This course is designed to introduce students to the intersection of tensor networks and quantum computing. It covers foundational concepts in tensor networks, their applications to quantum systems, and advanced techniques for quantum computation and simulation. The course is divided into two main sections: Preliminaries and Applications.

I. Preliminaries

The first part of the course provides a solid foundation in both tensor network basics and quantum physics concepts, essential for understanding how these tools are applied in quantum computing.

1. Tensor Network Basics
   • Tensors: Introduction to the fundamental building blocks of tensor networks, including specialized tensors such as rank-2 tensors, and their roles in high-dimensional data representation.
   • Tensor Networks: Discussion on how tensor networks organize and simplify computations, with an emphasis on popular architectures like Matrix Product States (MPS) and Tree Tensor Networks (TTN).
   • Tensor Network Decomposition: Techniques for decomposing tensors into network forms, improving computational efficiency.
2. Quantum Physics with Tensors
   • Quantum Mechanics Fundamentals: Key principles such as states, observables, and quantum evolution.
   • Quantum Systems: A deep dive into qubit dynamics, including the Bloch sphere and entanglement. It emphasizes tensor network representations of quantum states, including MPS and Matrix Product Operators (MPO), which are crucial for simulating quantum systems efficiently.
3. Quantum Computing with Tensor Networks
   • Quantum Circuits and Measurement: Transition from tensor network representations to quantum circuits, and exploration of quantum measurement protocols.
   • Real-time evolution: TEBD simulates the unitary time evolution of quantum systems by approximating the evolution operator exp(-iHt) using matrix product states (MPS), effectively managing entanglement growth.

II. Applications

The second part focuses on the practical applications of tensor networks in quantum computing, particularly in the context of quantum dynamics, optimization, and open-system dynamics.

4. Unitary Evolution of Hamiltonian Dynamics
   • Time-Dependent Variational Principle (TDVP): Application of TDVP for simulating time evolution in quantum systems using tensor networks.
   • Optimization Algorithms: Introduction to quantum-enhanced optimization algorithms like Quadratic Unconstrained Binary Optimization (QUBO) and Quantum Approximate Optimization Algorithm (QAOA).
   • Quantum Annealing: An exploration of quantum annealing, digitized quantum annealing (dQA), and how tensor networks contribute to efficient solutions.
5. Open Tensor Network
   • Open Systems Dynamics: The course covers how to model open systems with tensor networks, introducing key concepts such as Density Operators, Minimally Entangled Typical Thermal States (METTS), and Lindblad dynamics.
   • Lindblad Master Equation: Application of tensor networks to the Lindblad equation, used to describe non-unitary evolution in quantum open systems.
6. Tensor Networks and Quantum Magic
   • Stabilizer Formalism: Introduction to the stabilizer formalism, a mathematical framework for understanding quantum error correction and quantum magic.
   • Measuring Magic: Techniques for quantifying the "magic" of quantum states through tensor network methods, such as the computation of stabilizer Rényi entropies and the development of Clifford-enhanced Matrix Product States (𝒞MPS).
   • Clifford-Dressed TDVP: Incorporating Clifford operations into TDVP for improved simulation of Hamiltonian dynamics.

Learning Objectives:

By the end of the course, students will:

• Understand the fundamental principles of tensor networks and their role in quantum computing.
• Be proficient in applying tensor network methods to model and simulate quantum systems, including both unitary and non-unitary dynamics.
• Develop insights into the computational advantages of tensor networks in optimization, quantum dynamics.

This course will equip students with the tools and knowledge to explore cutting-edge research at the intersection of quantum computing and tensor network theory.

FINAL EXAM: Implementation of an algorithm to solve an interesting many-body/ quantum-computing problem.