Matrix product state

A matrix product state (MPS) is a representation of a quantum many-body state. It is at the core of one of the most effective algorithms for solving one dimensional strongly correlated quantum systems – the density matrix renormalization group (DMRG) algorithm.

For a system of ${\displaystyle N}$ spins of dimension ${\displaystyle d}$, the general form of the MPS for periodic boundary conditions (PBC) can be written in the following form:

$${\displaystyle |\Psi \rangle =\sum _{\{s\}}\operatorname {Tr} \left[A_{1}^{(s_{1})}A_{2}^{(s_{2})}\cdots A_{N}^{(s_{N})}\right]|s_{1}s_{2}\ldots s_{N}\rangle .}$$

For open boundary conditions (OBC), ${\displaystyle |\Psi \rangle }$ takes the form

${\displaystyle |\Psi \rangle =\sum _{\{s\}}A_{1}^{(s_{1})}A_{2}^{(s_{2})}\cdots A_{N}^{(s_{N})}|s_{1}s_{2}\ldots s_{N}\rangle .}$

Here ${\displaystyle A_{i}^{(s_{i})}}$ are the ${\displaystyle D_{i}\times D_{i+1}}$ matrices (${\displaystyle D}$ is the dimension of the virtual subsystems) and ${\displaystyle |s_{i}\rangle }$ are the single-site basis states. For periodic boundary conditions, we consider ${\displaystyle D_{N+1}=D_{1}}$, and for open boundary conditions ${\displaystyle D_{1}=1}$. The parameter ${\displaystyle D}$  is related to the entanglement between particles. In particular, if the state is a product state (i.e. not entangled at all), it can be described as a matrix product state with ${\displaystyle D=1}$. ${\displaystyle \{s_{i}\}}$ represents a ${\displaystyle d}$-dimensional local space on site ${\displaystyle i=1,2,...,N}$. For qubits, ${\displaystyle s_{i}\in \{0,1\}}$. For qudits (d-level systems), ${\displaystyle s_{i}\in \{0,1,\ldots ,d-1\}}$.

For states that are translationally symmetric, we can choose: $${\displaystyle A_{1}^{(s)}=A_{2}^{(s)}=\cdots =A_{N}^{(s)}\equiv A^{(s)}.}$$ In general, every state can be written in the MPS form (with ${\displaystyle D}$ growing exponentially with the particle number N). Note that the MPS decomposition is not unique. MPS are practical when ${\displaystyle D}$ 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.

For introductions see, and. In the context of finite automata see. For emphasis placed on the graphical reasoning of tensor networks, see the introduction.