Matrix product state

A matrix product state of five particles, as drawn with Penrose graphical notation (also known as tensor diagram notation).

Matrix product state (MPS) is a pure quantum state of many particles, written in the following form:

$|\Psi\rangle = \sum_{\{s\}} \text{Tr}[A_1^{(s_1)} A_2^{(s_2)} \cdots A_N^{(s_N)}] |s_1 s_2 \ldots s_N\rangle,$

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

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

For states that are translationally symmetric, we can choose:

$A_1^{(s)} = A_2^{(s)} = \cdots = A_N^{(s)} \equiv A^{(s)}.$

In general, every state can be written in the MPS form (with $\chi$ growing exponentially with the particle number N). However, MPS are practical when $\chi$ 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 thing is not possible. Though, in many cases it serves as a good approximation.

MPS decomposition is not unique.

Introductions in.^[1] and.^[2] In the context of finite automata:^[3]

Obtaining MPS

One method to obtain MPS is to use Schmidt decomposition N − 1 times.

Examples

Greenberger–Horne–Zeilinger state

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

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

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

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

or equivalently, using notation from:^[3]

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

This notation uses matrices with entries being wave functions (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

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

Note that tensor product is not commutative.

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

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

W state

W state, i.e. a being symmetric superposition of a single one among. Even through the state is permutation-symmetric, its simplest MPS representation is not.^[1] For example:

$A_1 = \begin{bmatrix} | 0 \rangle & 0\\ | 0 \rangle & | 1 \rangle \end{bmatrix} \quad A_2 = \begin{bmatrix} | 0 \rangle & | 1 \rangle\\ 0 & | 0 \rangle \end{bmatrix} \quad A_3 = \begin{bmatrix} | 1 \rangle & 0\\ 0 & | 0 \rangle \end{bmatrix}.$

AKLT model

The AKLT ground state wavefunction, which is the historical example of MPS approach:,^[4] corresponds to the choice^[5]

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

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

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

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

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

Majumdar–Ghosh model

Majumdar–Ghosh ground state can be written as MPS with

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

See also

• Density matrix renormalization group
• Variational method (quantum mechanics)
• Renormalization
• Markov chain
• Projected Entangled Pair States (PEPS)

External links

• State of Matrix Product States – Physics Stack Exchange
• A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States^[6]

References

1. Perez-Garcia, D.; Verstraete, F.; Wolf, M.M. (2008). "Matrix product state representations". arXiv:quant-ph/0608197. Bibcode:2006quant.ph..8197P.
2. Verstraete, F.; Murg, V.; Cirac, J.I. (2008). "Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems". Advances in Physics 57 (2): 143–224. arXiv:0907.2796. Bibcode:2008AdPhy..57..143V. doi:10.1080/14789940801912366.
3. 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.
4. Affleck, Ian; Kennedy, Tom; Lieb, Elliott H.; Tasaki, Hal (1987). "Rigorous results on valence-bond ground states in antiferromagnets". Physical Review Letters 59 (7): 799–802. Bibcode:1987PhRvL..59..799A. doi:10.1103/PhysRevLett.59.799. PMID 10035874.
5. 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.
6. Orus, Roman (2013). "A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States". Annals of Physics 349: 117–158. arXiv:1306.2164. Bibcode:2014AnPhy.349..117O. doi:10.1016/j.aop.2014.06.013.