Exploring quantum many-body systems with tensor networks

Quantum many-body systems are fascinating systems exhibiting many rich phenomena, and are central in areas of physics as diverse as Condensed Matter, Quantum Information Theory, High Energy Physics, and Quantum Chemistry. Their standard description is however extremely costly, as the size of the description doubles with every new particle in the system. This mathematical difficulty translates into a physical limitation of our understanding and predictability of these systems. Fortunately, quantum many-body systems that appear in nature are very special, as they contain only local correlations, and can hence be described in a much more efficient way. Tensor networks are a family of states that aim at describing these relevant states in a scalable manner. They have proven extremely successful for one-dimensional pure states, and some progress has been made for one-dimensional mixed states and for two-dimensional states. The present project, entitled "Exploring quantum many-body systems with tensor networks", aims at deepening our understanding of quantum many-body systems using tensor networks, in particular concerning two aspects. In Part A, we focus on one-dimensional mixed states, which are the general framerwork for describing thermal states, systems out of equilibrium or lack of knowledge of the system. Their most basic mathematical property is that the probabilities of measurement outcomes are positive. It is desirable, both for theoretical and numerical reasons, to enforce this positivity property on the local matrices, which are the building blocks of the tensor network formalism, but it had been unknown for 20 years what the cost of this enforcement is. Using connections to new results in convex algebraic geometry, we have recently shown that this cost can be arbitrarily high. At the same time, we showed that in the approximate case this problem can be substantially easier. Now we want to explore the consequences of these findings. This will allow us to obtain more efficient and robust descriptions of quantum many-body systems with tensor networks, and to design more stable and reliable numerical algorithms. Part B is centered around the question Which states have a continuum limit?. We want to systematically study, from a tensor perspective, which states are coarse-grained descriptions of other states describing the physics at a much finer scale. This is a very fundamental question in Condensed Matter and High Energy Physics. We have recently studied this question for one-dimensional pure states, and have established a mathematically rigorous classification of continuum limits using quantum information tools. We now want to extend this analysis to more general and realistic scenarios, and to two-dimensional systems.

Most things in this world are too complicated to be described exactly mathematically. For example, one cannot describe every air molecule in the atmosphere because there are too many variables. Instead, one must make effective models to simulate the weather. Clearly, better models will allow us to forecast the weather more accurately. The same happens with quantum systems: describing many quantum particles exactly is too complicated. Physicists have developed effective models, called tensor networks, which work very well for the simplest quantum states, called pure states. To incorporate our lack of knowledge of the system we must use mixed states. Mixed states are essential to describe quantum experiments (including experimental quantum computers), and condensed matter phenomena. Unfortunately, tensor networks descriptions of mixed states are much less developed, because it is a more difficult mathematical problem. For example, some states can be described very efficiently, but contain many correlations. In this project, we have shown that this problem is related to some other mathematical problems. With this connection, we have shown that if a state has the next-to-simplest description (i.e. one needs two matrices per site), then it can only contain classical correlations and very few of them. A different line of investigation concerns continuum limits of states. What is a continuum limit? A river is microscopically composed of many molecules of water, which are discrete. Nonetheless, the flow of a river is continuous, and thus the water molecules have a continuum limit. But not every system has a continuum limit: the many agents buying and selling stocks give rise to fluctuating stock prices, but these cannot be described globally in a continuous and simplified way. In our project, we have asked whether a quantum state has a continuum limit. We have answered this question for a simple class of tensor networks, and we have shown that the set of states in the continuum that can be described by a tensor network needs to be enlarged.