Matrix product state

It is at the core of the one of the most effective algorithms for solving one dimensional strongly correlated quantum systems – the density matrix renormalization group (DMRG) algorithm.

For open boundary conditions (OBC),

In general, every state can be written in the MPS form (with

growing exponentially with the particle number N).

is small – for example, does not depend on the particle number.

Except for a small number of specific cases (some mentioned in the section Examples), such a thing is not possible, though in many cases it serves as a good approximation.

[5] For emphasis placed on the graphical reasoning of tensor networks, see the introduction.

-dimensional Hilbert space, the completely general state can be written as

For example, the wave function of the system described by the Heisenberg model is defined by the

dimensional tensor, whereas for the Hubbard model the rank is

The main idea of the MPS approach is to separate physical degrees of freedom of each site, so that the wave function can be rewritten as the product of

matrices, where each matrix corresponds to one particular site.

The whole procedure includes the series of reshaping and singular value decompositions (SVD).

, which describes physical degrees of freedom of the first site.

So, the state vector takes the form

-matrices take place in the case of the exact decomposition, i.e., assuming for simplicity that

However, due to the exponential growth of the matrix dimensions in most of the cases it is impossible to perform the exact decomposition.

The dual MPS is defined by replacing each matrix

in the SVD is a semi-unitary matrix with property

Since matrices are left-normalized, we call the composition left-canonical.

Performing the series of reshaping and SVD, the state vector takes the form

in the SVD is a semi-unitary matrix with property

Assuming that the left-canonical decomposition was performed for the first n sites,

and proceed with the series of reshaping and SVD from the right up to site

Greenberger–Horne–Zeilinger state, which for N particles can be written as superposition of N zeros and N ones can be expressed as a Matrix Product State, up to normalization, with or equivalently, using notation from:[10] This notation uses matrices with entries being state vectors (instead of complex numbers), and when multiplying matrices using tensor product for its entries (instead of product of two complex numbers).

Such matrix is constructed as Note that tensor product is not commutative.

In this particular example, a product of two A matrices is: W state, i.e., the superposition of all the computational basis states of Hamming weight one.

Even though the state is permutation-symmetric, its simplest MPS representation is not.

[1] For example: The AKLT ground state wavefunction, which is the historical example of MPS approach,[11] corresponds to the choice[9] where the

are Pauli matrices, or Majumdar–Ghosh ground state can be written as MPS with