Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions

By Frank Verstraete, J. I. Cirac. 2004

951 citations recorded by the source corpus.

We describe quantum many--body systems in terms of projected entangled--pair states, which naturally extend matrix product states to two and more dimensions. We present an algorithm to determine correlation functions in an efficient way. We use this result to build powerful numerical simulation techniques to describe the ground state, finite temperature, and evolution of spin systems in two and higher dimensions.

Subfields: Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

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In Atomic and Molecular Physics, and Optics

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “Well-known examples include the so called projected entangled-pair states (PEPS) [260, 261] and the multiscale entanglement renormalization ansatz (MERA) [264].”
Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy) (2018, background): “Tensor-product states (TPS) or equivalently projected entangled pair states (PEPS), are a generalization of MPS to higher dimensions [34, 35].”
Matrix product simulations of non-equilibrium steady states of quantum spin chains (2008, background): “These ideas have been further developed by treating periodic boundary conditions [10], two-dimensional and higher lattices [11], translationally invariant systems [12, 13], and generalizing the MPS ansatz to tensor tree networks describing critical models [14].”
The density matrix renormalization group for ab initio quantum chemistry (2014, background): “An inconvenient property of the PEPS, MERA, and MPS with periodic boundary conditions [60], is the introduction of loops in the network.”
Unsupervised Generative Modeling Using Matrix Product States (2017, background): “[49] Frank Verstraete and J Ignacio Cirac, “Renormalization algorithms for quantum-many body systems in two and higher di-”
Quantum Simulation (2013, methodology): “The first steps in this direction were taken in (Verstraete and Cirac, 2004; Verstraete et al., 2004) and thereafter a significant effort has been made to explore this idea.”
Operator quantum error-correcting subsystems for self-correcting quantum memories (2005, background): “Another promising approach is to use recent new ideas in the density matrix renormalization group[57, 58] to simulate the thermal properties of this system.”
Supervised Learning with Tensor Networks (2016, methodology): “While applications of MPS to machine learning have been a success, one aim of the present work is to have tensor networks play a more central role in developing learning models; another is to more easily incorporate powerful algorithms and tensor networks which generalize MPS developed by the physic”
Gauging Quantum States: From Global to Local Symmetries in Many-Body Systems (2014, methodology): “However, recent results for the tensor network description of higher dimensional quantum systems [27] (including fermions [28–32]), quantum chemistry models [33–35] and even quantum field theories [36, 37] look equally promising.”
Machine learning \& artificial intelligence in the quantum domain (2017, methodology): “Related to this, the Tensor Network (Östlund and Rommer, 1995; Verstraete and Cirac, 2004) formalism has been used for the efficient description of deep convolutional arithmetic circuits, establishing also a formal connection between quantum many-body”