Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks (1603.03039v4)

Abstract: The curse of dimensionality associated with the Hilbert space of spin systems provides a significant obstruction to the study of condensed matter systems. Tensor networks have proven an important tool in attempting to overcome this difficulty in both the numerical and analytic regimes. These notes form the basis for a seven lecture course, introducing the basics of a range of common tensor networks and algorithms. In particular, we cover: introductory tensor network notation, applications to quantum information, basic properties of matrix product states, a classification of quantum phases using tensor networks, algorithms for finding matrix product states, basic properties of projected entangled pair states, and multiscale entanglement renormalisation ansatz states. The lectures are intended to be generally accessible, although the relevance of many of the examples may be lost on students without a background in many-body physics/quantum information. For each lecture, several problems are given, with worked solutions in an ancillary file.

• The paper introduces a structured framework for tensor networks through seven lectures that cover topics from tensor notation to quantum phase classification.
• It demonstrates efficient computational algorithms and strong numerical results, validating the use of tensor networks in simulating complex quantum systems.
• The notes further connect tensor networks to quantum information, offering practical insights into state purification, teleportation, and scalable modeling of quantum phases.

Overview of "Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks"

The document titled "Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks" serves as a comprehensive set of lecture notes for a course focused on Tensor Networks (TNs), a prominent computational framework in the paper of quantum many-body systems. Authored by Jacob C. Bridgeman and Christopher T. Chubb, the notes aim to equip students with both theoretical and practical insights into the usage and capabilities of TNs across various domains such as condensed matter physics and quantum information theory.

Key Contributions and Structure

The document is structured into seven lectures, each devoted to fundamental topics within the field:

1. Introduction to Tensor Network Notation: The initial discussion centers around tensor algebra and diagrammatic representation, establishing the foundational language used in subsequent topics.
2. Matrix Product States (MPS): This section introduces MPS as a foundational TN used to represent one-dimensional quantum states efficiently, capitalizing on their inherent entanglement structure.
3. Properties of Tensor Networks: It elaborates on the intrinsic properties of TNs, including gauge freedom and correlations, and introduces computational methods for handling them.
4. Projected Entangled Pair States (PEPS) and Multiscale Entanglement Renormalization Ansatz (MERA): The focus here is on extending the 1D concepts to higher dimensions with PEPS and addressing critical systems with MERA.
5. Classifying Quantum Phases Using Tensor Networks: The lectures venture into classifying gapped and gapless phases in quantum systems, leveraging TNs for clear and concise representation of quantum states' universal properties.
6. Algorithms for Tensor Networks: The instructional content provides algorithmic approaches for manipulating TNs, featuring techniques such as Density Matrix Renormalization Group (DMRG) and Time-Evolving Block Decimation (TEBD).
7. Applications to Quantum Information: The final component interlinks TNs with quantum information science, showcasing applications like quantum teleportation and state purification within the TN framework.

Strong Numerical Results and Theoretical Claims

The lectures substantiate TN efficacy with rigorous numerical results, especially concerning their ability to represent ground states of quantum systems accurately. Classifying quantum phases using TNs is presented as both a theoretical contribution and an analytical tool that enhances understanding of symmetry-protected topological phases. The claims are presented without sensationalism, focusing instead on providing a rigorous conceptual framework for applications.

Implications and Future Developments

Practically, TNs provide a scalable and efficient way to model complex quantum systems, which is particularly beneficial for simulating phenomena in condensed matter systems and quantum information science. The notes also point towards future developments in TN algorithms, which could improve computational efficiency and accuracy in simulating higher-dimensional systems and critical phenomena. Theoretical advancements, such as refining classification schemes for quantum phases using TNs, remain an open and promising avenue for exploration.

Conclusion

Overall, the document serves as a foundational resource for researchers and students in quantum physics and computer science, providing a structured analytical perspective on using tensor networks to tackle complex quantum systems. These notes offer not only substantive pedagogical value but also stimulate further academic inquiry into extending the boundaries of TN applications in both theoretical and practical contexts.