# Quantum Critical Dynamics of Tensor Networks

Lead Research Organisation:
University College London

Department Name: London Centre for Nanotechnology

### Abstract

One of the strangest aspects of quantum mechanics is the idea of action at a distance, whereby if two particles are prepared in a certain state relative to one another and then separated to opposite sides of the galaxy, a measurement on one of them will instantaneously tell one about the state of the other. This so worried Einstein that it gave him a deep suspicion about the theory of quantum mechanics. Nevertheless, experiment after experiment has confirmed that that is just how the universe is. If an intuitive understanding of this escapes us, we at least now know how to quantify the amount of this "quantumness" in a measure called entanglement.

The situation is even trickier when we think about systems with many particles. In this case we can imagine that every pair of particles might be entangled with one another. An important insight about the real world, however, is that for a variety of reasons usually pairs of particles are not entangled with one another if they are more than a certain distance apart. A certain type of wavefunction can be written down that encodes this idea of a finite range of entanglement mathematically. These states are known as tensor network states.

The use of these states is relatively new, but already their impact is profound and far-reaching. They have enabled the behaviour of many-particle quantum systems to be simulated on a computer with unprecedented accuracy. At the same time, insights flowing from the way in which these states behave has enabled us to make some hitherto unsuspected connections between quantum systems - including ultimately a link between the theory of black holes and the quantum mechanics of electrons in certain types of crystals known a quantum critical.

In this project we will use tensor network states to try to understand the dynamics of quantum critical systems - systems that in a certain sense are balanced between being classical and quantum mechanical. Our hope is that in doing so, we will find new ways of modelling quantum systems efficiently on computers, new ways of thinking about the quantum world and new ways to harness it to technological ends.

The situation is even trickier when we think about systems with many particles. In this case we can imagine that every pair of particles might be entangled with one another. An important insight about the real world, however, is that for a variety of reasons usually pairs of particles are not entangled with one another if they are more than a certain distance apart. A certain type of wavefunction can be written down that encodes this idea of a finite range of entanglement mathematically. These states are known as tensor network states.

The use of these states is relatively new, but already their impact is profound and far-reaching. They have enabled the behaviour of many-particle quantum systems to be simulated on a computer with unprecedented accuracy. At the same time, insights flowing from the way in which these states behave has enabled us to make some hitherto unsuspected connections between quantum systems - including ultimately a link between the theory of black holes and the quantum mechanics of electrons in certain types of crystals known a quantum critical.

In this project we will use tensor network states to try to understand the dynamics of quantum critical systems - systems that in a certain sense are balanced between being classical and quantum mechanical. Our hope is that in doing so, we will find new ways of modelling quantum systems efficiently on computers, new ways of thinking about the quantum world and new ways to harness it to technological ends.

### Planned Impact

Entanglement is rapidly proving to be a notion that has universal applicability in the study of quantum mechanics. The idea that it might be a resource that persists in real systems only up to a given scale is embodied in tensor network wavefunctions. Such states are proving powerful in generating both accurate and efficient numerical simulations and analytical insights. They provide a common language to discuss entanglement across a very broad range of situations and disciplines from quantum chemistry to string theory. By its very formulation, it provides a fixed point for discourse between analytical and numerical physicists and the potential to facilitate detailed calculations that can engage experiment.

As technology embraces the quantum limit, it will increasingly require this facility. The very subject of this project, then provides the main pathway to its impact. We will work to develop a fruitful interaction between analytical and numerical theorists - the PI and PDRA respectively - communicating the benefits of this to the UK condensed matter community (for example, through the Thomas Young Centre) and geographically and intellectually further afield though our network of collaborations.

This project is a pilot towards long-term goals that address some of the most fundamental problems in quantum mechanics. By generating new knowledge in the interplay between numerical and analytical theory, it aims to give an early impetus to such research and, by focusing on a few clearly defined questions, some concrete results to build upon. It is hoped that the generation of these new ideas and the international collaborations that we foster in the course of working on them will enhance the international profile of UK science.

A major impact of this project is the personnel that it will train. By guiding the PDRA towards work in tensor networks tied closely to analytical understanding, I will train him/her to engage productively with diverse research and researchers in quantum mechanics. The experience of the PhD student working alongside this PDRA will be salutary. The particular area of scientific expertise will fulfill a specific need in the UK physics community, however, the experience of this challenging and stimulating environment will allow them to develop a skill set needed to achieve across a range of walks of life.

As technology embraces the quantum limit, it will increasingly require this facility. The very subject of this project, then provides the main pathway to its impact. We will work to develop a fruitful interaction between analytical and numerical theorists - the PI and PDRA respectively - communicating the benefits of this to the UK condensed matter community (for example, through the Thomas Young Centre) and geographically and intellectually further afield though our network of collaborations.

This project is a pilot towards long-term goals that address some of the most fundamental problems in quantum mechanics. By generating new knowledge in the interplay between numerical and analytical theory, it aims to give an early impetus to such research and, by focusing on a few clearly defined questions, some concrete results to build upon. It is hoped that the generation of these new ideas and the international collaborations that we foster in the course of working on them will enhance the international profile of UK science.

A major impact of this project is the personnel that it will train. By guiding the PDRA towards work in tensor networks tied closely to analytical understanding, I will train him/her to engage productively with diverse research and researchers in quantum mechanics. The experience of the PhD student working alongside this PDRA will be salutary. The particular area of scientific expertise will fulfill a specific need in the UK physics community, however, the experience of this challenging and stimulating environment will allow them to develop a skill set needed to achieve across a range of walks of life.

## People |
## ORCID iD |

Andrew George Green (Principal Investigator) |

### Publications

Haegeman Jutho
(2017)

*Quantum Gross-Pitaevskii Equation*in SCIPOST PHYSICS
Hallam A
(2019)

*The Lyapunov spectra of quantum thermalisation.*in Nature communications
Joshi C
(2013)

*Quantum correlations in the one-dimensional driven dissipative X Y model*in Physical Review A
Stojevic Vid
(2016)

*Time evolution and deterministic optimization of correlator product states*in PHYSICAL REVIEW B
Zauner V
(2015)

*Transfer matrices and excitations with matrix product states*in New Journal of Physics
Ðuric T
(2014)

*Interaction-induced anomalous quantum Hall state on the honeycomb lattice*in Physical Review BDescription | This grant is aimed at developing new ways to understand the out of equilibrium dynamics of quantum systems and - ultimately - how the classical world of our experience emerges from this dynamics. As a starting point to this work, we have identified several areas where a mathematical framework that could support progress was lacking. We are developing the mathematical tools that will enable these barriers to be overcome. The initial goals were ambitious and the technical barriers towards achieving them were greater than anticipated. Nevertheless, we have made significant progress: We have developed an algorithm to time evolve a set of ansatz Hamiltonians where no such algorithm existed before. Key ideas have fed into the goals of my fellowship programme and feedback from these has developed a clearer perspective on the potential for describing the process of thermalisation (and extent to which on might do so) This work has now appeared. Understanding developed in this grant has continued to inform my research. A new work concerning the thermalisation and emergence of classical correlations in closed quantum systems has been published in nature communications, and another is in preparation concerning a new way to describe highly entangled quantum systems coupled to the environment and the hydrodynamics of eigenstate thermalisation. |

Exploitation Route | Our numerical results (published) provide a new way to evolve and optimise an important class of quantum states. Although an attempt to do so deterministically had been produced previously by other authors, ours has the advantage of much greater stability. Our algorithm will be a use to those investigating quantum out of equilibrium dynamics. This paper has now appeared. Insights gained in this work are feeding into current efforts to develop a higher dimensional version of our recently constructed path integral over entangled states. |

Sectors | Education,Electronics,Energy,Pharmaceuticals and Medical Biotechnology |

Description | Work under this project involved identifying what analytical tools must be developed in order to address the physical problems that we have in mind. We have developed the TDVP approach to understanding the time evolution of correlated plaquette states and will publish this shortly(now published Feb 2019). In addition, this work has benefited from efforts to develop a field theory over tensor networks. This has enabled a re-interpretation of the truncated Wigner approximation applied to systems that display an eigenstate thermalisation. We hope to use these insights to develop a hydrodynamic picture of late time eigenstate thermalisation. Aspects of this research are ongoing and the results obtained in this grant provide a tremendously useful underpinning. For example, arXiv:1806.05204 has since been published in Nature Communications and concerns analysis seeded by this work. |

Sector | Digital/Communication/Information Technologies (including Software),Education |

Impact Types | Societal |

Description | cMPS and CFT |

Organisation | Brookhaven National Laboratory |

Department | Condensed Matter Physics & Materials Science Department |

Country | United States |

Sector | Public |

PI Contribution | My post-doc under the quantum critical dynamics of tensor networks grant has been collaborating with Prof Robert Konik to compare predictions of out of equilibrium quantum dynamics under cMPS and conformal field theory. |

Collaborator Contribution | Prof Robert Konik has committed both his time and the time of his post-doc to this research |

Impact | Publications expected shortly |

Start Year | 2015 |

Company Name | Generative Tensor Networks Ltd |

Description | GTN combine tensor network descriptions of quantum mechanics and machine learning to discover new pharmaceuticals. The CTO was a postdoc on this grant and i am an advisor to the company. |

Year Established | 2017 |

Impact | Many observers (MIT tech review, Financial times, Forbes etc etc) have picked GTN as an up and coming company to watch |

Website | https://gtn.ai/ |