SciPost: SciPost Physics Lecture Notes

SciPost Physics Lecture Notes

Quantum Neural Network Classifiers: A Tutorial

Weikang Li, Zhide Lu, Dong-Ling Deng

SciPost Phys. Lect. Notes 61 (2022) · published 17 August 2022 | · pdf

Machine learning has achieved dramatic success over the past decade, with applications ranging from face recognition to natural language processing. Meanwhile, rapid progress has been made in the field of quantum computation including developing both powerful quantum algorithms and advanced quantum devices. The interplay between machine learning and quantum physics holds the intriguing potential for bringing practical applications to the modern society. Here, we focus on quantum neural networks in the form of parameterized quantum circuits. We will mainly discuss different structures and encoding strategies of quantum neural networks for supervised learning tasks, and benchmark their performance utilizing Yao.jl, a quantum simulation package written in Julia Language. The codes are efficient, aiming to provide convenience for beginners in scientific works such as developing powerful variational quantum learning models and assisting the corresponding experimental demonstrations.
Phase transitions above the upper critical dimension

Bertrand Berche, Tim Ellis, Yurij Holovatch, Ralph Kenna

SciPost Phys. Lect. Notes 60 (2022) · published 12 August 2022 | · pdf

These lecture notes provide an overview of the renormalization group (RG) as a successful framework to understand critical phenomena above the upper critical dimension $d_{\rm uc}$. After an introduction to the scaling picture of continuous phase transitions, we discuss the apparent failure of the Gaussian fixed point to capture scaling for Landau mean-field theory, which should hold in the thermodynamic limit above $d_{\rm uc}$. We recount how Fisher's dangerous-irrelevant-variable formalism applied to thermodynamic functions partially repairs the situation but at the expense of hyperscaling and finite-size scaling, both of which were, until recently, believed not to apply above $d_{\rm uc}$. We recall limitations of various attempts to match the RG with analytical and numerical results for Ising systems. We explain how the extension of dangerous irrelevancy to the correlation sector is key to marrying the above concepts into a comprehensive RG scaling picture that renders hyperscaling and finite-size scaling valid in all dimensions. We collect what we believe is the current status of the theory, including some new insights and results. This paper is in grateful memory of Michael Fisher who introduced many of the concepts discussed and who, half a century later, contributed to their advancement.
SIMP Dark Matter

Yonit Hochberg

SciPost Phys. Lect. Notes 59 (2022) · published 29 July 2022 | · pdf

These notes summarize the zoom-course on Strongly Interacting Massive Particle (SIMP) dark matter given at the Les Houches Summer School in the summer of 2021. An alternative title for this course would be `SIMPS, Cannibals and ELDERs', where our focus is on high-point interactions amongst dark matter particles. The spirit of the course is to give students a taste of these exciting developments in dark matter, while primarily teaching tricks and methods that are tough to captivate when reading textbooks or papers. These notes are written in similar spirit: I will put emphasis on how quantities scale and how to perform back of the envelope estimates, which should serve to help you when you develop the next great dark matter idea.
From classical to quantum regime of topological surface states via defect engineering

Maryam Salehi, Xiong Yao, Seongshik Oh

SciPost Phys. Lect. Notes 58 (2022) · published 21 July 2022 | · pdf

Since the notion of topological insulator (TI) was envisioned in late 2000s, topology has become a new paradigm in condensed matter physics. Realization of topology as a generic property of materials has led to numerous predictions of topological effects. Although most of the classical topological effects, directly resulting from the presence of the spin-momentum-locked topological surface states (TSS), were experimentally confirmed soon after the theoretical prediction of TIs, many topological quantum effects remained elusive for a long while. It turns out that native defects, particularly interfacial defects, have been the main culprit behind this impasse. Even after quantum regime is achieved for the bulk states, TSS still tends to remain in the classical regime due to high density of interfacial defects, which frequently donate mobile carriers due to the very nature of the topologically-protected surface states. However, with several defect engineering schemes that suppress these effects, a series of topological quantum effects have emerged including quantum anomalous Hall effect, quantum Hall effect, quantized Faraday/Kerr rotations, topological quantum phase transitions, axion insulating state, zeroth-Landau level state, etc. Here, we review how these defect engineering schemes have allowed topological surface states to pull out of the murky classical regime and reveal their elusive quantum signatures, over the past decade.
Quantum to classical mapping of the two-dimensional toric code in an external field

Sydney R. Timmerman, Zvonimir Z. Bandic, Roger G. Melko

SciPost Phys. Lect. Notes 57 (2022) · published 20 July 2022 | · pdf

Kitaev's toric code Hamiltonian in dimension D=2 has been extensively studied for its topological properties, including its quantum error correction capabilities. While the Hamiltonian is quantum, it lies within the class of models that admits a D+1 dimensional classical representation. In these notes, we provide details of a Suzuki-Trotter expansion of the partition function of the toric code Hamiltonian in the presence of an external magnetic field. By coupling additional degrees of freedom in the form of a matter field that can subsequently be gauged away, we explicitly derive a classical Hamiltonian on a cubic lattice which takes the form of a non-isotropic 3D Ising gauge theory.

See all Publications in SciPost Physics Lecture Notes

Physics: Condensed Matter Physics - Theory • High-Energy Physics - Theory

Quantum Field Theory Anomalies in Condensed Matter Physics

by R. Arouca, Andrea Cappelli, T. H. Hansson

Version 2 (current version)
Submitted 2022-08-03 11:12 to SciPost Physics Lecture Notes · latest activity: 2022-08-10 08:43
Physics: Condensed Matter Physics - Computational

Efficient ab initio many-body calculations based on sparse modeling of Matsubara Green's function

by Hiroshi Shinaoka, Naoya Chikano, Emanuel Gull, Jia Li, Takuya Nomoto, Junya Otsuki, Markus Wallerberger, Tianchun Wang, Kazuyoshi Yoshimi

Version 2 (current version)
Submitted 2022-06-10 08:33 to SciPost Physics Lecture Notes · latest activity: 2022-07-20 08:59

Series contained in this Journal

Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy)

Johannes Hauschild, Frank Pollmann

SciPost Phys. Lect. Notes 5 (2018) · published 8 October 2018 | · pdf

Tensor product state (TPS) based methods are powerful tools to efficiently simulate quantum many-body systems in and out of equilibrium. In particular, the one-dimensional matrix-product (MPS) formalism is by now an established tool in condensed matter theory and quantum chemistry. In these lecture notes, we combine a compact review of basic TPS concepts with the introduction of a versatile tensor library for Python (TeNPy) [https://github.com/tenpy/tenpy]. As concrete examples, we consider the MPS based time-evolving block decimation and the density matrix renormalization group algorithm. Moreover, we provide a practical guide on how to implement abelian symmetries (e.g., a particle number conservation) to accelerate tensor operations.

128 citations
Tangent-space methods for uniform matrix product states

Laurens Vanderstraeten, Jutho Haegeman, Frank Verstraete

SciPost Phys. Lect. Notes 7 (2019) · published 15 January 2019 | · pdf

In these lecture notes we give a technical overview of tangent-space methods for matrix product states in the thermodynamic limit. We introduce the manifold of uniform matrix product states, show how to compute different types of observables, and discuss the concept of a tangent space. We explain how to variationally optimize ground-state approximations, implement real-time evolution and describe elementary excitations for a given model Hamiltonian. Also, we explain how matrix product states approximate fixed points of one-dimensional transfer matrices. We show how all these methods can be translated to the language of continuous matrix product states for one-dimensional field theories. We conclude with some extensions of the tangent-space formalism and with an outlook to new applications.

63 citations
Lecture notes on Generalised Hydrodynamics

Benjamin Doyon

SciPost Phys. Lect. Notes 18 (2020) · published 28 August 2020 | · pdf

These are lecture notes for a series of lectures given at the Les Houches Summer School on Integrability in Atomic and Condensed Matter Physics, 30 July to 24 August 2018. The same series of lectures has also been given at the Tokyo Institute of Technology, October 2019. I overview in a pedagogical fashion the main aspects of the theory of generalised hydrodynamics, a hydrodynamic theory for quantum and classical many-body integrable systems. Only very basic knowledge of hydrodynamics and integrable systems is assumed.

49 citations
The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems

Pietro Silvi, Ferdinand Tschirsich, Matthias Gerster, Johannes Jünemann, Daniel Jaschke, Matteo Rizzi, Simone Montangero

SciPost Phys. Lect. Notes 8 (2019) · published 18 March 2019 | · pdf

We present a compendium of numerical simulation techniques, based on tensor network methods, aiming to address problems of many-body quantum mechanics on a classical computer. The core setting of this anthology are lattice problems in low spatial dimension at finite size, a physical scenario where tensor network methods, both Density Matrix Renormalization Group and beyond, have long proven to be winning strategies. Here we explore in detail the numerical frameworks and methods employed to deal with low-dimension physical setups, from a computational physics perspective. We focus on symmetries and closed-system simulations in arbitrary boundary conditions, while discussing the numerical data structures and linear algebra manipulation routines involved, which form the core libraries of any tensor network code. At a higher level, we put the spotlight on loop-free network geometries, discussing their advantages, and presenting in detail algorithms to simulate low-energy equilibrium states. Accompanied by discussions of data structures, numerical techniques and performance, this anthology serves as a programmer's companion, as well as a self-contained introduction and review of the basic and selected advanced concepts in tensor networks, including examples of their applications.

41 citations
An introduction to spontaneous symmetry breaking

Aron J. Beekman, Louk Rademaker, Jasper van Wezel

SciPost Phys. Lect. Notes 11 (2019) · published 4 December 2019 | · pdf

Perhaps the most important aspect of symmetry in physics is the idea that a state does not need to have the same symmetries as the theory that describes it. This phenomenon is known as spontaneous symmetry breaking. In these lecture notes, starting from a careful definition of symmetry in physics, we introduce symmetry breaking and its consequences. Emphasis is placed on the physics of singular limits, showing the reality of symmetry breaking even in small-sized systems. Topics covered include Nambu-Goldstone modes, quantum corrections, phase transitions, topological defects and gauge fields. We provide many examples from both high energy and condensed matter physics. These notes are suitable for graduate students.

30 citations

Physics: Condensed Matter Physics - Theory • High-Energy Physics - Theory

Physics: Condensed Matter Physics - Computational

Series contained in this Journal

128 citations

63 citations

49 citations

41 citations

30 citations