Abstract: In this thesis, we develop an interpretable machine-learning framework based on the Ising model on the Bethe lattice. The hierarchical structure of the lattice allows us to formulate both discriminative and generative learning tasks in terms of a holographic renormalization group (HRG) flow. We first construct a binary classifier on the Bethe lattice for supervised learning. In this setting, inference is naturally interpreted as an HRG flow of external fields from the boundary toward the bulk. We demonstrate that this procedure leads to a strong, nearly exponential suppression of noise between different classes of data, enabling effective classification. We then extend the framework to a generative model inspired by deep Boltzmann machines. Training is implemented through the HRG transformation, providing an efficient and analytically transparent alternative to standard optimization methods. We further show that the inference procedure obtained from alternating Gibbs sampling in the deep Boltzmann machine is consistent with the HRG-based description. Overall, this work establishes the Bethe lattice as a physically motivated and exactly tractable setting for building interpretable machine-learning models, where both learning and generation can be understood through renormalization-group flow.
Summer Internships
Summer Internship (3rd yr)
May - Jul 2024
Title: "On Quantum Master Equation and its application to driven dissipative two-level systems"
Abstract: This report begins with discussing the evolution of open quantum systems and corresponding Quantum Master Equation (QME). The main emphasis is given to the Fluctuation Regulated Quantum Master Equation (FRQME) and its applications on the dynamics of driven dissipative two-level systems (TLS) and quantum optimal control of that TLS. I have also presented the results of numerical simulations for the solutions of the FRQME in the report.
Title: "On Polarization Properties of Light, Gaussian Beams and Spin-Orbit Interaction of Light"
Contributor: Pritish Karmakar.
Supervisor: Prof. Ayan Banerjee, Department of Physical Sciences, IISER
Kolkata.
Abstract: I worked in
optics and polarization, delving into topics such as Jones and Stokes-Mueller formalism, Gaussian beam properties, and light momentum principles. My study covered spin-orbit interaction dynamics, including geometric phases and their manifestations, especially within anisotropic media.
Title: "Composite Fermion Theory of Fractional Quantum Hall Effect"
Contributor: Pritish Karmakar.
Supervisor: Prof. Amit Ghosal, Department of Physical Sciences, IISER
Kolkata.
Abstract: This report gives an overview of the integer and fractional quantum Hall effects, beginning with Landau level physics and the role of disorder. It then shows how strong interactions produce fractional states, and discusses both Laughlin’s theory and the composite fermion framework. Finally, it highlights how these two approaches are closely connected.
Abstract: We review the Hubbard model and its extensions as a framework to describe electron correlation and the Mott metal-insulator transition. Starting from the atomic and band limits, we explore the emergence of Hubbard sub-bands and the Mott transition through Hubbard-I decoupling. The strong-coupling expansion leads to the t-J model, highlighting spin-exchange mechanisms. We then study the extended Hubbard model with emphasis on orbital physics and transition metal oxides. The role of super-exchange, charge-transfer effects are discussed in relation to real materials. Finally, the Dynamical Mean Field Theory (DMFT) approach is introduced to capture certain features of the Hubbard model, including emergence of sub-bands, the evolution of spectral weight.
Abstract: This report introduces photonic topological insulators (PTIs), drawing parallels with electronic systems like the Integer and Quantum Spin Hall Effects. Key concepts include the Berry phase, Chern number, and bulk-edge correspondence. We explore the translation of topology to photonic systems, highlight experimental realizations, and discuss potential applications of PTIs.
Title: "Open Quantum Systems: Kraus Representation Theorem and Quantum Channels"
Contributors: Pritish Karmakar, Ravi
Inderpreet Singh and R. Pradeep.
Supervisor: Prof. Sourin Das, Department of Physical Sciences, IISER
Kolkata.
Abstract: This report begins by discussing the general dynamics of an open system. We explore the theory of quantum measurements and elaborate on the intricacies of operator sum representation including the Kraus representation theorem. Finally, the main emphasis is given to the quantum channels including amplitude-damping, dephasing, bit-flip, phase-flip, and depolarizing channels.
