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Bulletin of the American Physical Society

DMRG Approach to Optimizing Two-Dimensional Tensor Networks

Katharine Hyatt, Miles Stoudenmire

Published Mar 5, 2020• 0 citations
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DMRG Approach to Optimizing Two-Dimensional Tensor Networks

Katharine Hyatt, Edwin Miles Stoudenmire

Tensor network algorithms have been remarkably successful solving a variety of problems in quantum many-body physics. However, algorithms to optimize two-dimensional tensor networks known as PEPS lack many of the aspects that make the seminal density matrix renormalization group (DMRG) algorithm so powerful for optimizing one-dimensional tensor networks known as matrix product states. We implement a framework for optimizing two-dimensional PEPS tensor networks which includes all of steps that make DMRG so successful for optimizing one-dimension tensor networks. We present results for several 2D spin models and discuss possible extensions and applications.

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The multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) method and the density matrix renormalization group (DMRG) are powerful workhorses applied mostly in different scientific fields. Although both methods are based on tensor network states, very different mathematical languages are used for describing them. This severely limits knowledge transfer and sometimes leads to re-inventions of ideas well known in the other field. Here, we review ML-MCTDH and DMRG theory using both MCTDH expressions and tensor network diagrams. We derive the ML-MCTDH equations of motions using diagrams and compare them with time-dependent and time-independent DMRG algorithms. We further review two selected recent advancements. The first advancement is related to optimizing unoccupied single-particle functions in MCTDH, which corresponds to subspace enrichment in the DMRG. The second advancement is related to finding optimal tree structures and on highlighting similarities and differences of tensor networks used in MCTDH and DMRG theories. We hope that this contribution will foster more fruitful cross-fertilization of ideas between ML-MCTDH and DMRG.

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Tensor network algorithms have been remarkably successful solving a variety of problems in quantum many-body physics. However, algorithms to optimize two-dimensional tensor networks known as PEPS lack many of the aspects that make the seminal density matrix renormalization group (DMRG) algorithm so powerful for optimizing one-dimensional tensor networks known as matrix product states. We implement a framework for optimizing two-dimensional PEPS tensor networks which includes all of steps that make DMRG so successful for optimizing one-dimension tensor networks. We present results for several 2D spin models and discuss possible extensions and applications.
Tensor Networks are a relatively new machine learning approach. The architectures proposed initially are inspired by approaches from quantum many-body physics simulations. One common layout is the matrix product state (MPS) also known as a tensor train optimized with gradient descent techniques. We introduce a global normalization condition, so that the MPS represents a quantum state. We investigate two optimization methods that find the locally optimal tensors and compare them regarding their effectiveness. One is based on gradient descent and the other on an adaptation of DMRG.
The multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) method and the density matrix renormalization group (DMRG) are powerful workhorses applied mostly in different scientific fields. Although both methods are based on tensor network states, very different mathematical languages are used for describing them. This severely limits knowledge transfer and sometimes leads to re-inventions of ideas well known in the other field. Here, we review ML-MCTDH and DMRG theory using both MCTDH expressions and tensor network diagrams. We derive the ML-MCTDH equations of motions using diagrams and compare them with time-dependent and time-independent DMRG algorithms. We further review two selected recent advancements. The first advancement is related to optimizing unoccupied single-particle functions in MCTDH, which corresponds to subspace enrichment in the DMRG. The second advancement is related to finding optimal tree structures and on highlighting similarities and differences of tensor networks used in MCTDH and DMRG theories. We hope that this contribution will foster more fruitful cross-fertilization of ideas between ML-MCTDH and DMRG.