Tensor network states and algorithms in the presence of a global SU(2) symmetry

The benefits of exploiting the presence of symmetries in tensor network algorithms have been extensively demonstrated in the context of matrix product states (MPSs). These include the ability to select a specific symmetry sector (e.g., with a given particle number or spin), to ensure the exact preservation of total charge, and to significantly reduce computational costs. Compared to the case of a generic tensor network, the practical implementation of symmetries in the MPS is simplified by the fact that tensors only have three indices (they are trivalent, just as the Clebsch-Gordan coefficients of the symmetry group) and are organized as a one-dimensional array of tensors, without closed loops. Instead, a more complex tensor network, one where tensors have a larger number of indices and/or a more elaborate network structure, requires a more general treatment. In two recent papers, namely, (i) [Singh, Pfeifer, and Vidal, Phys. Rev. A 82, 050301 (2010)] and (ii) [Singh, Pfeifer, and Vidal, Phys. Rev. B 83, 115125 (2011)], we described how to incorporate a global internal symmetry into a generic tensor network algorithm based on decomposing and manipulating tensors that are invariant under the symmetry. In (i) we considered a generic symmetry group $\mathcal{G}$ that is compact, completely reducible, and multiplicity free, acting as a global internal symmetry. Then, in (ii) we described the implementation of Abelian group symmetries in much more detail, considering a U(1) symmetry (e.g., conservation of global particle number) as a concrete example. In this paper, we describe the implementation of non-Abelian group symmetries in great detail. For concreteness, we consider an SU(2) symmetry (e.g., conservation of global quantum spin). Our formalism can be readily extended to more exotic symmetries associated with conservation of total fermionic or anyonic charge. As a practical demonstration, we describe the SU(2)-invariant version of the multiscale entanglement renormalization ansatz and apply it to study the low-energy spectrum of a quantum spin chain with a global SU(2) symmetry.

Figure 1

Computational gain obtained by exploiting the symmetry in a MPS algorithm. Computation time (in seconds) for one iteration of the “infinite time-evolving block-decimation” (TEBD) (Ref. 3) algorithm, as a function of the MPS bond dimension $\chi$ is shown. Here, $\chi$ is a refinement parameter; a larger $\chi$ leads to a better accuracy of the method. For sufficiently large $\chi$, exploiting symmetry leads to reductions in computation time. The horizontal line on this graph shows that this reduction in computation time equates to the ability to evaluate MPSs with a higher bond dimension $\chi$: For the same cost per iteration incurred when optimizing a regular MPS in matlab with bond dimension $\chi =220$, one may choose instead to optimize a U(1)-symmetric MPS with $\chi =380$ or an SU(2)-symmetric MPS with $\chi =1300$.

Figure 2

The schematic organization of the paper.

Figure 3

Constraint fulfilled by (a) an SU(2)-invariant vector, (b) an SU(2)-invariant matrix, and (c) a rank-3 SU(2)-invariant tensor with two incoming and one outgoing indices.

Figure 4

(a) Each index $i=(j,t_j,m_j)$ of an SU(2)-invariant tensor that is written in the spin basis decomposes into a degeneracy index $(j,t_j)$ and a spin index $(j,m_j)$ in accordance with the decomposition (B14) of the vector space associated with it. (b) Only $j=0$ is relevant for an SU(2)-invariant tensor with one index. (c)–(e) For fixed values of $j_a$ and $j_b$, a rank-2 SU(2)-invariant tensor ${\widehat{T}}_{ab}$ decomposes into a degeneracy tensor ${\widehat{P}}_{j_a{j}_b}$ and a structural term that, depending on the directions of $a$ and $b$, is the identity (c), tensor $C_{j_a{m}_{j_a},j_b{m}_{j_b}\to 00}^{\mathrm{fuse}}$ (d), or tensor $C_{00\to j_a{m}_{j_a},j_b{m}_{j_b}}^{\mathrm{split}}$ (e).

Figure 5

Examples of the decomposition of rank-3 SU(2)-invariant tensors into degeneracy tensors $\widehat{P}$ and Clebsch-Gordan tensors.

Figure 6

Two different canonical decompositions of a rank-4 SU(2)-invariant tensor $\widehat{T}$ into $(\widehat{P},\widehat{Q})$ and $({\widehat{P}}^{\prime },{\widehat{Q}}^{\prime })$ tensors corresponding to two different choices of the fusion tree.

Figure 7

(a) A tensor network $\mathcal{N}$ made of SU(2)-invariant tensors represents an SU(2)-invariant tensor $\widehat{T}$. This is seen by means of two equalities. The first equality is obtained by inserting resolutions of identity $\widehat{I}={\widehat{W}}_{\mathbf{r}}{\widehat{W}}_{\mathbf{r}}^{†}$ on each index connecting two tensors in $\mathcal{N}$ (the small dark circles depict the transformation ${\widehat{W}}_{\mathbf{r}}$, whereas the small light circles depict the adjoint transformation ${\widehat{W}}_{\mathbf{r}}^{†}$). The second equality follows from the fact that each tensor in $\mathcal{N}$ is SU(2) invariant. (b) For fixed values of $j$'s on all indices, the tensor network $\mathcal{N}$ decomposes into a tensor network made degeneracy tensors and a spin network. The sum is over all $j$'s and the corresponding $t$'s and $m$'s that are associated with the contracted indices.

Figure 8

[Contrast with Fig. 20a] (a) The leftward bending of an outgoing index $a=(j_a,t_{j_a},m_{j_a})$ of an SU(2)-invariant tensor $\widehat{T}$ given in the canonical form $(\widehat{P},C^{\mathrm{split}})$ to obtain another SU(2)-invariant tensor ${\widehat{T}}^{\prime }$, shown in two steps. The first step shows that for fixed values of $j_a,j_b$, and $j_c$, the degeneracy index $(j_a,t_{j_a})$ and the spin index $(j_a,m_{j_a})$ can be bent separately. Degeneracy tensor ${\widehat{P}}_{j_a{j}_b{j}_c}$ remains unchanged (like a regular tensor) by bending its index $(j_a,t_{j_a})$. In contrast, bending the index $(j_a,m_{j_a})$ corresponds to “multiplying” the structural tensor $C^{\mathrm{split}}$ with the cup ${\Omega }_{j_a}^{\mathrm{cup}}$ such that $(j_a,m_{j_a})$ is contracted. This multiplication corresponds to applying the $F$ move of Fig. 29c. The second step corresponds to multiplying the resulting factor ${\mu }_{j_a{j}_c{j}_b}^{\mathrm{cup}}$ into the degeneracy tensors $\widehat{P}$ to obtain ${\widehat{T}}^{\prime }$ in the canonical form $({\widehat{P}}^{\prime },C^{\mathrm{fuse}})$ [Eq. (30)]. (b) Tensor $\widehat{T}$ is recovered from ${\widehat{T}}^{\prime }$ by bending $\overline{a}$ downward using the cap transformation ${\Omega }_{j_{\overline{a}}}^{\mathrm{cap}}$ as shown and restoring the canonical form by applying the inverse $F$ move [Fig. 29d], leading to Eq. (31).

Figure 9

[Contrast with Fig. 20b] Swapping indices $a=(j_a,t_{j_a},m_{j_a})$ and $b=(j_b,t_{j_b},m_{j_b})$ of an SU(2)-invariant tensor $\widehat{T}$ that is given in the canonical form $(\widehat{P},C^{\mathrm{split}})$ to obtain another SU(2)-invariant tensor ${\widehat{T}}^{\prime }$, shown in two steps. The first step shows that for fixed values of $j_a,j_b$, and $j_c$, the corresponding indices of the degeneracy tensor and the structural tensor can be swapped separately. Swapping indices $(j_a,t_{j_a})$ and $(j_b,t_{j_b})$ of degeneracy tensor ${\widehat{P}}_{j_a{j}_b{j}_b}$ is achieved by swapping the labels $j_a$ and $j_b$ and also the indices $t_{j_a}$ and $t_{j_b}$. The analogous swap of the indices $(j_a,m_{j_a})$ and $(j_b,m_{j_b})$ of the structural tensor is straightforward and can be performed algebraically [Eq. (B49)]: the resulting tensor is simply proportional to another Clebsch-Gordan tensor $C_{j_c{m}_{j_c}\to j_b{m}_{j_b},j_a{m}_{j_a}}^{\mathrm{split}}$ times the factor $R_{j_a,j_b\to j_c}^{\mathrm{swap}}$. The second step corresponds to multiplying the factor $R_{j_a,j_b\to j_c}^{\mathrm{swap}}$ into the degeneracy tensors ${\widehat{P}}_{j_a{j}_b{j}_c}$ to obtain ${\widehat{T}}^{\prime }$ in the canonical form [Eq. (35)].

Figure 10

[Contrast with Figs. 20d, 20e] (a) Fusion, by means of the transformation ${{\rm Y}}^{\mathrm{fuse}}$, of two indices of an SU(2)-invariant tensor $\widehat{T}$ given in the canonical form $(\widehat{P},C^{\mathrm{split}})$ to obtain an SU(2)-invariant matrix ${\widehat{T}}^{\prime }$. For fixed values of $j_a,j_b$, and $j_c$, the fusion is addressed separately for the degeneracy tensors and structural tensors by decomposing the transformation ${{\rm Y}}^{\mathrm{fuse}}$ into $X^{\mathrm{fuse}}$ and $C^{\mathrm{fuse}}$ parts. Degeneracy indices are fused by multiplying tensors ${\widehat{P}}_{j_a{j}_b{j}_c}$ with $X^{\mathrm{fuse}}$ as shown. For a fixed value of $j_c$, tensor ${\widehat{P}}_{j_c{j}_d}^{\prime }$ ($j_d=j_c$) is the sum of all reshaped degeneracy tensors ${\widehat{P}}_{j_a{j}_b{j}_c}$ with compatible labels $j_a,j_b$, and $j_c$. The fusion of the spin indices is trivial since the structural tensor cancels out with the applied $C^{\mathrm{fuse}}$ [Fig. 26c], leading to Eq. (38). (b) Tensor $\widehat{T}$ is recovered by analogously splitting back the index $(j_d,t_{j_d})$ of ${\widehat{T}}^{\prime }$ using the transformation ${{\rm Y}}^{\mathrm{split}}$ which decomposes into $X^{\mathrm{split}}$ and $C^{\mathrm{split}}$ parts.

Figure 11

When reshaping incoming indices, transformations ${{\rm Y}}^{\mathrm{fuse}}$ and ${{\rm Y}}^{\mathrm{split}}$ reverse roles. (a) Fusion of incoming indices is by means of transformation ${{\rm Y}}^{\mathrm{split}}$. (b) Splitting of an incoming index is by means of transformation ${{\rm Y}}^{\mathrm{fuse}}$.

Figure 12

Computation times (in seconds) required to permute indices of a rank-4 tensor $\widehat{T}$ as a function of the size of the indices. All four indices of $\widehat{T}$ have the same size $9d$, and therefore the tensor contains $|\widehat{T}|=9^4{d}^4$ coefficients. The figures compare the time required to perform these operations using a regular tensor and an SU(2)-invariant tensor, where in the second case each index contains three different values of spin $j=0,1,2$, each with degeneracy $d$, and the canonical form of Eq. (24) is used. The upper figure shows the time required to permute two indices: For large $d$, exploiting the symmetry of an SU(2)-invariant tensor by using the canonical form results in shorter computation times. The lower figure shows the time required to fuse two adjacent indices. In this case, maintaining the canonical form requires more computation time. Notice that in both figures the asymptotic cost scales as $O\left(d^4\right)$, or the size of $\widehat{T}$, since this is the number of coefficients which need to be rearranged. We note that the fixed-cost overheads associated with symmetric manipulations could potentially vary substantially with choice of programming language, compiler, and machine architecture. The results given here show the performance of our matlab implementation of SU(2) symmetry.

Figure 13

Computation times (in seconds) required to multiply two matrices (upper panel) and to perform a singular value decomposition (lower panel) as a function of the size of the indices. Matrices of size $9d\times 9d$ are considered. The figures compare the time required to perform these operations using regular matrices and SU(2)-invariant matrices, where for the SU(2) matrices each index contains three different values of the spin $j=0,1,2$, each with degeneracy $d$, and the canonical form of Eq. (11) is used. That is, each matrix decomposes into three blocks of size $d\times d$. For large $d$, exploiting the block-diagonal form of SU(2)-invariant matrices results in shorter computation time for both multiplication and singular value decomposition. The asymptotic cost scales with $d$ as $O\left(d^3\right)$, while the size of the matrices grows as $O\left(d^2\right)$. [For matrix multiplication. a tighter bound of $O\left(d^{2.5}\right)$ for the scaling of computational time with $d$ is seen in this example.] We note that the fixed-cost overheads associated with symmetric manipulations could potentially vary substantially with choice of programming language, compiler, and machine architecture. The results given here show the performance of our matlab implementation of SU(2) symmetry.

Figure 14

MERA for a system of $L=2\times 3^2=18$ sites made of two layers of disentanglers $\widehat{u}$ and isometries $\widehat{w}$ and a top tensor $\widehat{t}$.

Figure 15

Error in ground-state energy $\Delta E$ (in the singlet sector $J=0$) as a function of $\chi$ for the Heisenberg model with $2L=108$ spins and periodic boundary conditions. The error is calculated with respect to the exact solutions and is seen to decay polynomially with $\chi$ for the particular choice of spins listed in Table .

Figure 16

Low-energy spectrum of $\widehat{H}$ with $L=54$ sites (=108 spins). Depicted states have spin $J$ of zero ($\times$, blue loops), one (+, red loops), or two ($\circ$, green loop). The superscript “2” close to the boundary of a loop indicates that the loop encloses twofold-degenerate states, e.g., the second, third, and fourth spin-1 triplets are twofold degenerate. The inset shows a zoom in of the region enclosed within the box. It compares the energies of the twofold-degenerate spin-one states within the box with those obtained using the regular MERA (black asterisk points). Since the symmetry is not protected, the states obtained with the regular MERA corresponding to different $m_J$ do not have the same energies.

Figure 17

Memory cost (in number of components) for storing the MERA as a function of the bond dimension $\chi$. The horizontal line on this graph shows that this reduction in memory cost equates to the ability to store MERAs with a higher bond dimension $\chi$: For the same amount of memory required to store a MERA with bond dimension $\chi =15$, one may choose instead to store a U(1)-symmetric MERA with $\chi =26$ or an SU(2)-symmetric MERA with $\chi =39$.

Figure 18

Computation time (in seconds) for one iteration of the MERA energy minimization algorithm as a function of the bond dimension $\chi$. For sufficiently large $\chi$, exploiting the SU(2) symmetry leads to reductions in computation time. The horizontal line on this graph shows that this reduction in computation time equates to the ability to evaluate MERAs with a higher bond dimension $\chi$: For the same cost per iteration incurred when optimizing a regular MERA in matlab with bond dimension $\chi =18$, one may choose instead to optimize a U(1)-symmetric MERA with $\chi =26$ or an SU(2)-symmetric MERA with $\chi =33$.

Figure 19

Graphical representation of tensors by means of a shape, e.g., circle or a “blob.” Indices of the tensor correspond to lines emerging perpendicular to the boundary of the shape. Indices may be incoming or outgoing as indicated by arrows. (a) Graphical representation of tensors with rank $0,1$, and 2, corresponding to a complex number $c\in \mathbb{C}$, a vector $|v\rangle \in {\mathbb{C}}^{\left[i\right]}$, and a matrix $\widehat{M}\in {\mathbb{C}}^{\left|a\right|\times \left|b\right|}$. (b) Graphical representation of a tensor $\widehat{T}$ with components ${\widehat{T}}_{abcd}$ and directions $\vec{D}=\left\{\text{in,}\text{out,}\text{in,}\text{out}\right\}$. Indices emerge in a counterclockwise order; the first index is the one emerging closest to a mark (dot) inside the blob. By convention, all arrows in a tensor diagram point downward in the page.

Figure 20

Transformations of a tensor: (a) Reversing the direction of an index [Eq. (A1)] is depicted by bending lines. (b) Swapping two outgoing indices [Eq. (A2)]. (c) Cyclic permutation of indices, e.g., ${\left({\widehat{T}}^{\prime }\right)}_{cab}={\widehat{T}}_{abc}$, is depicted by simply shifting the starting mark (black dot) close to the first index of the resulting tensor. (d) Fusion of indices $a$ and $b$ into $d=a\times b$ [Eq. (A3)]; splitting of index $d=a\times b$ into $a$ and $b$ [Eq. (A4)].

Figure 21

(a) Swapping an incoming and outgoing index as decomposed into two reversals and a swap. (b) Fusion of an incoming and an outgoing index as decomposed into a reversal and a fusion of outgoing indices.

Figure 22

(a) Graphical representation of the matrix multiplication of two matrices $\widehat{R}$ and $\widehat{S}$ into a new matrix $\widehat{T}$ [Eq. (A5)]. (b) Graphical representation of an example of the multiplication of two tensors $\widehat{R}$ and $\widehat{S}$ into a new tensor $\widehat{T}$ [Eq. (A6)].

Figure 23

Graphical representations of the five elementary steps 1–5 into which one can decompose the multiplication of the tensors of Eq. (A6).

Figure 24

(a) Factorization of a matrix $\widehat{T}$ according to a singular value decomposition [Eq. (A12)]. (b) Factorization of a rank-4 tensor $\widehat{T}$ according to one of several possible singular value decompositions.

Figure 25

(a) Example of a tensor network $\mathcal{N}$. (b) Tensor $\widehat{T}$ of which the tensor network $\mathcal{N}$ could be a representation. (c) Tensor $\widehat{T}$ can be obtained from $\mathcal{N}$ through a sequence of contractions of pairs of tensors. Shading indicates the two tensors to be multiplied together at each step. The product tensor at each step is conveniently depicted by the blob that covers the two tensors that are multiplied.

Figure 26

(a) The graphical representation of the transformations $C^{\mathrm{fuse}}$ and $C^{\mathrm{split}}$. (b) Depiction of Eq. (B47). The sum is over all values of $j_{AB}$ (and $m_{j_{AB}}$) that are compatible with $j_A$ and $j_B$. (c) Depiction of Eq. (B48). The identity holds for all compatible values of $j_A,j_B,$, and $j_{AB}$. Any contraction involving $C^{\mathrm{fuse}}$ and $C^{\mathrm{split}}$ corresponds to summing the $m$'s and, if required, also the $j$'s. We explicitly indicate a summation only when the $j$'s are summed. (d) The tensor obtained by swapping indices $(j_A,m_{j_A})$ and $(j_B,m_{j_B})$ of tensor $C^{\mathrm{split}}$ is proportional to another $C^{\mathrm{split}}$ tensor [Eq. (B49)].

Figure 27

The graphical representation of (a) the fuse tensor ${{\rm Y}}^{\mathrm{fuse}}$ [Eq. (B55)] and (b) the split tensor [Eq. (B60)], and their decomposition into $X$ and $C$ tensors. (c) Tensors ${{\rm Y}}^{\mathrm{fuse}}$ and ${{\rm Y}}^{\mathrm{split}}$ are unitary and thus yield the identity when contracted pairwise as shown. The sum is over all the $j$'s (and implicitly over the corresponding $t$'s and $m$'s) on the contracted indices that are compatible with the given $j$'s on the open indices.

Figure 28

The $F$ move that relates two different ways of fusing three spins $j_A,j_B,j_C$ into a total spin $j_{ABC}$ [Eq. (B75)].

Figure 29

(a) An $F$ move that relates two different ways of fusing two incoming and two outgoing spins. (b) A special instance of (a) corresponding to setting $j_D=0$ and the only compatible value $j_G=j_C$ and $j_H=j_A$ [Eq. (B79)]. (c) Depicting the $F$ move (b) in terms of a left-directed cup [Eq. (B79)] to explicitly indicate the upward bending of index $(j_A,m_{j_A})$. (d) The inverse of (c), i.e., the downward bending of index $(j_A,m_{j_A})$, as explicitly depicted by a left-directed cap [Eq. (B80)]. Contrast the bending of indices of the Clebsch-Gordan tensors $C^{\mathrm{fuse}}$ and $C^{\mathrm{split}}$ in (c) and (d) with Fig. 20a. (e) Right-directed cups and caps are related to their left counterparts by a factor $R_{j_A,j_A\to 0}^{\mathrm{swap}}$.

Figure 30

A tree decomposition of an SU(2)-invariant tensor $\widehat{T}$ with components ${\left(\widehat{T}\right)}_{i_1{i}_2{i}_3{i}_4}$ and directions $\left\{\text{in,}\text{out,}\text{out,}\text{in}\right\}$. It comprises of an SU(2)-invariant vector $\widehat{v}$, three split tensors ${{\rm Y}}^{\mathrm{split}}$, and two cups. Tensor ${\widehat{T}}^{\prime }$ is obtained by bending the incoming indices ($i_1$ and $i_4$) of $\widehat{T}$. The tree decomposition of tensor $\widehat{T}$ is obtained by applying a resolution of identity $\mathcal{I}\left(\mathit{\tau }\right)$ on the indices of ${\widehat{T}}^{\prime }$ where $\mathcal{I}\left(\mathit{\tau }\right)$ corresponds to a tensor network made of fuse and split tensors that are interconnected according to a given fusion tree $\mathit{\tau }$.

Figure 31

(a) Two different tree decompositions of a rank-4 SU(2)-invariant tensor corresponding to the choice of two different fusion trees ${\mathit{\tau }}^X$ and ${\mathit{\tau }}^Y$. The two tree decompositions are obtained from the tensor by means of the resolutions of identity (b) $\mathcal{I}\left({\mathit{\tau }}^X\right)$ and (c) $\mathcal{I}\left({\mathit{\tau }}^Y\right)$ as illustrated in Fig. 30.

Figure 32

Mapping between two tree decompositions (Fig. 31) ${\mathcal{D}}^X\left(\widehat{T}\right)$ and ${\mathcal{D}}^Y\left(\widehat{T}\right)$ of an SU(2)-invariant tensor $\widehat{T}$. Tree decomposition ${\mathcal{D}}^Y\left(\widehat{T}\right)$ is obtained from ${\mathcal{D}}^X\left(\widehat{T}\right)$ in two steps. First, the resolution of identity $\mathcal{I}\left({\mathit{\tau }}^Y\right)$ is applied on ${\mathcal{D}}^X\left(\widehat{T}\right)$ as shown and a matrix $\widehat{\Gamma }$ is obtained by multiplying together the split tensors in ${\mathcal{D}}^X\left(\widehat{T}\right)$ and the fuse tensors in $\mathcal{I}\left({\mathit{\tau }}^Y\right)$. Then, vector ${\widehat{v}}^Y\in {\mathcal{D}}^Y\left(\widehat{T}\right)$ is obtained by multiplying $\widehat{\Gamma }$ with ${\widehat{v}}^X$ [Eq. (C1)].

Figure 33

Bending indices of an SU(2)-invariant tensor $\widehat{T}$ that is given in the tree decomposition $\mathcal{D}\left(\widehat{T}\right)$ to obtain another SU(2)-invariant tensor ${\widehat{T}}^{\prime }$. The only nontrivial cases are bending down a “left” index from the right (shown here) and bending a “right” index from the left, which results in a “loop” as shown. A tree decomposition of the tensor ${\widehat{T}}^{\prime }$ is obtained by subsuming the loop into $\mathcal{D}\left(\widehat{T}\right)$. This is achieved by applying the resolution of identity $\mathcal{I}\left(\mathit{\tau }\right)$ as shown, straightening the loop (inset), then multiplying together the resulting swap factors, the split tensors in $\mathcal{D}\left(\widehat{T}\right)$, and the fuse tensors in $\mathcal{I}\left(\mathit{\tau }\right)$ to obtain a matrix ${\widehat{\Gamma }}^{\mathrm{bend}}$. The vector ${\widehat{v}}^{\prime }$ that comprises the tree decomposition of ${\widehat{T}}^{\prime }$ is obtained by multiplying ${\widehat{\Gamma }}^{\mathrm{bend}}$ with $\widehat{v}$ [Eq. (C2)].

Figure 34

Permuting (or intercrossing) indices of an SU(2)-invariant tensor $\widehat{T}$ that is given in a tree decomposition $\mathcal{D}\left(\widehat{T}\right)$ to obtain another SU(2)-invariant tensor ${\widehat{T}}^{\prime }$. A tree decomposition of the tensor ${\widehat{T}}^{\prime }$ is obtained by subsuming the intercrossings into the decomposition $\mathcal{D}\left(\widehat{T}\right)$. This is achieved by applying a resolution of identity $\mathcal{I}\left({\mathit{\tau }}^{\prime }\right)$ (for a specified fusion tree ${\mathit{\tau }}^{\prime }$) as shown and then multiplying together the split tensors in $\mathcal{D}\left(\widehat{T}\right)$ and the fuse tensors in $\mathcal{I}\left({\mathit{\tau }}^{\prime }\right)$ to obtain a matrix ${\widehat{\Gamma }}^{\mathrm{perm}}$. The vector ${\widehat{v}}^{\prime }$ that comprises the tree decomposition of ${\widehat{T}}^{\prime }$ is obtained by multiplying ${\widehat{\Gamma }}^{\mathrm{perm}}$ and the vector $\widehat{v}$ [Eq. (C3)].

Figure 35

Reshaping indices of an SU(2)-invariant tensor $\widehat{T}$ that is given in a tree decomposition $\mathcal{D}\left(\widehat{T}\right)$ to obtain another SU(2)-invariant tensor ${\widehat{T}}^{\prime }$. Fusion and splitting of straight indices is by using transformations ${{\rm Y}}^{\mathrm{fuse}}$ and ${{\rm Y}}^{\mathrm{split}}$, respectively. (a) Fusion of two straight indices of $\widehat{T}$ that belong to the same split tensor in the tree decomposition $\mathcal{D}\left(\widehat{T}\right)$. A tree decomposition of ${\widehat{T}}^{\prime }$ is obtained by simply deleting the split tensor [Eq. (27)]. (b) Tensor $\widehat{T}$ is recovered from ${\widehat{T}}^{\prime }$ by splitting back the fused index. That is, the tree decomposition $\mathcal{D}\left(\widehat{T}\right)$ is recovered by reattaching the split tensor to the tree decomposition of ${\widehat{T}}^{\prime }$.

Figure 36

(a) Obtaining the block-diagonal form [i.e., the $(\widehat{P},\widehat{Q})$ form] of an SU(2)-invariant matrix from its tree decomposition (left) by performing two multiplications (highlighted by shading). First, the vector $\widehat{v}$ is multiplied with the split tensor to obtain an intermediate SU(2)-invariant tensor ${\widehat{T}}^{\prime }$. Then, ${\widehat{T}}^{\prime }$ is multiplied with the cup to obtain the block-diagonal matrix $\widehat{T}$. (b) The two multiplications of (a) as performed in the canonical form at each step. The “multiplication” with the cup simply corresponds to applying the $F$ move of Fig. 29b.

Figure 37

The matrix $\widehat{\Gamma }$ of Eq. (C1) is obtained by contracting a tensor network $\mathcal{M}$ made of fuse and split tensors. Only the $j=0$ block [${\widehat{D}}_{j=0}$ in Eq. (C4)] of $\widehat{\Gamma }$ is relevant. For fixed values of $j$'s on all the contracted indices, compatible with $j=0$ on the two open indices, each tensor ${{\rm Y}}^{\mathrm{fuse}}$ and ${{\rm Y}}^{\mathrm{split}}$ decomposes into $X$ and $C$ parts. Subsequently, the tensor network $\mathcal{M}$ factorizes into two terms tensors. The first one is a tensor network made of $X$ tensors that can be contracted to obtain a matrix made of 0${}^{\prime }$s and 1${}^{\prime }$s. The second term is a spin network which, here, can be contracted to obtain a number (the “value” of the spin network) since the open indices take only one value: $j=0,m=0$, i.e., have size one. This “evaluation” can be achieved by applying a sequence of $F$ moves (instead of actually multiplying the Clebsch-Gordan tensors). Therefore, the value of the spin network is given in terms of $\widehat{F}$ coefficients (and possibly also the $R^{\mathrm{swap}}$ coefficients when considering the contractions for obtaining the matrices ${\widehat{\Gamma }}^{\mathrm{bend}}$ or ${\widehat{\Gamma }}^{\mathrm{perm}}$).