טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentElkabetz Roy
SubjectUsing the Belief Propagation Algorithm for Finding Tensor
Networks Approximations of Many-Body Ground States
DepartmentDepartment of Physics
Supervisor Dr. Itai Arad
Full Thesis textFull thesis text - English Version


Abstract

Tensor Networks (TN) is a mathematical framework that enables an efficient description of many-body quantum states using a network of inter-connected tensors. In the last decade, there has been significant progress in using tensor networks in many fields such as condensed matter, cosmology, quantum informa- tion, machine learning, etc. One of the main challenges in the tensor networks community is the contraction of networks in high spatial dimensions (>1D), which can take exponential time. This presents critical issues in tensor networks optimizations, such as finding a tensor network approximation to a ground state of a many-body Hamiltonian. Over the years, many tensor networks algorithms have been developed to deal with this problem; all have their strengths and weaknesses. In this work, I will present a new approach for dealing with ten- sor networks contractions. It is based on a well-known method from statistical mechanics and computer science, which is called Belief Propagation (BP). Origi- nally, this method was developed for approximating the marginals of many-body classical Gibbs distributions, or, more generally, of Probabilistic Graphical Mod- els (PGM). As both problems involve a summation over an exponential number of configurations, it follows that the BP algorithm can be imported to the quantum setup. To that aim, I will introduce the Double-Edge Factor Graphs (DEFG), a new type of probabilistic graphical model that supports the quantum behavior of tensor networks. Based on this connection, I will present the Belief Propagation Update (BPU) algorithm for the contraction of tensor networks and compare it with a well-known tensor networks algorithm called the Simple Update (SU) algorithm on a 4x4 and 10x10 Antiferromagnetic Heisenberg model. As a final step, I will give a mathematical proof of the equivalence between BP and the trivial-SU, which is a variation of the SU algorithm and closely relates to the canonical representation in tensor networks.