Matrix product state
Matrix product state (MPS) is a pure quantum state of many particles, written in the following form:
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  and . In the context of finite automata see . For emphasis placed on the graphical reasoning of tensor networks, see the introduction .
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.
can be expressed as a Matrix Product State, up to normalization, with
or equivalently, using notation from:
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:
where the are Pauli matrices, or
Majumdar–Ghosh ground state can be written as MPS with
- Density matrix renormalization group
- Variational method (quantum mechanics)
- Markov chain
- Perez-Garcia, D.; Verstraete, F.; Wolf, M.M. (2008). "Matrix product state representations". arXiv:quant-ph/0608197.
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- Crosswhite, Gregory; Bacon, Dave (2008). "Finite automata for caching in matrix product algorithms". Physical Review A. 78 (1): 012356. arXiv:0708.1221. Bibcode:2008PhRvA..78a2356C. doi:10.1103/PhysRevA.78.012356.
- Biamonte, Jacob; Bergholm, Ville (2017). "Tensor Networks in a Nutshell": 35. arXiv:1708.00006.
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- Schollwöck, Ulrich (2011). "The density-matrix renormalization group in the age of matrix product states". Annals of Physics. 326: 96–192. arXiv:1008.3477. Bibcode:2011AnPhy.326...96S. doi:10.1016/j.aop.2010.09.012.
- State of Matrix Product States – Physics Stack Exchange
- A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
- Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks
- Tensor Networks in a Nutshell: An Introduction to Tensor Networks