Lode Pollet

We introduce ParaToric, a C++ package for simulating the toric code in a parallel field (i.e., X X - and Z 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 either periodic or smooth open boundaries. The package is expandable to arbitrary lattice geometries and custom observables diagonal in either the X X - or Z 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.

We introduce ParaToric, a C++ package for simulating the toric code in a parallel field (i.e., X X - and Z 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 either periodic or smooth open boundaries. The package is expandable to arbitrary lattice geometries and custom observables diagonal in either the X X - or Z 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.

Fracton codes have been intensively studied as novel topological states of matter, yet their fault-tolerant properties remain largely unexplored. Here, we investigate the optimal thresholds of self-dual fracton codes, in particular, the checkerboard code, against stochastic Pauli noise. By utilizing a statistical-mechanical mapping combined with large-scale parallel tempering Monte Carlo simulations, we calculate the optimal code-capacity threshold of the checkerboard code to be p th ≃ 0.107 ( 3 ) . This value is the highest among known three-dimensional codes and nearly saturates the theoretical limit for topological codes. Our results further validate the generalized entropy relation for two mutually dual models, H ( p th ) + H ( p ̃ th ) ≈ 1 , and extend its applicability beyond standard topological codes. This verification indicates the Haah's code also possesses a code-capacity threshold near the theoretical limit p th ≈ 0.11 . These findings highlight fracton codes as highly resilient quantum memory and demonstrate the utility of duality techniques in analyzing intricate error-correcting codes.

Mapping finite-temperature dynamical phase diagrams of quantum many-body models is a necessary step towards establishing a framework of far-from-equilibrium quantum many-body universality. However, this is quite difficult due, in part, to the severe challenges in representing the volume-law entanglement that is generated under nonequilibrium dynamics at finite temperatures. Here, we address these challenges with an efficient equilibrium quantum Monte Carlo (QMC) framework for computing the finite-temperature dynamical phase diagram. Our method uses energy conservation and the self-thermalizing properties of ergodic quantum systems to determine observables at late times after a quantum quench. We use this technique to chart the dynamical phase diagram of the $2+1$D quantum Ising model generated by quenches of the transverse field in initial thermal states. Our approach allows us to track the evolution of dynamical phases as a function of both the initial temperature and transverse field. Surprisingly, we identify quenches in the ordered phase that cool the system as well as an interval of initial temperatures where it is possible to quench from the paramagnetic (PM) to ferromagnetic (FM) phases. Our method gives access to dynamical properties without explicitly simulating unitary time evolution, and is immediately applicable to other lattice geometries and interacting many-body systems. Finally, we propose a quantum simulation experiment on state-of-the-art digital quantum hardware to directly probe the predicted dynamical phases and their real-time formation.

Kitaev’s toric code has become one of the most studied models in physics and is highly relevant to the fields of both quantum error correction and condensed matter physics. Most notably, it is the simplest known model hosting an extended, deconfined topological bulk phase. To this day, it remains challenging to reliably and robustly probe topological phases, as many state-of-the-art order parameters are sensitive to specific models and even specific parameter regimes. With the emergence of powerful quantum simulators which are approaching the regimes of topological bulk phases, there is a timely need for experimentally accessible order parameters. Here we study the ground state physics of the parallel field toric code on the honeycomb, triangular, and cubic lattices using continuous-time quantum Monte Carlo. By extending the concept of experimentally accessible percolation-inspired order parameters (POPs) we show that electric and magnetic anyons are independently confined on the honeycomb and triangular lattices, unlike on the square lattice. Our work manifestly demonstrates that, even in the ground state, we must make a distinction between topological order and (de-)confinement. Moreover, we report multicritical points in the aforementioned confinement phase diagrams. Finally, we map out the topological phase diagrams on the honeycomb, triangular, and cubic lattices and compare the performance of the POPs with other topological order parameters. Our work paves the way for studies of confinement involving dynamical matter and the associated multicritical points in contemporary quantum simulation platforms for Z 2 lattice gauge theories.

The low-energy physics of quantum spin ice is known to support an emergent form of quantum electrodynamics (QED), where magnetic monopoles exist and the fine-structure constant is material dependent. In this article, we show how this QED is modified via a coupling to dynamical matter on the centered pyrochlore lattice, a structure that has recently been synthesized using metal-organic frameworks. Specifically, we study the low-energy properties of the S = 1 / 2 quantum XXZ model on the centered pyrochlore lattice, with a focus on the sign-problem-free region. At fourth order in degenerate perturbation theory, this model hosts a quantum spin liquid distinct from the well-known U(1) quantum spin ice on the pyrochlore because of the presence of dynamical matter in the ground state. Exact diagonalization results are consistent with this quantum spin liquid over an extended region of the ground-state phase diagram, although potential quantum critical points within this region could indicate a richer phase structure. Our work thus expands the physics of quantum spin ice in an experimentally motivated geometry, showing how the emergent QED can be coupled to dynamical matter at zero temperature.

Machine-learning algorithms thrive on large data sets of good quality. Here we show that they can also excel in a typical research setting with little data of limited quality, through an interplay of insights coming from machine and human researchers. The question we address is the unsolved problem of ordering out of a spin-liquid phase described by an emergent rank-2 U ( 1 ) gauge theory, as described by H. Yan []. Published Monte Carlo simulations for this problem are consistent with a strong first-order phase transition, but were too noisy for the form of low-temperature order to be identified. Using a highly interpretable machine-learning approach based on a support vector machine with a tensorial kernel, we reanalyze these Monte Carlo data, gaining new information about the form of order that could in turn be interpreted by traditionally trained physicists. We find that the low-temperature ordered phase is a form of magnetic order analogous to a smectic liquid crystal. This arises due to a subtle thermal order-by-disorder mechanism, that can be understood from the fluctuations of the tensor electric field of the parent rank-2 gauge theory. These results were obtained by a back-and-forth process which closely resembles a collaboration between human researchers and machines. We argue that this “collaborative” approach may provide a blueprint for solving other problems that have not yielded to human insights alone.

Fracton models host unconventional topological orders in three and higher dimensions and provide promising candidates for quantum memory platforms. Understanding their robustness against quantum fluctuations is an important task but also poses great challenges due to the lack of efficient numerical tools. In this work, we establish neural quantum states (NQS) as new tools to study phase transitions in these models. Exact and efficient parametrizations are derived for three prototypical fracton codes — the checkerboard and X-cube model, as well as Haah’s code — both in terms of a restricted Boltzmann machine (RBM) and a correlation-enhanced RBM. We then adapt the correlation-enhanced RBM architecture to a perturbed checkerboard model and reveal its strong first-order phase transition between the fracton phase and a trivial field-polarizing phase. To this end, we simulate this highly entangled system on lattices of up to 512 qubits with high accuracy, representing a cutting-edge application of variational neural-network methods. In addition, we reproduce the phase transition of the X-cube model previously obtained with quantum Monte Carlo and high-order series expansion methods. Our work demonstrates the remarkable potential of NQS in studying complicated three-dimensional problems and highlights physics-oriented constructions of NQS architectures.

The recent experimental detection of the onset of a dynamically prepared, gapped Z 2 quantum spin liquid on the ruby lattice brought the physics of frustrated magnetism and lattice gauge theory to Rydberg tweezer arrays [Semeghini , Probing topological spin liquids on a programmable quantum simulator, ]. The thermodynamic properties of such models remain inadequately addressed, yet knowledge thereof is indispensable if one wants to prepare large, robust, and long-lived quantum spin liquids. Using large scale quantum Monte Carlo simulations we find in the model a renormalized classical spin liquid with constant entropy density S / N approaching ln ( 2 ) / 6 in the thermodynamic limit for all moderate and large values of the detuning δ and starting from T / Ω ∼ 0.5 (in units of the Rabi frequency Ω ) down to the lowest temperatures we could simulate, T / Ω ∼ 0.01 . With van der Waals interactions, constant entropy plateaus are still found but its value shifts with δ . We comment on the implications of the adiabatic approximation to the dynamical ramps for the electric degrees of freedom, which leads to a reinterpretation of the experimental observations.