Tensor networks for complex quantum systems (1812.04011v2)

Abstract: Tensor network states and methods have erupted in recent years. Originally developed in the context of condensed matter physics and based on renormalization group ideas, tensor networks lived a revival thanks to quantum information theory and the understanding of entanglement in quantum many-body systems. Moreover, it has been not-so-long realized that tensor network states play a key role in other scientific disciplines, such as quantum gravity and artificial intelligence. In this context, here we provide an overview of basic concepts and key developments in the field. In particular, we briefly discuss the most important tensor network structures and algorithms, together with a sketch on advances related to global and gauge symmetries, fermions, topological order, classification of phases, entanglement Hamiltonians, AdS/CFT, artificial intelligence, the 2d Hubbard model, 2d quantum antiferromagnets, conformal field theory, quantum chemistry, disordered systems, and many-body localization.

• The paper introduces tensor networks as an approach to encode quantum many-body states by reducing exponential complexity to a manageable polynomial scale using bond dimensions.
• The paper examines key architectures such as MPS, PEPS, TTN, MERA, and bMERA, demonstrating their roles in capturing entanglement in one- and two-dimensional systems.
• The paper details advanced algorithms like DMRG, TEBD, and CTM for optimizing and contracting tensor networks, highlighting their impact on simulating dynamic and critical quantum phenomena.

Overview of Tensor Networks for Complex Quantum Systems

The paper "Tensor Networks for Complex Quantum Systems" presents a comprehensive exploration of tensor networks (TNs) as a methodological approach for representing quantum many-body states. Originating from condensed matter physics, TNs have come to serve as vital tools for tackling complex quantum systems due to their capacity for encapsulating quantum states in exponentially-large Hilbert spaces with remarkable computational efficiency.

Tensor networks capitalize on the inherent entanglement structure of quantum states. Tensors are interconnected in a network that reflects the many-body entanglement properties of the system. Within practical scenarios, especially in numerically intensive tasks, TNs revolutionize computational accessibility by reducing the parameter space from exponential to polynomial order, parameterized by the "bond dimension" $\chi$, which is a pivotal determinant of the network's expressiveness.

Core Tensor Network Structures

The paper dissects several prime tensor network structures instrumental in physical and computational domains:

1. Matrix Product States (MPS): Predominantly used in one-dimensional (1D) systems, MPS are particularly suited for low-energy states of gapped 1D Hamiltonians and prove efficient given their adherence to a 1D area law.
2. Projected Entangled Pair States (PEPS): Applied to two-dimensional (2D) quantum systems, PEPS accommodate the entanglement structures in low-energy states of 2D local Hamiltonians, although they require advanced approximation techniques for efficient contraction.
3. Tree Tensor Networks (TTN) and Multiscale Entanglement Renormalization Ansatz (MERA): These architectures provide alternative representations which incorporate scale-dependent analysis, essential for understanding systems under renormalization group frameworks, and are particularly useful in critical systems that exhibit scale invariance.
4. Branching MERA (bMERA): This extension of MERA allows for the representation of systems that demonstrate entanglement features beyond simple scale invariance, facilitating analysis of systems with diverse entanglement entropy scaling.

Algorithms and Their Computational Costs

The paper outlines sophisticated algorithms for optimizing and contracting tensor networks, critical for practical implementations:

• Density Matrix Renormalization Group (DMRG): As a factorization method of choice for 1D systems, DMRG governs the optimization of MPS through efficient sweeping algorithms.
• Time-Evolving Block Decimation (TEBD): Used for dynamic simulations of 1D systems, TEBD enables both real and imaginary-time evolution, harnessing the local entanglement dynamics.
• Boundary MPS and Corner Transfer Matrices (CTM): They are effective strategies for handling the contraction of 2D PEPS, with CTMs particularly favorable due to their computational tractability.

The algorithms discussed diagnose a variety of systems ranging from quantum phase transitions to topologically ordered states, proving essential for understanding emergent phenomena in complex quantum systems.

Implications and Future Prospects

Theoretical advancements outlined in this paper have substantial implications for the paper of quantum entanglement and many-body physics. Tensor networks not only provide a framework for understanding quantum field theories, particularly those associated with the AdS/CFT correspondence but also offer a toolkit for exploring quantum computation and simulation.

Future research directions include the further integration of TNs with machine learning methodologies, as suggested by their analogous structures, and the unfolding of richer physical phenomena in higher-dimensional quantum systems and complex materials. The paper posits that as the understanding of quantum correlations deepens, tensor networks will remain pivotal in decoding the layers of complexity in quantum states, continually enhancing their applicability in interdisciplinary scientific pursuits.