Description

The purpose of this project is to learn about tensor networks and their applications to quantum many-body systems, quantum field theory, and holography. Tensor networks provide a powerful mathematical language for describing highly entangled quantum states in a compressed and geometrically suggestive way. They have become important both as practical tools and as conceptual models for understanding how spacetime geometry may emerge from quantum entanglement.

A quantum many-body system has a Hilbert space whose dimension grows exponentially with the number of degrees of freedom. This makes many direct calculations impossible except for very small systems. Tensor networks address this problem by exploiting the structure of entanglement: rather than storing every coefficient of a many-body wavefunction, we decompose the state into a network of smaller tensors. This construction has been used to enhance our understanding of quantum field theory and holographic duality, as well as condensed matter physics and quantum information theory.

Group Project

You will begin by learning the basic language of quantum mechanics and its connection with tensor networks: beginning with the knowledge you will be gaining in the QMIII course (or equivalent), you will learn about encoding many body states with tensors, their contractions and graphical notation, Schmidt decomposition, entanglement entropy, matrix product states and tree tensor networks. You will then study how tensor networks represent simple quantum spin chains and how their efficiency is related to entanglement structure. The project will include both theoretical and computational components.

Mode of operation and evidence of learning for the Group Project

Individual Project

In the individual project, you will have the opportunity to explore one of the many fascinating aspects of tensor networks and their applications. Possible directions include implementing simple tensor-network algorithms in Python or Julia in order to perform Hamiltonian simulation, time evolution, or variational optimisation; studying matrix product states for one-dimensional spin systems; exploring the relationship between entanglement entropy and area laws; investigating MERA as a discrete model of scale transformations; or studying holographic tensor-network models such as the so-called HaPPY code. A more theoretical project could focus on the mathematical structure of tensor contractions, entanglement measures, and renormalisation. A more computational project could involve writing code to manipulate small tensor networks and compute entanglement entropies or correlation functions.

Mode of operation and evidence of learning for the Individual Project

By the end of the project you should have developed a working understanding of how tensor networks compress quantum states, why entanglement is the key organising principle, and how these ideas lead naturally towards modern connections between quantum information, quantum field theory, and holography.

Prerequisites

Mathematical Physics II. Programming experience in Python or Julia will be useful, especially for students who wish to take a computational route.

Corequisites

QMIII or QCIII or equivalent.

Resources

A good first introduction to tensor networks is Bridgeman and Chubb, Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks. A more detailed review, especially useful for matrix product states and projected entangled pair states, is Orús, A Practical Introduction to Tensor Networks. For the connection between entanglement renormalisation and holography see Swingle, Entanglement Renormalization and Holography. For holographic tensor networks and quantum error-correcting toy models of bulk/boundary duality see Pastawski, Yoshida, Harlow and Preskill, Holographic Quantum Error-Correcting Codes. For a very nice review video see this video of a talk by Tamara Kohler at the 2020 Simons Symposium on Quantum Fields and Geometry, which gives a nice introduction to the connection between tensor networks and holography, and also includes some discussion of the role of quantum error correction in holography.