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Tensor Network

Matrix

A matrix has two indices. Consider a $3 \times 4$ matrix $A$. It has $3$ rows and $4$ columns. The entry in the $i$-th row and $j$-th column is denoted as $A_{i,j}$ or $A[i,j]$. The first index represents the row index, while the second index represents the column index. It is commonly written in box brackets.

$$ [A] = \left[ \begin{array}{cccc} A_{0,0} & A_{0, 1} & A_{0,2} & A_{0,3} \\ A_{1,0} & A_{1, 1} & A_{1,2} & A_{1,3} \\ A_{2,0} & A_{2, 1} & A_{2,2} & A_{2,3} \end{array} \right] $$

A matrix of size $1 \times D$ represents a row vector

$$ [A] = \left[ \begin{array}{cccc} A_{0,0} & A_{0, 1} & A_{0,2} & A_{0,3} \end{array} \right] $$

A matrix of size $D \times 1$ represents a column vector.

$$ [A] = \left[ \begin{array}{c} A_{0,0} \\ A_{1,0} \\ A_{2,0} \end{array} \right] $$

When we use index notation,

  • The index before the comma "," is the row-index.
  • The index after the comma "," is the column-index.
  • The size $D_1 \times D_2$ is the shape $(D_1, D_2)$ of the matrix.

Rank-N tensor or N-dimensional array

A rank-N tensor or N-dimensional array $A$ has $N$ indices. The entry is denoted as

$$ A_{i_1 i_2 i_3 \cdots i_N}. $$ Its shape $(D_1, D_2, \cdots, D_N)$ determine the range of the index $i_j$, i.e., $ 0 \le i_j < D_j$.

In order to map naturally a rank-N tensor to a matrix we put a comma "," among the indices. A rank-N tensor with $N_c$ row indices is denoted as

$$ A_{i_1 i_2 i_3 \cdots i_{N_c}, i_{N_c+1}, \cdots i_N}. $$

Note that the $j$-th shape $D_j$ can be 1 and $i_j=0$ is a constant.

For example,

  • A rank-2 tensor with shape (1, D) represents a row vector, with entries $A_{0,i}$.
  • A rank-2 tensor with shape (D, 1) represents a column vector.

Combined index

We can combined several (subsequent) indices into a single combined index. Consider three indices $ijk$ with shape $(D_i, D_j, D_k)$. We can combine (reshape) them into a single index $\alpha$ with shape $D_\alpha=D_iD_jD_k$. The process is denoted as $(ijk) \rightarrow \beta$. Equivalently we can simply use $(ijk)$ to represents a combined-index.

We use the following convention: For $(D_i, D_j, D_k)=(2,2,2)$ so that $D_\beta=8$, one has

  • $ (0,0,0) \rightarrow 0$
  • $ (0,0,1) \rightarrow 1$
  • $ (0,1,0) \rightarrow 2$
  • $ (0,1,1) \rightarrow 3$
  • $ (1,0,0) \rightarrow 4$
  • $ (1,0,0) \rightarrow 5$
  • $ (1,1,0) \rightarrow 6$
  • $ (1,1,1) \rightarrow 7$

Graphical representation

We represent a rank-2 tensor $A_{i,j}$ as

  ┏━━━╳━━━┓
i─┨   A   ┠─j
  ┗━━━━━━━┛

We put the name of the tensor at the centor of the box. If we want to specific the shape information, we can put them inside the box near the bond.

  ┏━━━━╳━━━━┓
i─┨Di  A  Dj┠─j
  ┗━━━━━━━━━┛

The ╳ on top is to break the rotational symmetry. So that even if we rotate the figure, it still uniquely represents the same tensor and will not be confused with its transpose.

  ┏━━━╳━━━┓       ┏━━━━━━━┓
i─┨   A   ┠─j = j─┨   A   ┠─i
  ┗━━━━━━━┛       ┗━━━╳━━━┛

When a line connects two tensors, it implies a summation. So the following figure corresponds to $\sum_j A_{i,j} B_{j,k}$, which is nothing but the matrix multiplication $[A][B]$.

  ┏━━━╳━━━┓     ┏━━━╳━━━┓
i─┨   A   ┠──j──┨   B   ┠─j
  ┗━━━━━━━┛     ┗━━━━━━━┛

Actually, we can remove the dummy index and represent $[C]=[A][B]$ as

  ┏━━━╳━━━┓     ┏━━━╳━━━┓  ┏━━━╳━━━┓
 ─┨   C   ┠─ = ─┨   A   ┠──┨   B   ┠─
  ┗━━━━━━━┛     ┗━━━━━━━┛  ┗━━━━━━━┛

Tensor Network Notation

Introduction to tensor Network

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#    ┏━━━╳━━━┓    └─┨d      ┃      ┏━━━╳━━━┓                 └─┨d       ┃
# 1──┨d MT1 d┠─101──┨d  T  d┠──102─┨d  M2 d┠──2    =     1 ────┨d TOUT d┠──── 2
#    ┗━━━━━━━┛      ┃      d┠─┐    ┗━━━━━━━┛                   ┃       d┠─┐
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