Tensor network algorithms in computational physics and numerical analysis
- Philippe Corboz (Swiss Federal Institute of Technology Zurich (ETHZ), Switzerland)
- Daniel Kressner (EPF Lausanne, Switzerland)
- Matthias Troyer (Swiss Federal Institute of Technology Zurich (ETHZ), Switzerland)
One of the biggest challenges in Condensed Matter Physics is the study of strongly correlated quantum many-body systems, which give rise to fascinating phenomena such as high-Tc superconductivity, Mott insulators, quantum magnetism, or the fractional quantum Hall effect. Quantum Monte Carlo (QMC) is one of the most powerful methods to simulate these systems, however, it fails for important classes of models (fermionic and frustrated models) due to the so-called negative sign problem, which is the reason why the physics of many models is still controversial. Similar challenges are faced in Quantum Chemistry, where standard density functional theory fails to capture strong correlation effects in molecules, and exact calculations are limited to only a small number of orbitals. In order to make substantial progress in the understanding of strongly correlated systems, it is crucial to develop new accurate and efficient numerical tools to simulate these systems.
An enormous progress in the simulation of strongly correlated systems has been achieved with so-called tensor network algorithms. The main idea is to efficiently represent quantum many-body states by a tensor network ansatz, where the accuracy can be systematically controlled by the so-called bond dimension of the tensors. Examples of tensor networks for (quasi-) one-dimensional systems are matrix product states (MPS), which form the class of states underlying the famous density-matrix renormalization group (DMRG) method, and the multi-scale entanglement renormalization ansatz (MERA). Progress in Quantum Information Theory, in particular a better understanding of entanglement in quantum many-body systems, led to the development of tensor networks for two-dimensional systems, including e.g. projected entangled-pair states (PEPS) or the 2D MERA. Various algorithms exist and are being further developed to simulate ground state properties, systems at finite temperature, dissipative systems, real-time evolution and more.
In numerical analysis, the development of efficient algorithms for solving high-dimensional partial differential equations (PDEs) is currently one of the most active research directions. The discretization of such PDEs by standard techniques, such as the finite element method, leads to an exponentially growing number of degrees of freedom as the dimension increases. To a certain extent, this growth can be limited by the use of more sophisticated discretization techniques, most notably sparse grids. In recent years, an alternative approach has been pursued that aims to address the challenge of high dimensionality in a different manner, after the discretization. For this purpose, the solution of the discretized problem (for example, a linear system or eigenvalue problem) is approximated in a low-rank tensor format, analogous to the tensor network ansatz mentioned above. The success of this approach has been demonstrated for an impressive range of different applications in computational engineering, finance, and physics. Still, the understanding of the precise scope and limitations for such low-rank tensor techniques is far from complete and more research is urgently needed. For example, very few a priori approximation results are available, allowing for insights into the expected decrease in the effective number of degrees of freedom and guiding the appropriate choice of low-rank tensor format.
Although some of the low-rank tensor techniques in the numerical analysis community have been inspired by tensor network techniques in computational chemistry and physics, most of these partially overlapping developments have taken place independently. This is partly due to different objectives in both communities, but there would also be a clear benefit gained from increased interdisciplinary communication and
Matrix product states and the DMRG method have revolutionized the simulation of one dimensional strongly correlated systems [1,2] and had a remarkable impact also in Quantum Chemistry . Even for two-dimensional systems, MPS belong to the state-of-the-art tools [3,4], despite the exponential scaling of the computational cost with the width of the system. Two-dimensional tensor networks like PEPS [5,6] and the 2D MERA  do not suffer from this exponential scaling, i.e. the cost is polynomial, however, with a large power of the bond dimension. Nevertheless, already today - still at an early stage of development - they can compete with the best known variational methods , and they thus provide, together with MPS, one of the most promising routes to solve longstanding problems such as e.g. the 2D Hubbard model. Another key model in Condensed Matter Physics, the Heisenberg model on the Kagome lattice, has recently been solved using large scale DMRG simulations [3,4], which is one of the recent milestones in the field.
Low-rank tensor techniques have been successfully used to address discretized high-dimensional, parametrized, stochastic, and multi-scale PDEs from different application domains, see  for an overview. Algorithms inspired by the DMRG method [10,11] as well as combinations of classical iterative methods with low-rank truncation [12,13] have been proposed. Recently, connections to algebraic geometry  and differential geometry [14,] have been explored that not only enhance the theoretical understanding but can also be expected to lead to more robust algorithms.
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