A Matrix product state (MPS) is a quantum state of many particles (in N sites), 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 ,}$

where ${\displaystyle A_{i}^{(s_{i})))$ are complex, square matrices of order ${\displaystyle \chi }$ (this dimension is called local dimension). Indices ${\displaystyle s_{i))$ go over states in the computational basis. For qubits, it is ${\displaystyle s_{i}\in \{0,1\))$. For qudits (d-level systems), it is ${\displaystyle s_{i}\in \{0,1,\ldots ,d-1\))$.

It is particularly useful for dealing with ground states of one-dimensional quantum spin models (e.g. Heisenberg model (quantum)). The parameter ${\displaystyle \chi }$ 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 ${\displaystyle \chi =1}$.

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 ${\displaystyle \chi }$ growing exponentially with the particle number N). However, MPS are practical when ${\displaystyle \chi }$ 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.

The MPS decomposition is not unique. For introductions see ^[1] and.^[2] In the context of finite automata see.^[3] For emphasis placed on the graphical reasoning of tensor networks, see the introduction.^[4]

Obtaining MPS

One method to obtain an MPS representation of a quantum state is to use Schmidt decomposition N − 1 times. Alternatively if the quantum circuit which prepares the many body state is known, one could first try to obtain a matrix product operator representation of the circuit. The local tensors in the matrix product operator will be four index tensors. The local MPS tensor is obtained by contracting one physical index of the local MPO tensor with the state which is injected into the quantum circuit at that site.

Examples

Greenberger–Horne–Zeilinger state

Greenberger–Horne–Zeilinger state, which for N particles can be written as superposition of N zeros and N ones

${\displaystyle |\mathrm {GHZ} \rangle ={\frac {|0\rangle ^{\otimes N}+|1\rangle ^{\otimes N)){\sqrt {2))))$

can be expressed as a Matrix Product State, up to normalization, with

${\displaystyle A^{(0)}={\begin{bmatrix}1&0\\0&0\end{bmatrix))\quad A^{(1)}={\begin{bmatrix}0&0\\0&1\end{bmatrix)),}$

or equivalently, using notation from:^[3]

${\displaystyle A={\begin{bmatrix}|0\rangle &0\\0&|1\rangle \end{bmatrix)).}$

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

${\displaystyle A\equiv |0\rangle A^{(0)}+|1\rangle A^{(1)}+\ldots +|d-1\rangle A^{(d-1)}.}$

Note that tensor product is not commutative.

In this particular example, a product of two A matrices is:

${\displaystyle AA={\begin{bmatrix}|00\rangle &0\\0&|11\rangle \end{bmatrix)).}$

W state

W state, i.e., the superposition of all the computational basis states of Hamming weight one.

${\displaystyle |\mathrm {W} \rangle ={\frac {1}{\sqrt {3))}(|001\rangle +|010\rangle +|100\rangle )}$

Even though the state is permutation-symmetric, its simplest MPS representation is not.^[1] For example:

${\displaystyle A_{1}={\begin{bmatrix}|0\rangle &0\\|0\rangle &|1\rangle \end{bmatrix))\quad A_{2}={\begin{bmatrix}|0\rangle &|1\rangle \\0&|0\rangle \end{bmatrix))\quad A_{3}={\begin{bmatrix}|1\rangle &0\\0&|0\rangle \end{bmatrix)).}$

AKLT model

The AKLT ground state wavefunction, which is the historical example of MPS approach:,^[5] corresponds to the choice^[6]

${\displaystyle A^{+}={\sqrt {\frac {2}{3))}\ \sigma ^{+}={\begin{bmatrix}0&{\sqrt {2/3))\\0&0\end{bmatrix))}$

${\displaystyle A^{0}={\frac {-1}{\sqrt {3))}\ \sigma ^{z}={\begin{bmatrix}-1/{\sqrt {3))&0\\0&1/{\sqrt {3))\end{bmatrix))}$

${\displaystyle A^{-}=-{\sqrt {\frac {2}{3))}\ \sigma ^{-}={\begin{bmatrix}0&0\\-{\sqrt {2/3))&0\end{bmatrix))}$

where the ${\displaystyle \sigma {\text{'s))}$ are Pauli matrices, or

${\displaystyle A={\frac {1}{\sqrt {3))}{\begin{bmatrix}-|0\rangle &{\sqrt {2))|+\rangle \\-{\sqrt {2))|-\rangle &|0\rangle \end{bmatrix)).}$

Majumdar–Ghosh model

Majumdar–Ghosh ground state can be written as MPS with

${\displaystyle A={\begin{bmatrix}0&\left|\uparrow \right\rangle &\left|\downarrow \right\rangle \\{\frac {-1}{\sqrt {2))}\left|\downarrow \right\rangle &0&0\\{\frac {1}{\sqrt {2))}\left|\uparrow \right\rangle &0&0\end{bmatrix)).}$