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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

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.

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.

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.

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.

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