A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States (1306.2164v3)

Abstract: This is a partly non-technical introduction to selected topics on tensor network methods, based on several lectures and introductory seminars given on the subject. It should be a good place for newcomers to get familiarized with some of the key ideas in the field, specially regarding the numerics. After a very general introduction we motivate the concept of tensor network and provide several examples. We then move on to explain some basics about Matrix Product States (MPS) and Projected Entangled Pair States (PEPS). Selected details on some of the associated numerical methods for 1d and 2d quantum lattice systems are also discussed.

• The paper presents tensor network states as a framework for efficiently simulating quantum many-body systems using MPS and PEPS.
• It details numerical methods such as variational optimization and imaginary time evolution to approximate ground states and capture entanglement.
• The study highlights the computational challenges of contracting PEPS and outlines practical strategies to mitigate these issues.

An Introduction to Tensor Network States: Matrix Product States and Projected Entangled Pair States

The paper offers a comprehensive introduction to tensor network (TN) methods, specifically focusing on Matrix Product States (MPS) and Projected Entangled Pair States (PEPS), which have become essential tools in the numerical simulation of quantum many-body systems. The paper positions TNs as a crucial framework for representing and simulating quantum states of many-body systems efficiently, distinguishing them by their ability to capture the entanglement properties inherent in many quantum systems.

Overview of Tensor Networks

Tensor networks serve as a pivotal technique in condensed matter physics and quantum information because they offer a structured representation of quantum states that focus on entanglement properties. They are particularly powerful for addressing problems in quantum many-body physics, where the sheer size of the Hilbert space makes naive representations infeasible. A tensor network represents a quantum state through a network of tensors that are interconnected, allowing for the decomposition of complex wave functions into more manageable components.

Matrix Product States (MPS)

Matrix Product States (MPS) are a class of TN states particularly suited for one-dimensional quantum systems. An MPS efficiently captures the area-law scaling of entanglement entropy observed in ground states of gapped 1D local Hamiltonians. The paper outlines the fundamental properties of MPS, highlighting its representation power which, with finite bond dimensions, captures the significant correlations in non-critical 1D systems. The exact calculation of expectation values for open boundary conditions is a notable advantage of using MPS, which plays into algorithms like the Density Matrix Renormalization Group (DMRG).

Projected Entangled Pair States (PEPS)

For two-dimensional lattice systems, Projected Entangled Pair States (PEPS) extend the matrix product state formalism. PEPS are instrumental in simulating 2D systems, taking account of the area law in 2D systems and accommodating states with polynomially-decaying correlations. Unlike MPS, exact contraction of PEPS networks is computationally demanding and falls within the complexity class $\sharp$P-Hard, necessitating approximate methods to compute physical quantities. The paper elucidates this complexity and showcases methods like boundary MPS, Corner Transfer Matrix, and tensor coarse-graining to efficiently handle these contractions in practical simulations.

Numerical Methods and Practical Implications

The paper explores variational optimization and imaginary time evolution as approaches to determine the tensors for MPS and PEPS. These techniques are paramount for simulating ground states and dynamics of many-body systems. Variational methods leverage the TN structure to optimize energy expectation values iteratively, leading to efficient ground state approximations. Meanwhile, the imaginary time evolution is a dynamic method that exploits the approximations of e^{-H\Delta\tau} to efficiently evolve TN states toward ground states, necessitating sophisticated truncation strategies to manage bond dimensions effectively.

Conclusion and Future Directions

Tensor networks, with their rich ability to approximate complex quantum states, are highly relevant for exploring both practical and theoretical aspects of quantum many-body physics. The paper delineates how these methods transcend mere numerical utility, offering insights into quantum information metrics like entanglement structure and scaling properties. As challenges persist, especially in higher dimensions and near critical points, innovations in algorithmic strategies and computational efficiency will continue to push the boundaries of what is solvable within the TN framework. Future explorations might center around handling fermions in two dimensions, incorporating symmetries more robustly, and exploring connections with high-energy physics concepts like the holographic principle.