A Matrix product state (MPS) is a quantum state of many particles (in N sites), written in the following form:
where are complex, square matrices of order (this dimension is called local dimension). Indices go over states in the computational basis. For qubits, it is . For qudits (d-level systems), it is .
It is particularly useful for dealing with ground states of one-dimensional quantum spin models (e.g. Heisenberg model (quantum)).
The parameter 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 .
For states that are translationally symmetric, we can choose:
In general, every state can be written in the MPS form (with growing exponentially with the particle number N). However, MPS are practical when 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 decompositionN − 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.
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
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