Isometric Tensor Network States in Two Dimensions

TL;DR

This paper introduces an isometric tensor network state approach for 2D quantum systems, enabling efficient contraction and simulation of ground states, demonstrated on the 2D transverse field Ising model.

Contribution

The paper presents a novel isometric restriction of tensor network states for 2D systems, improving contraction efficiency and enabling new algorithms like TEBD$^2$ for ground state approximation.

Findings

Efficient contraction of 2D isometric tensor network states.

Transformation of 2D quantum states into isometric TNS.

Successful application of TEBD$^2$ to the 2D transverse field Ising model.

Abstract

Tensor network states (TNS) are a promising but numerically challenging tool for simulating two-dimensional (2D) quantum many-body problems. We introduce an isometric restriction of the TNS ansatz that allows for highly efficient contraction of the network. We consider two concrete applications using this ansatz. First, we show that a matrix-product state representation of a 2D quantum state can be iteratively transformed into an isometric 2D TNS. Second, we introduce a 2D version of the time-evolving block decimation algorithm (TEBD$^2$) for approximating the ground state of a Hamiltonian as an isometric TNS, which we demonstrate for the 2D transverse field Ising model.

Equations

$$\Psi =A^1{A}^2\cdots A^{\ell -1}{\Lambda }^{\ell }{B}^{\ell +1}\cdots B^N$$

$$\frac{d}{d{A_y^{\ell }}^{†}}⟨A^{\ell }\Lambda ⟩{\Lambda }^{\ell }=UsV=(UV)(V^{†}sV)$$

$${\lambda }_{\alpha \beta \gamma }$$

$${\Theta }_{\alpha {\beta }_L{\beta }_R\gamma }$$

Full text

Overcoming the exponential growth of complexity when simulating quantum many-body systems is one of the most challenging goals in computational physics. For ground state properties of one-dimensional systems (1D) this challenge was answered by the density matrix renormalization group (DMRG) algorithm, which provides an essentially exact numerical solution of gapped 1D lattice models White (1992) and field theories Verstraete and Cirac (2010). Subsequently understood as a variational method over the class of matrix product states (MPS) Ostlund and Rommer (1995); J. Dukelsky. M.A. Martın-Delgado and Sierra (1998) its success follows from the ability of MPS to adequately capture the area-law entanglement characteristic of gapped ground states Hastings (2007). A central goal has been to generalize the success of DMRG to higher dimensions. For certain classes of states, this is achieved by so called tensor network states (TNS) whose connectivity reflects the geometry of many-body entanglement Maeshima et al. (2001); Verstraete and Cirac (2004). However, while evaluating properties of 1D MPS is highly efficient (scaling with the tensor dimension $\chi$ and system size $N$ as $N\chi^{3}$), exactly evaluating properties of TNS in higher dimensions is generically exponentially hard. Consequently, there has been a long-standing effort to determine the best way to numerically approximate TNS contractions in order to minimize the variational energy of TNS for given a Hamiltonian. Progress has been made for two-dimensional (2D) systems by introducing a number of algorithms to manipulate and optimize TNS for various lattice models Nishino and Okunishi (1996, 1997); Maeshima et al. (2001); Verstraete and Cirac (2004); Levin and Nave (2007); Jordan et al. (2008); Jiang et al. (2008); Xie et al. (2009, 2012); Corboz (2016); Vanderstraeten et al. (2016); Bañuls et al. (2008); Fishman et al. (2018); Zheng et al. (2017); Corboz and Mila (2013, 2013); Liao et al. (2017). However, at this point it is fair to say that the “right” way to generalize 1D DMRG is not yet agreed upon.

In this work, we study a restriction of the TNS ansatz, which we dub “isoTNS,” which allows for highly efficient contraction of the network. When collapsing either the rows or columns of the 2D network it reduces to the canonical form of a 1D MPS Vidal (2004); Perez-Garcia et al. (2008). As a result, any 1D MPS algorithm, such as DMRG White (1992) or the time dependent block decimation (TEBD) Vidal (2004), can be turned into a 2D algorithm by applying it in a nested loop with respect to the rows and columns of the 2D isoTNS. While the ansatz we discuss is known to some practitioners Pri , and is related to a previous work on correlated contour states, Richter (1994) it does not seem to have been studied in practice. Here we introduce a key procedure for manipulating isoTNS, the “Moses Move” (MM), and demonstrate its utility with two concrete applications: First we show that a 1D MPS representation of a 2D quantum state can be iteratively transformed into an isoTNS, and examine the resulting entanglement properties. Second we implement a “TEBD${}^{2\,}$” algorithm and use it to approximate the ground state of the 2D transverse field Ising model as an isoTNS.

The isometric tensor network ansatz.

We first review the canonical form of a 1D MPS (see Ref. Schollwöck (2011) for more details). Suppressing the indices of all tensors, the MPS for an $N$-site chain takes the form $\Psi=T^{1}T^{2}\cdots T^{N}$. Here each $T^{a}$ is a rank-3 tensor which we view as a $\chi_{a-1}\times\chi_{a}$ matrix in an “ancilla space” whose entries are vectors in the $d$-dimensional single-site Hilbert space of site $a$. Multiplication of the matrices implicitly comes with a tensor product over the single-site Hilbert spaces, producing an $N$-site wavefunction. At the boundaries, $\chi_{0}=\chi_{N}=1$. For any contiguous region of spins $V=a:b$, the partial contraction $T^{V\to\partial V}\equiv T^{a}\cdots T^{b}$ is a linear map from the Hilbert space $\mathcal{H}_{V}$ of the subregion to the $\chi_{a-1}\times\chi_{b}$ dimensional Hilbert space $\mathcal{H}_{\partial V}$ of the ancillas dangling from the boundary of the region. The “canonical form with $\ell$-site center” is defined by requiring that the boundary map $T^{V\to\partial V}$ is an isometry if $V=1$:$a$ for $a<\ell$ or $V=a$:$N$ for $a>\ell$. Recall a map is an isometry if $T^{\dagger}_{V\to\partial V}T_{V\to\partial V}=\mathds{1}_{\partial V}$, while $T_{V\to\partial V}T^{\dagger}_{V\to\partial V}=P_{V}$ is a projection operator. The isometry condition ensures the ancillas on $\partial V$ form an orthonormal sub-basis for $V$. In what follows we denote the isometry conditions graphically by assigning arrows to the tensors as shown in Fig. 1a Stoudenmire and White (2013); Haegeman et al. (2016); Bal et al. (2016). A convenient notation for the representation of MPS with $\ell$-site center is to distinguish the tensors $A,\Lambda,B$ and write

[TABLE]

as shown in Fig. 1b. It is easy to verify that the canonical form is satisfied if and only if each $A^{a},B^{a}$ is individually an isometry from the left/right respectively. $T_{V\to\partial V}$ is an isometric boundary map if and only if the boundary $\partial V$ has only outgoing arrows. On the other hand, a region with only incoming arrows, like the $\Lambda$, is precisely the wavefunction of the system expressed in a orthonormal basis, so it is called an orthogonality center. In particular, $\|\Psi\|=\|\Lambda\|$ and any site-$\ell$ expectation value can be locally computed as $\bra{\Psi}O^{\ell}\ket{\Psi}=\bra{\Lambda}O^{\ell}\ket{\Lambda}$, as seen in Fig. 1c, because the $A,B$ tensors in its exterior contract to $\mathds{1}$ by the isometry condition.

Once the canonical form is understood as a restriction on the boundary maps, it can naturally be generalized to higher dimensions. By analogy to Eq. (1), we demand that each row and column of the TNS is an isometry, as indicated in Fig. 1d. This constraint can be satisfied by further demanding that each tensor is an isometry from a physical and two ancilla legs to the remaining two ancillas according to the direction of the arrows indicated. This gives a causal structure to the tensor-network, though in our convention time flows opposite to the direction of the arrows. As in 1D, there is a set of space-like hypersurfaces with only outgoing arrows whose “past” defines the wavefunction in a orthonormal basis and whose “future” is an isometric boundary map. An expectation value $\bra{\Psi}O\ket{\Psi}$ depends only on the tensors in the past of the insertion $O$. This is more than an analogy: the network between two space-like surfaces defines a Kraus decomposition of a quantum channel (time evolution) relating the boundary ancilla.

There are a special row and column $\Lambda$ (highlighted in red in Fig. 1d) which has only incoming arrows, which is hence the 1D “orthogonality hypersurface” of the TNS (our nomenclature anticipates the generalization to higher dimensions, where $\Lambda$ has codimension one.) Because its exterior is an isometry from the physical to the incoming ancillas, $\Lambda$ is the wavefunction of the system in an orthonormal basis. Hence $\Lambda$ can be treated just like an MPS and can itself put into 1D canonical form (consequently its orthogonality center tensor, $\lambda$, can be moved freely using the standard 1D algorithm). Tracing over the left or right ancillas of $\Lambda$ results in a density matrix which is iso-spectral to the reduced density matrix of the right or left, e.g. $\rho_{L}\sim\Lambda\Lambda^{\dagger}$, so $\Lambda$ encodes the entanglement spectrum. For any operator $O$ inside $\Lambda$, $\bra{\Psi}O\ket{\Psi}=\bra{\Lambda}O\ket{\Lambda}$, e.g., there is a dimensional reduction to a 1D expectation value which can be computed efficiently without further approximations via standard MPS algorithms. This is in stark contrast to generic TNS where expectation values require an approximate contraction of the entire network using, e.g., boundary MPS F. Verstraete (2004) or corner transfer matrices Nishino and Okunishi (1996, 1997). Furthermore, any variationally optimal compression of the orthogonality hypersurface $\Lambda$ (such as truncation of its entanglement spectrum via SVD) is variationally optimal for the global state. Note that by our choice of isometries, the resulting orthogonality hypersurface $\Lambda$ has minimal entanglement for each vertical cut and is expected to follow a 1D area law. This entanglement is different from the vertical entanglement of the full many-body wave function: it differs by the action of the isometries, which contain the 2D area law entanglement.

The great utility of both properties will become clear in the TEBD${}^{2\,}$algorithm we propose below.

It is an interesting and open question how the variational power of an isoTNS differs from that of a generic TNS. One restriction is that many of its correlations must decay exponentially, because any two-point function along the orthogonality hypersurface can be reduced to that of the MPS $\Lambda$, which must have exponentially decaying correlations. In contrast, a generic 2D TNS can represent power-law correlations. On the other hand, we have shown that any string-net state, thought to represent all 2D topological orders with gappable edges, can explicitly be put into isoTNS form.Soejima et al. (2019)

Shifting the orthogonality hypersurface. The canonical form is only useful for computational purposes if the orthogonality hypersurface $\Lambda$ can be moved throughout the network efficiently. In 1D, for example, the basic move $\Lambda^{\ell}B^{\ell+1}=A^{\ell}\Lambda^{\ell+1}$ can be accomplished by any orthogonal matrix factorization, i.e. QR or a singular value decomposition (SVD). In 2D we need to solve the same equation but with $A,\Lambda,B$ entire columns of the TNS. Using QR or SVD is hopeless, as it will destroy the locality required to express $\Lambda$ as an MPS. The key insight is that the canonical form can be preserved under a unitary insertion $(A^{\ell}U^{\dagger})(U\Lambda^{\ell+1})$. We propose to use this ambiguity to choose $A^{\ell}$ such that it “disentangles” $\Lambda^{\ell+1}$, so that $\Lambda^{\ell+1}$ has an efficient (low rank) MPS form.

It is actually sufficient to solve a simpler auxiliary problem: decompose $\Lambda^{\ell}=A^{\ell}\Lambda$, where $\Lambda$ is a wavefunction with only ancilla degrees of freedom (a “zero-column” wavefunction). The start and end points of the problem are shown in Fig. 2a (i) and (v). This move will be sufficient to move $\Lambda^{\ell}$ throughout the network, because we can tack the zero-column wavefunction onto the right in order to obtain the one-column wavefunction, $\Lambda^{\ell+1}=\Lambda B^{\ell+1}$.

We can solve $\Lambda^{\ell}\approx A^{\ell}\Lambda$ as a variational problem, sweeping back and forth through the tensors to minimize $|\Lambda^{\ell}-A^{\ell}\Lambda|$ while respecting the isometry condition on $A$ and reducing the bond dimension of $\Lambda$ sup . Interestingly, however, we find a single unzipping sweep based on disentangling provides a solution very close to the variational one, sup but is far quicker. This “Moses Move” (MM) is illustrated in sequence (i) to (v) of Fig. 2a.

The central subproblem of the MM (Fig. 2b) takes in the orthogonality center $\ket{\lambda}$, which by grouping legs is a tripartite state $\ket{ABC}$ on the top, lower left, and lower right degrees of freedom, and “splits” it into a four-partite state $\ket{AB_{L}B_{R}C}$. More precisely, we look for the splitting isometry $a^{\dagger}:B\to B_{L}\otimes B_{R}$, where $a^{\dagger}a=\mathds{1}$ and $B_{L/R}$ have dimension $\chi$, such that $\ket{AB_{L}B_{R}C}=a^{\dagger}\ket{ABC}$ has minimal entanglement $S_{AB_{L}:B_{R}C}$. This is closely related to finding the entanglement of purificationTerhal et al. (2002) of $\rho_{AC}$. To do so we make an initial (suboptimal) guess for the isometry $a_{0}$ chosen so that $B_{L}B_{R}$ includes the $\chi^{2}$ highest weight states in $B$, and then parameterize the optimal choice as $a=a_{0}U^{{\dagger}}$ for a unitary $U$ acting on $B_{L}B_{R}$. We choose $U$ to minimize $S_{AB_{L}:B_{R}C}$ or it’s Renyi generalization, 111Results closest to the variational optimum are obtain if we use Renyi index $\alpha<1$, see Supp. a well defined optimization problem. Evenbly and Vidal (2014); Hauschild et al. (2018) The resulting $a$ comprise the isometries in $A^{\ell}$, allowing us to successively unzip $\Lambda^{\ell}$ into $A^{\ell}\Lambda$.

It would be interesting to find a necessary and sufficient many-body entanglement criteria for the success of the MM, in order to better understand why it finds a solution so close to the variationally optimal one. In the absence of such rigorous results, we consider two practical numerical tests.

MPS to isoTNS. Given a ground state wavefunction $\ket{\Psi}$ on an $L_{x}\times L_{y}$ strip, we propose an iterative algorithm to put $\ket{\Psi}$ into an isoTNS which we test for the transverse Ising model $H=-\sum_{\langle i,j\rangle}\sigma^{z}_{i}\sigma_{j}^{z}-g\sum_{i}\sigma^{x}$ with Pauli matrices $\sigma^{\mu}$. To implement it numerically, we consider a strip with $L_{y}\gg L_{x}$ and use DMRG to obtain the ground state as a 1D MPS $\Lambda^{1:L_{x}}$ where each “site” contains the $L_{x}$ spins of the corresponding row (Fig. 3a). As described in Fig. 3, the MM can then be used to iteratively peal off columns of the wavefunction, $\Lambda^{\ell:L_{x}}=A^{\ell}\Lambda^{\ell+1:L_{x}}$, producing an isoTNS. The algorithm is exponentially difficult in $L_{x}$ (since $\Psi$ is obtained as an MPS!), but serves as a check on the ansatz independent of a ground state search scheme. Using an ancilla dimension $\chi=6$ for the isometries, the error $\||\Psi_{\mathrm{MPS}}\rangle-|\Psi_{\mathrm{isoTNS}}\rangle\|^{2}$ is $2\cdot 10^{-6}$ per site at $g=3.5$ (in the gapped, paramagnetic phase), $L_{y}=20,L_{x}=6,\chi_{\mathrm{MPS}}=128$, obtained in about 10 minutes on a laptop.

More interesting is the behavior of the “vertical” (top/bottom) and “horizontal” (left/right) entanglement of the resulting isoTNS. At each step $\ell$ the orthogonality hypersurface $\Lambda^{\ell+1:L_{x}}$ makes a “$\urcorner$” shape, running up the right and over the top. In Fig. 3b we show the entanglement entropy $S_{\ell}(y)$ for cuts along $\Lambda$, and find $S_{\ell}$ decrease with $\ell$. If the underlying phase has area law $S_{R}=s|\partial R|+\cdots$, for $y\sim L_{y}/2$ we hope $S_{\ell}$ goes as $S_{\ell}\approx s(L_{x}-\ell)+\cdots$. If not, the isometric columns $A^{\ell}$ aren’t removing their share of the entanglement and the algorithm will fail in the thermodynamic limit. In Fig. 3c, we see that after the initial delay the algorithm begins to remove remarkably close to $s$ entanglement per iteration. The initial delay is expected, because any two vertically-entangled degrees of freedom will individually have some horizontal extent. Until their entire support is to the left of $\Lambda$, the isometries $A$ cannot remove them.

The residual horizontal entanglement is left behind in the top-region of $\Lambda$. As hoped for, the horizontal entanglement is of order $s$, and for $\ell=L_{x}$ we find $S_{\ell}$ smoothly matches up between the right / top regions, despite the seemingly anisotropic nature of the algorithm.

TEBD${}^{2\,}$algorithm. We now propose a Trotterized time stepper for isoTNS which can be used to obtain the ground state by imaginary-time evolution. Assuming a nearest-neighbor interaction, we split the Hamiltonian into terms acting on columns and rows, $H=\sum^{L_{x}}_{c=1}H_{c}+\sum^{L_{y}}_{r=1}H_{r}$. We Trotterize according to $e^{-\tau H}\approx\prod_{r}e^{-\tau H_{r}}\prod_{c}e^{-\tau H_{c}}$ as illustrated in Fig. 4a. As for the TEBD update in 1D, the TEBD${}^{2\,}$can be easily improved to second order. We start in canonical form with the orthogonality center $\lambda^{1,1}$ at site $c,r=1,1$. The evolution $e^{-\tau H_{c=1}}$ is then applied to column $\Lambda^{1}$ by calling the standard 1D TEBD algorithm Vidal (2004) at a cost $\propto\chi^{6}$. We then use the MM to bring the orthogonality center over by one column, to $\lambda^{2,1}$ at a cost $\propto\chi^{7}$, and apply $H_{c=2}$, and so on, bringing the orthogonality center to $\lambda^{L_{x},1}$ (in contrast, the full update of an unconstrained PEPS costs $\chi^{12}$ Verstraete and Cirac (2004)). Applying $e^{-\tau H_{r}}$ analogously brings the center to $\lambda^{L_{x},L_{y}}$, and we repeat to bring $\lambda$ counter-clockwise around the four corners to complete the time step. Within a sweep the algorithm is literally two nested versions of 1D TEBD (with the MM replacing QR/SVD in the outer $c$-loop), hence the name “TEBD${}^{2\,}$”.

To benchmark TEBD${}^{2\,}$, we return to the transverse field Ising model. Fig. 4c shows the energy density obtained from TEBD${}^{2\,}$relative to numerically exact results from large scale 1D-DMRG simulations at $g=3.5$. If the evolution were exact the energy would decrease monotonically as the Trotter step $d\tau$ is decreased. However, the MM has a small truncation error $\epsilon_{\textrm{MM}}$, and we see that the resulting energy has a minimum in $d\tau$. For a $p$-th order Trotter step, the energy error should be $\Delta E=a\,\epsilon_{\textrm{MM}}/d\tau+b\,d\tau^{2p}$ (in our implementation $p=2$), 222Letting $\Delta$ be a typical energy scale and $\Delta E$ the energy density, one step of imaginary time evolution decreases the energy density by $\Delta E\,\Delta\,d\tau$, while the Moses Move truncation increases it by $\epsilon_{MM}\Delta$. Thus at long times the MM introduces $\Delta E\propto\epsilon_{\textrm{MM}}/d\tau$. in agreement with the observed minima. A similar effect is also observed in the full update of TNS, and can be partially remedied by using a variational update instead of imaginary time evolution Vanderstraeten et al. (2016); Corboz (2016). The minimum energy converges towards the exact result as the bond dimension $\chi$ is increased.

Conclusions. We introduced an isometric TNS ansatz which results in a canonical form that allows for 1D MPS algorithms to be efficiently adapted to 2D. To numerically benchmark the ansatz, we first demonstrated that an MPS representation of the ground state of the 2D transverse field Ising model can be efficiently transformed into an isoTNS. Second, we implemented a TEBD${}^{2\,}$algorithm and showed that it efficiently finds an approximation of the ground state of the 2D TFI model within the isoTNS form. Future directions include theoretically understanding the variational power of the isoTNS ansatz, as well as implementing a variational ground state algorithm, DMRG2, by nesting the standard 1D algorithm.

Supplemental Material

Appendix A Variational solution of $\Lambda^{\ell}=A^{\ell}\Lambda$

The Moses Move is not strictly variational: it sweeps only once through the network and chooses $A^{\ell}$ according to an entanglement criteria, rather than maximization of the global overlap. Here we describe a complementary variational procedure which can be used to further optimize the decomposition after the MM, though in practice we find the MM is surprisingly close to optimal (see below).

We optimize the overlap $\braket{A^{\ell}\Lambda}{\Lambda^{\ell}}$ by a variant of alternating least squares, e.g., optimize the tensors one at a time, subject to the constraints. We do so by first fixing $A^{\ell}$ and viewing the problem as maximization of $\textrm{max}_{\Lambda}\braket{\Lambda}{{A^{\ell}}^{\dagger}\Lambda^{\ell}}$ over the zero-column MPS $\Lambda$. This is the completely standard problem of variational MPS compression which is described elsewhere Verstraete et al. (2008); Schollwöck (2011). We then hold $\Lambda$ fixed and sweep up through the network to optimize each $A^{\ell}_{y}$ on site $y$. Here the isometric constraint comes in, so we use the polar-decomposition algorithm for optimizing over isometries Evenbly and Vidal (2014). We consider the variation of the overlap with respect to $A^{\ell}_{y}$, and reorganize the SVD decomposition of the variation as follows:

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To decode the index structure, we first view $A^{\ell}_{y}$ as a rank-two matrix by grouping together the incoming / outgoing arrows, so the variation is a matrix we can SVD. The isometry is then updated according to $A^{\ell}_{y}\to UV$. Using the MM as an initial guess, we find a slow but monotonic convergence to an optimal solution.

Appendix B Comparison of Moses Move with variationally optimal solution

Here we compare the Moses Move (MM) solution of $\Lambda^{\ell}=A^{\ell}\Lambda$ with the variational method just discussed. The difference varies from case by case, so we consider some representative data for the ground state $\Psi$ of the transverse field Ising model on a $2\times 20$ site ladder obtained from DMRG. The basic structure of the MM is to factorize a wavefunction with some “left” degrees of freedom $i$ and right degrees of freedom $j$ according to $\Psi_{ij}=\sum_{k}A_{ik}\Lambda_{kj}$ where $A$ is an isometry, and $A,\Lambda$ have the usual MPO structure. Here left / right will correspond to the first / second column of $\Psi$. We choose the isometric MPO $A$ to have dimensions $D_{H}=D_{V}=2$ in the bulk of the strip, while for $\Lambda$, $D_{H}=2,D_{V}=\eta$, and we’ll measure how $|\Psi-A\Lambda|$ depends on $\eta$.

In the MM, recall the tensor $T$ which splits $T\ket{ABC}=\ket{AB_{L}:B_{R}C}$ is chosen to minimize the entanglement $S_{AB_{L}:B_{R}C}$. A priori, there is no reason we need to use the von-Neumann entropy for our measure, and in fact, because we want to minimize the bond dimension $\eta$ of $\Lambda$, it is better to use a Renyi entropy $S^{[\alpha]}$ with $\alpha<1$. It is known that $S^{[\alpha]}$ bounds the accuracy of a matrix product representation for any $\alpha<1$, while $\alpha\geq 1$ does not. Verstraete and Cirac (2006) Indeed, we find that the difference between the MM and variational error decreases over the range $\frac{1}{2}\lesssim\alpha\lesssim 2$, but (for the cases we have examined) there isn’t a significant difference beyond this. So here we use $\alpha=\frac{1}{2}$. The resulting error is shown in Fig. 5. We compare it with a factorization in which the MM is further optimized by the previously discussed variational sweeps, which presumably finds the optimal solution modulo issues of local minima. The difference between the MM error and the optimal one is below a factor of 2 for all $\eta$. Practically speaking, this is a very small difference because the error is reduced by an order of magnitude just by increasing $\eta$ slightly. So (we believe) it is more efficient to use the MM, which is drastically faster, and increase $\eta$ slightly, rather than obtain the variationally optimal solution at each step.

Appendix C Index structure of the “splitting” step of the MM

Here we make explicit the index structure of the central “splitting” step of the MM. By grouping legs, the central site wavefunction $\lambda$ is viewed a tripartite state $\lambda_{\alpha\beta\gamma}$ where indices $\alpha,\beta,\gamma$ run over parties $A,B,C$. $\lambda$ is then split into the product of three tensors $a,\lambda^{\prime},V$ as follows:

[TABLE]

Given a choice for the isometry $a^{\dagger}:B\to B_{L}B_{R}$, we define the state $\Theta$ on $AB_{L}B_{R}C$ by $\Theta=a^{\dagger}\lambda$. The error at this step is $|\lambda-a\Theta|^{2}=|(1-aa^{\dagger})\lambda|^{2}$. This is just the truncated weight from the projection $a^{\dagger}:B\to B_{L}B_{R}$. Next, we bipartition according to $(AB_{L})(B_{R}C)$ and Schmidt decompose $\Theta=UsV$. The tensor $\lambda^{\prime}=Us$ is then defined by truncating the Schmidt spectrum to the $\chi$-most important states. In order to maximize the fidelity, the splitting isometry $a$ is chosen to minimize an $\alpha$-Renyi entropy $S^{[\alpha]}$ of the Schmidt spectrum $s_{k}$ ($\alpha=\frac{1}{2}$ gives good results). The tensor $a$ becomes the constituent tensor of $A^{\ell}$, $V$ is the constituent tensor of $\Lambda$, while $\lambda^{\prime}$ is the new orthogonality center which gets merged upwards into the next step of the MM.