Algorithms for Matrix Product States / Tensor Trains

A wide variety of efficient algorithms have been developed for MPS/TT tensor networks.

Elementary MPS/TT Algorithms

• Retrieving a Single MPS/TT Component
• Inner Product of Two MPS/TT
• Compression of MPS/TT (Using Density Matrix Algorithm)
• Sampling from MPS/TT

Solving Linear Equations

The following algorithms involve solving equations such as $A x = \lambda x$ or $A x = b$ where $x$ is a tensor in MPS/TT form.

• DMRG — Density Matrix Renormalization Group. Adaptive algorithm for finding eigenvectors in MPS form.

Summing MPS/TT networks

The following are algorithms for summing two or more MPS/TT networks and approximating the result by a single MPS/TT.

• Density Matrix Algorithm (coming soon)
• Direct Algorithm (coming soon)

Multiplying a MPS/TT by an MPO

The following are algorithms for multiplying a given MPS/TT tensor network by an MPO tensor network, resulting in a new MPS/TT that approximates the result.

• Density Matrix Algorithm
• Zip-Up Algorithm
• Direct Algorithm (coming soon)
• Fitting Algorithm (coming soon)

Time Evolution Algorithms

One reason MPS are very useful in quantum physics applications is that they can be efficiently evolved in real or imaginary time. This capability is useful for studying quantum dynamics and thermalization, and directly simulating finite-temperature systems.

• Overview
• Trotter Gate Time Evolution (TEBD)
• MPO Time Evolution (MPO $W^\mathrm{II}$)
• MPS-local methods
  • Time-Step Targeting Method (Local Krylov)
  • Time-Dependent Variational Principle (TDVP)
• Krylov Time Evolution (Global Krlyov)
• Additional tricks

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