimation. These features make ParaToric a practical tool for exploring topological order, finite-temperature transitions, and benchmarking data for lattice gauge theories, quantum simulators, and quantum error correction studies, all under an MIT-compatible license.

Abstract

We introduce ParaToric, a C++ package for simulating the toric code in a parallel field (i.e., $X$- and $Z$-fields) at finite temperature. We implement and extend the continuous-time quantum Monte Carlo algorithm of Wu, Deng, and Prokof'ev on the square, triangular, honeycomb, and cubic lattices with open and periodic boundaries, respectively. The package is expandable to arbitrary lattice geometries and custom observables diagonal in either the $X$- or $Z$-basis. ParaToric also supports snapshot extraction in both bases, making it ideal for generating training/benchmarking data for other methods, such as lattice gauge theories, cold atom or other quantum simulators, quantum spin liquids, artificial intelligence, and quantum error correction. The software provides bindings to C/C++ and Python, and is thus almost universally integrable into other software projects.

Figures (3)

• Figure 1: Implemented lattices. We implement the extended toric code (\ref{['eq:eTC']}) on the square (a), honeycomb (b), triangular (c), and cubic lattices (d). For each lattice, we show the star ($\hat{A}_v$) and plaquette ($\hat{B}_p$) terms. The cubic lattice importantly features star interactions of six links and plaquette interactions of four links on the faces of cubes.
• Figure 2: Good and bad thermalization plots produced by the Python command-line interface. We show the gauge field energy $\propto \lambda$. (a) The system is well thermalized; after its initial drop, the energy fluctuates around the expectation value. (b) The system is not yet thermalized; the floating average is still decreasing.
• Figure 3: Topological phase transition of the extended toric code (\ref{['eq:eTC']}) on the square lattice. The critical field is known and located at $h_\mathrm{c}(\lambda=0.2) \approx 0.33$Wu2012Linsel2024. Our results agree with the value published in the literature, within error bars. (a) The percolation probability Binder ratio $U_{\Pi^x}$Linsel2024 features a crossing point around $h=0.33$. (b) The percolation probability $\Pi^x$ is non-zero in the topological phase and zero in the trivial phase. The transition gets sharper with increasing system size. (c) The staggered imaginary time order parameter $O^x_\mathrm{SI}$ is zero in the topological phase and non-zero in the trivial phase. (d) The Fredenhagen-Marcu order parameter $O^x_\mathrm{FM}$ is zero in the topological phase and non-zero in the trivial phase. The loop length grows with $\mathcal{O}(L)$. Compared to the other order parameters, it is very noisy because it is a multi-body correlator and on top of that a division of two exponentially small numbers.