# Kobe Satellite Workshop

**CMSI Kobe Satellite Workshop 2013**

* Recent Progress in Tensor Network Algorithms*

### Tensor network̶a new framework for computational scienc

A variety of calculation methods are used in computational materials science simulations, including the first-principles density-functional method and the molecular dynamics method. These algorithms have been developed and improved in order to quickly and accurately solve Schroedinger equations for electron states and Newtonian equations that describe molecular motion. There are also known algorithms that can be applied to a wider range of general problems,rather than to specific problems and basic equations. It might be more appropriate to call the Monte Carlo method, for example,“framework” that is used to solve complex problems on a computer in a realistic amount of time, rather than calling it “calculationmethods.”

Tensor networks have attracted a great deal of attention in recent years as one such framework. To rigorously express the state of a system that obeys quantum mechanics,an exponentially large number of variables with respect to the size (volume) of the system are needed. Even the world-largest supercomputer would be completely inadequate in terms of memory capacity and calculation capability. In the meantime, in an actual system the enormous number of variables would not all be equally important. Tensor networks are able to express the state of the system as the product of low rank tensors (generalization of matrices), which enables these networks to express accurately and at a low cost only the sections that contribute to the properties of the system.“Identifying only what is important in the midst of enormous degrees of freedom and describing the essence of a system” is an approach that is common to all science.

### Kobe, a place of interchange between disciplines

The CMSI International Workshop 2013“Recent Progress in Tensor Network Algorithms”was one of the satellite workshops held as part of the CMSI International Symposium 2013. The workshop was held for three days, October 16-18, 2013, at the RIKEN Advanced Institute for Computational Science (AICS) in Kobe, Japan. AICS is where the K computer is located, and it is the place where the CMSI Kobe Division as well as the divisions of five strategic fields are concentrated. It is therefore the most appropriate venue for a workshop such as this that transcends the boundaries between different fields.

The workshop featured presentations rangingfrom basic tensor network theory to applications in a wide variety of fields such as quantum chemistry calculations, strongly- correlated quantum systems, phase transition and critical phenomena, photo-induced phase transition, the relationship between black holes and tensor network renormalization groups, machine learning, data mining and wavelets. The specific topics included entanglement (quantum correlations)in the ground state of many-body quantum systems, matrix product states, density matrix renormalization groups, Projected Entangled Pair State (PEPS) and Multi-scale Entanglement Renormalization(MERA). There were also many intriguing lectures from a computer science perspective, on topics such as optimization of the computational cost of simulations, detailed analysis of tensor network accuracy, large-scale parallelization using the K computer, and the development of tensor network libraries.

The workshop was attended by more than 40 persons, greatly exceeding the number of participants that had initially been expected. There were ten participantsfrom overseas including some who were invited speakers, and each day there was an air of excitement in the AICS seminar room on the first floor where the lectures were held. A significant period of time had been allotted for questions and answers after each lecture, but even afterward during the break periods the spiriteddebate continued. In between the sessions,participants also visited the K computer(Fig.2) and enjoyed lunch in the Kobe Kachoen (flower and bird) Park, and there was also a get-together where participants could establish closer relationships as they enjoyed the view of the Akashi Kaikyo Bridge at night.

Tensor networks are a new type of algorithm that has experienced rapid development since the beginning of this century.Both the lecturers and the participants at the workshop were overwhelmingly young, and most of the invited speakers were young researchers in their 20s or 30s. In the poster session, established researchers could be seen posing questions to young researchers in order to take the discussion to a deeper level. The workshop was one that was filled with a sense of purpose that spanned the boundaries between generations and disciplines, and aimed at the further development of this new method. Especially, the workds by G. Chan, “a new framework for quantum mechanics, second only to Feynman's path integral ,” and R.Orús , “ a new language and component for computational science” expressed in their invited talks have left a strong impression on our mind.

(Synge Todo: Graduate School of Science/ISSP,The University of Tokyo)