One of the main challenges of quantum many‐body physics is the exponential growth in the dimensionality of the Hilbert space with system size. This growth makes solving the Schrödinger equation of the system extremely difficult. Nonetheless, many physical systems have a simplified internal structure that typically makes the parameters needed to characterize their ground states exponentially smaller. Many numerical methods then become available to capture the physics of the system. Among modern numerical techniques, neural networks, which show great power in approximating functions and extracting features of big data, are now attracting much interest. In this work, the progress in using artificial neural networks to build quantum many‐body states is reviewed. The Boltzmann machine representation is taken as a prototypical example to illustrate various aspects of the neural network states. The classical neural networks are also briefly reviewed, and it is illustrated how to use neural networks to represent quantum states and density operators. Some physical properties of the neural network states are discussed. For applications, the progress in many‐body calculations based on neural network states, the neural network state approach to tomography, and the classical simulation of quantum computing based on Boltzmann machine states are briefly reviewed.

In the classical world, physical events always happen in a fixed causal order. However, it was recently revealed that quantum mechanics allows events to occur with indefinite causal order (ICO). In this study, we use an optical quantum switch to experimentally investigate the application of ICO in thermodynamic tasks. Specifically, we simulate the working system interacting with two identical thermal reservoirs in an ICO, observing the quantum heat extraction even though they are in thermal equilibrium where heat extraction is inaccessible by traditional thermal contact. Using such a process, we simulate an ICO refrigeration cycle and investigate its properties. We also show that by passing through the ICO channel multiple times, one can extract more heat per cycle and thus obtain a higher refrigeration performance. Our results suggest that the causal nonseparability can be a powerful resource for quantum thermodynamic tasks.

Machine-learning representations of many-body quantum states have recently been introduced as an ansatz to describe the ground states and unitary evolutions of many-body quantum systems. We investigate one of the most important representations, the restricted Boltzmann machine (RBM), in the stabilizer formalism. A general method to construct RBM representations for stabilizer code states is given, and exact RBM representations for several types of stabilizer groups with the number of hidden neurons equal to or less than the number of visible neurons are presented. The result indicates that the representation is extremely efficient. Then we analyze a surface code with boundaries, defects, domain walls, and twists in full detail and find that almost all the models can be efficiently represented via the RBM ansatz: the RBM parameters of the perfect case, boundary case, and defect case are constructed analytically using the method we provide in the stabilizer formalism, and the domain wall and twist case is studied numerically. In addition, the case for Kitaev's $D\left({\mathbb{Z}}_d\right)$ model, which is a generalized model of the surface code, is also investigated.

Cluster states are crucial resources for measurement-based quantum computation (MBQC). It exhibits symmetry-protected topological (SPT) order, thus also playing a crucial role in studying topological phases. We present the construction of cluster states based on Hopf algebras. By generalizing the finite group valued qudit to a Hopf algebra valued qudit and introducing the generalized Pauli-X operator based on the regular action of the Hopf algebra, as well as the generalized Pauli-Z operator based on the irreducible representation action on the Hopf algebra, we develop a comprehensive theory of Hopf qudits. We demonstrate that non-invertible symmetry naturally emerges for Hopf qudits. Subsequently, for a bipartite graph termed the cluster graph, we assign the identity state and trivial representation state to even and odd vertices, respectively. Introducing the edge entangler as controlled regular action, we provide a general construction of Hopf cluster states. To ensure the commutativity of the edge entangler, we propose a method to construct a cluster lattice for any triangulable manifold. We use the 1d cluster state as an example to illustrate our construction. As this serves as a promising candidate for SPT phases, we construct the gapped Hamiltonian for this scenario and provide a detailed discussion of its non-invertible symmetries. We demonstrate that the 1d cluster state model is equivalent to the quasi-1d Hopf quantum double model with one rough boundary and one smooth boundary. We also discuss the generalization of the Hopf cluster state model to the Hopf ladder model through symmetry topological field theory. Furthermore, we introduce the Hopf tensor network representation of Hopf cluster states by integrating the tensor representation of structure constants with the string diagrams of the Hopf algebra, which can be used to solve the Hopf cluster state model.

Spatial and temporal quantum correlations can be unified in the framework of the pseudo-density operators (PDOs), and quantum causality between the involved events in an experiment is encoded in the corresponding PDO. We study the relationship between local causal information and global causal structure. A space-time marginal problem is proposed to infer global causal structures from given marginal causal structures where causal structures are represented by the reduced PDOs; we show that there almost always exists a solution in this case. By imposing the corresponding constraints on this solution set, we could obtain the required solutions for special classes of marginal problems, like a positive semidefinite marginal problem, separable marginal problem, etc. We introduce a space-time entropy and propose a method to determine the global causal structure based on the maximum entropy principle. The notion of quantum pseudo-channel (QPC) is also introduced and we demonstrate that the QPC marginal problem can be solved by transforming it into a PDO marginal problem via the channel-state duality.

A bstract: We propose weak Hopf symmetry as a general framework to explore (1+1)D topological phases that exhibit non-invertible symmetries. Inspired by the Symmetry Topological Field Theory (SymTFT) description of quantum phases with non-invertible symmetry, we construct a lattice model by introducing two distinct topological boundary conditions for a weak Hopf lattice gauge theory. One boundary encodes the topological symmetry information, while the other incorporates the non-topological dynamics. The resulting model is termed the cluster ladder model. We demonstrate that the cluster state model is a special case of this broader class of lattice models exhibiting weak Hopf symmetry H × Ĥ , where H is a weak Hopf algebra and Ĥ is its dual weak Hopf algebra. On a closed manifold, the symmetry reduces to Cocom( H ) × Cocom( Ĥ ), corresponding to the cocommutative subalgebras of H × Ĥ . An essential weak Hopf sub-symmetry is Cocom( H ) × Rep( H ), which, in the finite group case, reduces to the familiar symmetry G × Rep( G ). To exactly solve the lattice model, we introduce a weak Hopf tensor network. Furthermore, we demonstrate how to construct the lattice realization of an arbitrary fusion category symmetry $$ \mathcal{S} $$ S via combining Tannaka-Krein reconstruction or weak Hopf tube algebra and the cluster ladder model.