&Bullet; *physics* 14, 60

Three new studies provide analytical descriptions and precise solutions for various aspects of thermodynamics in quantum many-body systems.

The concept of thermodynamics is approaching its 175th birthday. Despite the longevity of the field, many questions related to thermodynamics remain open. One of these questions is exactly how the laws of thermodynamics arise in a many-body quantum system. Another is how to characterize chaos in the same system. Both problems involve understanding how quantum entanglement develops in a complex quantum system, which has been difficult to study because solving the models used to describe such systems is time-consuming and computationally intensive. Now three research groups are presenting theoretical advances that will help circumvent this challenge by providing analytical descriptions and accurate solutions for various aspects of thermodynamics and chaos in these quantum systems [1–3] .

The new studies take into account all quantum many-body systems, which are systems that contain large numbers of interacting particles. These particles all obey the laws of quantum mechanics and their repeated interactions lead to quantum correlations and quantum entanglements in the system. The systems also all obey the laws of thermodynamics.

The second law of thermodynamics says that the state of the system evolves over time when an isolated quantum many-body system is “erased” so that it becomes unbalanced to achieve a state of thermal equilibrium and maximum entropy . A precise description of this so-called thermalization process is difficult due to several complicating factors. One factor is the large number of degrees of freedom in the interaction of quantum many-body systems that can make models unsolvable. Another factor is that the parameters that describe the entanglement and correlations of the system change over time. These two factors usually limit analysis to small quantum systems and short development times, but even in these cases the studies are still extremely challenging.

A third complicating factor is chaos, which manifests itself in quantum many-body systems as the non-local spread of quantum information. The presence of chaos adds an extra layer of complexity to the models, but it can also allow some simplifications. In a chaotic classical system, for example, researchers can ignore the individual movements of particles and instead describe the system using macroscopic features such as temperature and specific heat that are provided by thermodynamics. Chaos is also closely linked to thermodynamics through the entanglement dynamics of a quantum system. This means that models that describe the dynamics of quantum many-body systems can also describe their thermodynamic and chaotic properties.

Previously developed methods address some of the complications mentioned above and allow the modeling of larger systems and longer development times. However, these methods involve approximations that can limit accuracy. Researchers are therefore developing alternative methods. A promising approach is to interpret the system as a series of quantum circuits. These circuits approximate the temporal dynamics of the quantum system with gates and wires. The gates perform uniform transformations of the system state and the wires are used to represent the passage of time. Quantum circuit models are oversimplified models, but they still retain most of the essential physics of the complex phenomena. The three new studies all use this approach, although they work with different circuits and consider different aspects of the problem.

The first study, conducted by Pieter Claeys and Austen Lamacraft of the University of Cambridge, UK, uses quantum circuits to predict the evolution of quantum correlations in a many-body system – something relevant to both thermodynamics and chaos [2] . In particular, the duo demonstrates a method for generating so-called dual-unit circuits, the quantum correlations of which can be explicitly obtained spatially and temporally [4] . You consider systems with arbitrary dimensions of the Hilbert space, which includes all possible quantum mechanical states of a given system. They show that their approach leads to analytically comprehensible solutions for every ergodicity level, a property that relates to the decay of the correlations in the system.

In the second study, Katja Klobas from Oxford University, UK, and colleagues use quantum circuits to investigate the thermalization of quantum many-body systems [1] . You are looking at a specific type of circuit known as rule 54. These circuits are made up of qubit chains, and as other studies have shown, their properties are precisely solvable. The team erases the system and then monitors the dynamics after the erasure using so-called tensor network methods [5] . They show that their approach allows for a full and accurate description of the thermalization process, and they find accurate predictions for asymptotic growth in a number of parameters known as Rényi entropies, often used in quantum information to measure the degree of entanglement in to measure a system.

As with the Oxford study, the third study deals with Rényi entropies. Zongping Gong and colleagues from the Max Planck Institute for Quantum Optics are studying the development of these entropies in a quantum circle known as the quantum cell automaton [3] . This circuit has a topological index that is believed to characterize its ability to transport quantum information [6] . This index can be obtained mathematically, but the calculated index typically lacks an explicit physical interpretation. Instead, Gong and colleagues obtain the index using Rényi entropies, which can be measured experimentally [7] confirming the quantum information interpretation of the index. The team also sets a lower bound for the entropy of entropy of its quantum circuit, which is twice the topological index, and interprets it as the lower bound for the growth rate of the quantum chaos.

Together, these three studies offer remarkable advances in predicting the non-equilibrium dynamics in isolated, interacting quantum many-body systems. But they are not the end of the journey. For further progress, it is crucial that researchers develop methods and approaches that examine other aspects of dynamics and can be linked to state-of-the-art experiments with ultracold atoms. In particular, a deeper understanding of the time scales of non-equilibrium dynamics is required to uncover non-trivial features. This topic is fundamentally relevant to condensed matter physics, but also to the development of future technologies such as quantum computers, which in some cases use quantum circuits to perform quantum simulations [8] .

## References

- K. Klobas
*et al.*, “Exact thermalization dynamics in the quantum cellular automaton ‘Rule 54′” Phys. Rev. Lett.**126**160602 (2021). - PW Claeys and A. Lamacraft, “Ergodic and non-ergodic dual-unit quantum circuits with any local Hilbert space dimension”, Phys. Rev. Lett.
**126**100603 (2021). - Z. Gong
*et al.*, “Topological lower limit of quantum chaos through entanglement growth” Phys. Rev. Lett.**126**160601 (2021). - B. Bertini
*et al.*, “Exact correlation functions for lattice models with two units in 1 + 1 dimensions” Phys. Rev. Lett.**123**210601 (2019). - MC Bañuls
*et al.*, “Matrix product states for the dynamic simulation of infinite chains” Phys. Rev. Lett.**102**240603 (2009). - D. Gross
*et al.*, “Index theory of one-dimensional quantum runs and cellular automata” Comm. Mathematics. Phys.**310**419 (2012). - R. Islam
*et al.*, “Measurement of entropy entropy in a quantum many-body system” nature**528**77 (2015). - F. Arute
*et al.*, “Quantum Superiority Using a Programmable Superconducting Processor” nature**574**505 (2019).