Matrix product state

For a system of $N$ lattice sites each of which has a $d$-dimensional Hilbert space, the completely general state can be written as

$|\backslash Psi\backslash rangle\; =\; \backslash sum\_\{\backslash \{s\backslash \}\}\; \backslash psi\_\{s\_1...s\_N\}\; |s\_1\; \backslash ldots\; s\_N\backslash rangle\; ,$

where $\backslash psi\_\{s\_1...s\_N\}$ is a $d^N$-dimensional tensor. For example, the wave function of the system described by the Heisenberg model is defined by the $2^N$ dimensional tensor, whereas for the Hubbard model the rank is $4^N$.

The main idea of the MPS approach is to separate physical degrees of freedom of each site, so that the wave function can be rewritten as the product of $N$ matrices, where each matrix corresponds to one particular site. The whole procedure includes the series of reshaping and singular value decompositions (SVD).{{Cite journal |last1=Baker |first1=Thomas E. |last2=Desrosiers |first2=Samuel |last3=Tremblay |first3=Maxime |last4=Thompson |first4=Martin P. |title=Méthodes de calcul avec réseaux de tenseurs en physique |url=https://cdnsciencepub.com/doi/10.1139/cjp-2019-0611 |journal=Canadian Journal of Physics |date=2021 |language=en |volume=99 |issue=4 |pages=207–221 |doi=10.1139/cjp-2019-0611 |arxiv=1911.11566 |bibcode=2021CaJPh..99..207B |issn=0008-4204}}{{Citation |last1=Baker |first1=Thomas E. |title=Build your own tensor network library: DMRjulia I. Basic library for the density matrix renormalization group |date=2021-09-07 |url=https://arxiv.org/abs/2109.03120 |access-date=2024-11-03 |arxiv=2109.03120 |last2=Thompson |first2=Martin P.}}

There are three ways to represent wave function as an MPS: left-canonical decomposition, right-canonical decomposition, and mixed-canonical decomposition.

The decomposition of the $d^N$-dimensional tensor starts with the separation of the very left index, i.e., the first index $s\_1$, which describes physical degrees of freedom of the first site. It is performed by reshaping $|\backslash Psi\backslash rangle$ as follows

$|\backslash Psi\backslash rangle\; =\; \backslash sum\_\{\backslash \{s\backslash \}\}\; \backslash psi\_\{s\_1,\; (s\_2...s\_N\; )\}\; |s\_1\; \backslash ldots\; s\_N\backslash rangle.$

In this notation, $s\_1$ is treated as a row index, $(\; s\_2\; \backslash ldots\; s\_N)$ as a column index, and the coefficient $\backslash psi\_\{s\_1,(s\_2...s\_N\; )\}$ is of dimension $(d\; \backslash times\; d^\{N-1\})$. The SVD procedure yields

$\backslash psi\_\{s\_1,(s\_2...s\_N\; )\}\; =\; \backslash sum\_\{\backslash alpha\_1\}^\{r\_1\}\; U\_\{s\_1,\backslash alpha\_1\}\; D\_\{\backslash alpha\_1,\backslash alpha\_1\}\; (V^\{\backslash dagger\})\_\{\backslash alpha\_1,(s\_2...s\_N\; )\}\; =\; \backslash sum\_\{\backslash alpha\_1\}^\{r\_1\}\; U\_\{s\_1,\backslash alpha\_1\}\backslash psi\_\{\backslash alpha\_1,\; (s\_2...s\_N\; )\}\; =\; \backslash sum^\{r\_1\}\_\{\backslash alpha\_1\}\; A^\{s\_1\}\_\{\backslash alpha\_1\}\; \backslash psi\_\{\backslash alpha\_1,(s\_2...s\_N\; )\}.$

File:Left_mps_1.png

In the relation above, matrices $D$ and $V^\{\backslash dagger\}$ are multiplied and form the matrix $\backslash psi\_\{\backslash alpha\_1,(s\_2...s\_N\; )\}$ and $r\_1\; \backslash leq\; d$. $A^\{s\_1\}\_\{\backslash alpha\_1\}$ stores the information about the first lattice site. It was obtained by decomposing matrix $U$ into $d$ row vectors $A^\{s\_1\}$ with entries $A^\{s\_1\}\_\{\backslash alpha\_1\}=U\_\{s\_1,\backslash alpha\_1\}$. So, the state vector takes the form

$|\backslash Psi\backslash rangle\; =\; \backslash sum\_\{\backslash \{s\backslash \}\}\; \backslash sum\_\{\backslash alpha\_1\}\; A^\{s\_1\}\_\{\backslash alpha\_1\}\; \backslash psi\_\{\backslash alpha\_1,(s\_2...s\_N\; )\}\; |s\_1\; \backslash ldots\; s\_N\backslash rangle.$

The separation of the second site is performed by grouping $s\_2$ and $\backslash alpha\_1$, and representing $\backslash psi\_\{\backslash alpha\_1,(s\_2...s\_N\; )\}$ as a matrix $\backslash psi\_\{(\backslash alpha\_1\; s\_2),(s\_3...s\_N\; )\}$ of dimension $(r\_1\; d\; \backslash times\; d^\{N-2\})$. The subsequent SVD of $\backslash psi\_\{(\backslash alpha\_1\; s\_2),(s\_3...s\_N\; )\}$ can be performed as follows:

$\backslash psi\_\{\; (\backslash alpha\_1\; s\_2),\; (s\_3...s\_N\; )\; \}\; =\; \backslash sum^\{r\_2\}\_\{\backslash alpha\_2\}\; U\_\{\; (\backslash alpha\_1\; s\_2),\; \backslash alpha\_2\}\; D\_\{\backslash alpha\_2,\backslash alpha\_2\}\; (V^\{\backslash dagger\})\_\{\backslash alpha\_2,(s\_3\; ...\; s\_N)\}\; =\; \backslash sum^\{r\_2\}\_\{\backslash alpha\_2\}\; A^\{s\_2\}\_\{\backslash alpha\_1,\backslash alpha\_2\}\backslash psi\_\{\backslash alpha\_2,(s\_3\; ...\; s\_N)\}$.

File:Left_mps_2.png

Above we replace $U$ by a set of $d$ matrices of dimension $(r\_1\; \backslash times\; r\_2)$ with entries $A^\{s\_2\}\_\{\backslash alpha\_1,\backslash alpha\_2\}\; =\; U\_\{\; (\backslash alpha\_1\; s\_2),\; \backslash alpha\_2\}$. The dimension of $\backslash psi\_\{\backslash alpha\_2,(s\_3\; ...\; s\_N)\}$ is $(r\_2\; \backslash times\; d^\{N-2\})$ with $r\_2\; \backslash leq\; r\_1\; d\; \backslash leq\; d^2$. Hence,

$|\backslash Psi\backslash rangle\; =\; \backslash sum\_\{\backslash \{s\backslash \}\}\; \backslash sum\_\{\backslash alpha\_1\}\; A^\{s\_1\}\_\{\backslash alpha\_1\}\; \backslash psi\_\{(\backslash alpha\_1\; s\_2),(s\_3...s\_N\; )\}\; |s\_1\; \backslash ldots\; s\_N\backslash rangle\; =\; \backslash sum\_\{\backslash \{s\backslash \}\}\; \backslash sum\_\{\backslash alpha\_1,\; \backslash alpha\_2\}\; A^\{s\_1\}\_\{\backslash alpha\_1\}\; A^\{s\_2\}\_\{\backslash alpha\_1,\; \backslash alpha\_2\}\; \backslash psi\_\{\backslash alpha\_2,(s\_3...s\_N\; )\}\; |s\_1\; \backslash ldots\; s\_N\backslash rangle.$

Following the steps described above, the state $|\backslash Psi\backslash rangle$ can be represented as a product of matrices

$|\backslash Psi\backslash rangle\; =\backslash sum\_\{\backslash \{s\backslash \}\}\; \backslash sum\_\{\backslash alpha\_1,\; \backslash ldots,\; \backslash alpha\_\{N-1\}\}\; A^\{s\_1\}\_\{\backslash alpha\_1\}\; A^\{s\_2\}\_\{\backslash alpha\_1,\backslash alpha\_2\}\backslash ldots\; A^\{s\_\{N-1\}\}\_\{\backslash alpha\_\{N-2\},\backslash alpha\_\{N-1\}\}\; A^\{s\_N\}\_\{\backslash alpha\_\{N-1\}\}\; |s\_1\; \backslash ldots\; s\_N\backslash rangle.$

The maximal dimensions of the $A$-matrices take place in the case of the exact decomposition, i.e., assuming for simplicity that $N$ is even, $(1\backslash times\; d),(d\backslash times\; d^2),\backslash ldots,(d^\{N/2-1\}\backslash times\; d^\{N/2\}),(d^\{N/2\}\backslash times\; d^\{N/2-1\}),\backslash ldots,(d^2\backslash times\; d),(d\backslash times\; 1)$ going from the first to the last site. However, due to the exponential growth of the matrix dimensions in most of the cases it is impossible to perform the exact decomposition.

The dual MPS is defined by replacing each matrix $A$ with $A^*$:

$\backslash langle\backslash Psi|$

=\sum\limits_{\{s\}}\sum\limits_{\alpha'_1,...,\alpha'_{N-1}}A^{*s'_1}_{\alpha'_1}A^{*s'_2}_{\alpha'_1,\alpha'_2}...A^{*s'_{N-1}}_{\alpha'_{N-2},\alpha'_{N-1}}A^{*s'_{N}}_{\alpha'_{N-1}}\langle s'_1...s'_N|.

Note that each matrix $U$ in the SVD is a semi-unitary matrix with property $U^\backslash dagger\; U\; =\; I$. This leads to

$\backslash delta\_\{\backslash alpha\_i,\backslash alpha\_j\}\; =\; \backslash sum\_\{\backslash alpha\_\{i-1\}s\_i\}\; (U^\backslash dagger)\_\{\backslash alpha\_i,\; (\backslash alpha\_\{i-1\}s\_i)\}\; U\_\{(\backslash alpha\_\{i-1\}s\_i),\; \backslash alpha\_j\}\; =\; \backslash sum\_\{\backslash alpha\_\{i-1\}s\_i\}\; (A^\{s\_i\; \backslash dagger\})\_\{\backslash alpha\_i,\; \backslash alpha\_\{i-1\}\}\; A^\{s\_i\}\_\{\backslash alpha\_\{i-1\},\; \backslash alpha\_j\}\; =\; \backslash sum\_\{s\_i\}\; (A^\{s\_i\; \backslash dagger\}\; A^\{s\_i\})\_\{\backslash alpha\_i,\; \backslash alpha\_j\}$.

To be more precise, $\backslash sum\_\{s\_i\}\; A^\{s\_i\; \backslash dagger\}\; A^\{s\_i\}\; =\; I$. Since matrices are left-normalized, we call the composition left-canonical.

Similarly, the decomposition can be started from the very right site. After the separation of the first index, the tensor $\backslash psi\_\{s\_1...s\_N\}$ transforms as follows:

$\backslash psi\_\{s\_1...s\_N\}\; =\; \backslash psi\_\{(s\_1\; ...\; s\_\{N-1\}),s\_N\}\; =\; \backslash sum\_\{\backslash alpha\_\{N-1\}\}\; U\_\{(s\_1\; ...\; s\_\{N-1\}),\backslash alpha\_\{N-1\}\}\; D\_\{\backslash alpha\_\{N-1\},\backslash alpha\_\{N-1\}\}(V^\backslash dagger)\_\{\backslash alpha\_\{N-1\},\; s\_N\}\; =\; \backslash sum\_\{\backslash alpha\_\{N-1\}\}\; \backslash psi\_\{(s\_1\; ...\; s\_\{N-1\}),\backslash alpha\_\{N-1\}\}\; B^\{s\_N\}\_\{\backslash alpha\_\{N-1\}\}$.

The matrix $\backslash psi\_\{(s\_1\; ...\; s\_\{N-1\}),\backslash alpha\_\{N-1\}\}$ was obtained by multiplying matrices $U$ and $D$, and the reshaping of $(V^\backslash dagger)\_\{\backslash alpha\_\{N-1\},\; s\_N\}$ into $d$ column vectors forms $B^\{s\_N\}\_\{\backslash alpha\_\{N-1\}\}$. Performing the series of reshaping and SVD, the state vector takes the form

$|\backslash Psi\backslash rangle\; =\backslash sum\_\{\backslash \{s\backslash \}\}\; \backslash sum\_\{\backslash alpha\_1,\; \backslash ldots,\; \backslash alpha\_\{N-1\}\}\; B^\{s\_1\}\_\{\backslash alpha\_1\}\; B^\{s\_2\}\_\{\backslash alpha\_1,\backslash alpha\_2\}\backslash ldots\; B^\{s\_\{N-1\}\}\_\{\backslash alpha\_\{N-2\},\backslash alpha\_\{N-1\}\}\; B^\{s\_N\}\_\{\backslash alpha\_\{N-1\}\}\; |s\_1\; \backslash ldots\; s\_N\backslash rangle.$

Since each matrix $V$ in the SVD is a semi-unitary matrix with property $V^\backslash dagger\; V\; =\; I$, the $B$-matrices are right-normalized and obey $\backslash sum\_\{s\_i\}\; B^\{s\_i\}\; B^\{s\_i\; \backslash dagger\}\; =\; I$. Hence, the decomposition is called right-canonical.

The decomposition performs from both the right and from the left. Assuming that the left-canonical decomposition was performed for the first n sites, $\backslash psi\_\{s\_1...s\_N\}$ can be rewritten as

$\backslash psi\_\{s\_1...s\_N\}\; =\; \backslash sum\_\{\backslash alpha\_1,\; \backslash ldots,\; \backslash alpha\_\{n\}\}\; A^\{s\_1\}\_\{\backslash alpha\_1\}\; A^\{s\_2\}\_\{\backslash alpha\_1,\backslash alpha\_2\}\backslash ldots\; A^\{s\_\{n\}\}\_\{\backslash alpha\_\{n-1\},\backslash alpha\_\{n\}\}\; D\_\{\backslash alpha\_n,\; \backslash alpha\_n\}\; (V^\backslash dagger)\_\{\backslash alpha\_n,\; (s\_\{n+1\}...s\_N)\}$.

File:Mixed_mps.png

In the next step, we reshape $(V^\backslash dagger)\_\{\backslash alpha\_n,\; (s\_\{n+1\}...s\_N)\}$ as $\backslash psi\_\{(\backslash alpha\_n\; s\_\{n+1\}\; ...\; s\_\{n-1\}),\; s\_N\}$ and proceed with the series of reshaping and SVD from the right up to site $s\_\{n+1\}$:

$\backslash begin\{align\}$

\psi_{(\alpha_n s_{n+1} ... s_{n-1}), s_N} &=& \sum_{\alpha_{n+1} ... \alpha_N} U_{(\alpha_n s_{n+1}), \alpha_{n+1}} D_{\alpha_{n+1}, \alpha_{n+1}}B^{s_{n+2}}_{\alpha_{n+1},\alpha_{n+2}}\ldots B^{s_{N-1}}_{\alpha_{N-2},\alpha_{N-1}} B^{s_N}_{\alpha_{N-1}} \\

&=& \sum_{\alpha_{n+1} ... \alpha_N} B^{s_{n+1}}_{\alpha_n, \alpha_{n+1}} B^{s_{n+2}}_{\alpha_{n+1},\alpha_{n+2}}\ldots B^{s_{N-1}}_{\alpha_{N-2},\alpha_{N-1}} B^{s_N}_{\alpha_{N-1}}

\end{align}.

As the result,

$\backslash psi\_\{s\_1...s\_N\}\; =\; \backslash sum\_\{\backslash alpha\_1,\; \backslash ldots,\backslash alpha\_\{N\}\}\; A^\{s\_1\}\_\{\backslash alpha\_1\}\; A^\{s\_2\}\_\{\backslash alpha\_1,\backslash alpha\_2\}\backslash ldots\; A^\{s\_\{n\}\}\_\{\backslash alpha\_\{n-1\},\backslash alpha\_\{n\}\}\; D\_\{\backslash alpha\_n,\; \backslash alpha\_n\}B^\{s\_\{n+1\}\}\_\{\backslash alpha\_n,\; \backslash alpha\_\{n+1\}\}\; B^\{s\_\{n+2\}\}\_\{\backslash alpha\_\{n+1\},\backslash alpha\_\{n+2\}\}\backslash ldots\; B^\{s\_\{N-1\}\}\_\{\backslash alpha\_\{N-2\},\backslash alpha\_\{N-1\}\}\; B^\{s\_N\}\_\{\backslash alpha\_\{N-1\}\}$.

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

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

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:

:$$

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

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

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

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

Even though the state is permutation-symmetric, its simplest MPS representation is not. 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}.

{{Main|AKLT model}}

The AKLT ground state wavefunction, which is the historical example of MPS approach, corresponds to the choice

: $A^\{+\}\; =\; \backslash sqrt\{\backslash frac\{2\}\{3\}\}\backslash \; \backslash sigma^\{+\}$

=

\begin{bmatrix}

0 & \sqrt{2/3}\\

0 & 0

\end{bmatrix}

: $A^\{0\}\; =\; \backslash frac\{-1\}\{\backslash sqrt\{3\}\}\backslash \; \backslash sigma^\{z\}$

=

\begin{bmatrix}

-1/\sqrt{3} & 0\\

0 & 1/\sqrt{3}

\end{bmatrix}

: $A^\{-\}\; =\; -\backslash sqrt\{\backslash frac\{2\}\{3\}\}\backslash \; \backslash sigma^\{-\}$

=

\begin{bmatrix}

0 & 0\\

-\sqrt{2/3} & 0

\end{bmatrix}

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

: $$

A =

\frac{1}{\sqrt{3}}

\begin{bmatrix}

- | 0 \rangle & \sqrt{2} | + \rangle\\

- \sqrt{2} | - \rangle & | 0 \rangle

\end{bmatrix}.

{{Main|Majumdar–Ghosh model}}

Majumdar–Ghosh ground state can be written as MPS with

:$$

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}.

• Density matrix renormalization group
• Variational method (quantum mechanics)
• Renormalization
• Markov chain
• Tensor network

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• [https://tensornetwork.org Open-source review article focused on tensor network algorithms, applications, and software]
• [http://physics.stackexchange.com/questions/27043/state-of-matrix-product-states State of Matrix Product States – Physics Stack Exchange]
• [https://arxiv.org/abs/1306.2164 A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States]
• [https://arxiv.org/abs/1603.03039 Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks]
• [https://arxiv.org/abs/1708.00006 Tensor Networks in a Nutshell: An Introduction to Tensor Networks]

Category:Mathematical physics

Category:Quantum states