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

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

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.

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)

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
- Fitting Algorithm (coming soon)
- Zip-Up Algorithm (coming soon)

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.

- Trotter Gate Time Evolution (coming soon)
- Time-Step Targeting Method (coming soon)
- Time-Dependent Variational Principle (TDVP) (coming soon)
- MPO Time Evolution (coming soon)
- Krylov Time Evolution (coming soon)

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