Ideas and methods from the field of machine learning are currently having a significant impact in many areas of physics research [1]. Machine learning offers powerful new tools for classifying phases of matter [2–7], for processing experimental results [8, 9], and for modeling quantum many-body systems [10–12], to name but a few of the plethora of applications. With this crossing of fields has come the intriguing realization that the neural networks [13, 14] used in machine learning share extensive similarities with the tensor networks [15] used in modeling quantum many-body systems [16]. These connections are perhaps not so surprising since both types of network have the primary function of encoding large sets of correlated data: neural networks encode ensembles of training data, while tensor networks encode superpositions of quantum states. Currently there is great interest in exploring the potential applications of this relation, both from the directions of 1) using ideas from neural networks and machine learning to improve methods for modeling quantum wave-functions [17–20] and 2) examining tensor networks as a new approach for tasks in machine learning [21–31].

In this manuscript we focus on the second direction (ii), and explore the use of tensor networks as classifiers for supervised learning problems. Research in this area has already produced encouraging early results, with examples where tensor networks have been trained to produce relatively competitive classifiers in both supervised and unsupervised learning tasks [21, 25–27, 30, 31]. However there are some significant issues with respect to the use of tensor networks as classifiers. One such issue is that of interpretability. Usually, when applying a tensor network as a classifier, each sample from the (classical) dataset is associated to a product state. However, under generic tensor transformations, product states can be mapped to entangled quantum states, which can no longer be re-interpreted classically. One can understand this as a problem of generic tensor networks being overly-broad when used as classifiers: they are designed to carry information about phases and/or signs between superposition states, which are necessary for describing wave-functions but seem to be extraneous from the perspective of characterizing classical datasets. A second issue is that of computational efficiency. Most previous studies have utilized only relatively simple classes of tensor networks, such as matrix product states [32, 33] (MPS) and tree tensor networks [34, 35] (TTN), as classifiers. The more formidable weapons in the arsenal of tensor networks, such as the multi-scale entanglement renormalization ansatz [36–39] (MERA), which are seen as the direct analogues to the high successful convolutional neural networks [40–42] (CNNs), have yet to be deployed in earnest for challenging problems. The primary reason being that, in order for a tensor network to be of use as a classifier, ones needs to be able to compute scalar products between the network and product states (representing the training data); this can be done efficiently for simple networks such as MPS and TTN, but is generally computationally intractable for more sophisticated networks like MERA.

The main motivation for this manuscript is to help resolve the two issues discussed above. In particular, we propose to use networks built from a restricted class of tensor, those which act to preserve number-states, as classifiers for supervised learning tasks. Such number-state preserving networks automatically resolve the issue of interpretability, provided that each sample of the training data is encoded as a number state. Moreover, the restriction to number-state preserving tensors endows networks with a causal cone structure when contracted against number states, similar to the causal cone structure present in isometric networks when contracted against themselves. This property allows for a broad class of number-state preserving networks, including versions of MERA, to be efficiently trained as classifiers for supervised learning problems. Furthermore, we demonstrate numerically that networks built from this restricted class of number-state preserving tensor perform well for several example classification problems. The above considerations indicate that number-state preserving tensors are a natural restriction to impose when applying tensor methods to learn from sets of classical data.

This manuscript is organized as follows. Firstly in Section 2, we characterize number-state preserving tensors and some of their properties, then in Section 3 we formulate how problems in supervised learning can be approached using tensor networks. In Section 4 we propose an algorithm for training number-state preserving tensor networks to correctly classify a labeled dataset, while Section 5 we describe how single tensor environments can be efficiently evaluated, a key ingredient in the proposed training algorithm. Benchmark numerical results for number-state preserving versions of MPS, TTN and MERA applied to example classification problems are presented in Section 6, and conclusions are presented in Section 7.

Let L be a lattice of sites, with each site described by a local Hilbert space of some dimension d. We label the basis states for each site by integers, |z⟩ ∈ {|0⟩, |1⟩, …, |d − 1⟩}, which are interpreted as particle number and are represented as unit vectors, 0 = 1 0 0 0 ⋮ , 1 = 0 1 0 0 ⋮ , 2 = 0 0 1 0 ⋮ , … (1)

A number state | Z L 〉 (or, equivalently, a Fock state) on lattice L is a product state with well-defined particle number, Z L = | z 0 〉 | z 1 〉 | z 2 〉 … , (2) where superscripts are here used to denote lattice position. Alternatively, if one is thinking in terms of spin degrees of freedom, a number state can be defined as a product state with a well-defined z-component of spin.

We now turn our considerations to transformations of number-states implemented by certain types of oriented tensor: these are tensors where each index has been fixed as either incoming or outgoing. Any oriented tensor can be interpreted as a mapping between states defined on an input lattice L , whose sites match the incoming tensor indices, to states on an output lattice L ′ , whose sites match the outgoing tensor indices. We define an oriented tensor as number-state preserving if it maps any number state defined on L to another number state on L ′ . Several examples of number-state preserving tensors are given in Figure 1. Let u i j k l be a four index tensor, with subscripts denoting incoming indices and superscripts denoting outgoing indices, as depicted in Figure 1b. Consider the reshape of u into an input-output matrix, i.e., where the rows of the matrix enumerate over the tensor product (i ⊗ j) of incoming indices and columns enumerate over the tensor product of the outgoing indices (k ⊗ l). It is easily understood that the property of u being number-state preserving is equivalent to the property that each row of the corresponding input-output matrix must have at most a single non-zero entry. Note that we also include in the definition of number-state preserving tensors those where the input-output matrix has rows with only zero entries; equivalently these are tensors which can map some number states to the null (or norm-zero) state. An important property of number-state preserving tensors is that networks formed from their composition, where outputs from one tensor are properly matched with inputs to other tensors, are also number-state preserving, as depicted in Figure 2A. This allows us to form number-state preserving versions of commonly used tensor networks, such as MERA, as shown in Figure 2B. However, it is vital to realize that number-state preserving tensors do not necessarily remain number-state preserving if the orientation of their indices is reversed (i.e., the incoming and outgoing indices are switched); thus number-state preserving networks can still generate interesting superpositions and entangled states when “run” in reverse.

For the main text of this paper we shall further restrict our consideration to unital number-state preserving tensors, where each tensor entry must be either a zero or a one, and each row of the corresponding input-output matrix is required to have a single non-zero entry. Note that this class of tensor maps incoming number-states to outgoing number-states of the same normalization and phase. The restriction to unital tensors will be useful in simplifying their application to supervised learning problems, although the formalism and optimization algorithms that we present are still general for all number-state preserving networks. There are many reasons why one may also wish to consider networks comprised of non-unital number-state preserving tensors, where entries can take any real or complex value, and thus change the normalization of states and introduce phases; the interested reader is directed to Supplementary Appendix SA for further discussion.

Given that number-state preserving networks represent a severely restricted class of tensor network states it may be interesting to consider how much of their power has been lost, for instance, in describing ground states of quantum many-body systems. Although this remains to be explored, it seems likely that majority of many-body systems will not have ground-states that can be well-approximated by number-state preserving tensor networks. However, there does exist several examples of non-trivial quantum many-body systems related to Motzkin paths [43], whose ground states possess interesting entanglement and yet can be exactly represented by number-state preserving networks [44, 45]. Investigation of the ability of number-state preserving networks to describe general quantum ground states remains an intriguing direction for future research.

In this section we discuss how the task of supervised learning can be formulated in terms of tensor networks. We consider problems where each training sample Z ⃗ is represented as a length N vector, with the ith component z ⁱ an element of Z d (the set of integers modulo d), i.e., such that Z ⃗ k = z k 0 , z k 1 , z k 2 , … , z k N − 1 , (3) where k is a label over the set of training samples. Every training sample is assumed to be paired with a corresponding label y ∈ Z c , where c represents the number of distinct categories for the classification problem. The goal of the supervised learning problem is to construct a function f that maps each sample of the training set to its correct label, f : Z ⃗ k ↦ y k . (4)

Although classifiers based on linear functions f have some considerable utility [46], many non-trivial classification problems require non-linear functions f in order to achieve good accuracy.

We now describe how a tensor network can be implemented as the classifying function in Eq. 4. At this point, one could be tempted to believe that tensor networks would have limited utility as classifiers as, given that tensors simply are extensions of matrices to higher dimensions, they are inherently linear constructs. However, in order to recast the supervised learning problem into a problem amenable to tensor networks, we first (non-linearly) embed the training data into a higher dimensional space, similar to a kernel method [47]. By using an appropriate non-linear embedding, a linear classifier acting the higher dimension space can reproduce the classifying power of non-linear functions in the original space; thus it remains possible that tensor network approaches could be competitive with classifiers based on (non-linear) neural networks. Indeed, as will be argued later in this manuscript, it can be understood that a tensor network of sufficiently large bond dimension χ can, in principle, obtain perfect accuracy for any training set of a supervised learning problem as formulated above.

Let us recast each training sample Z ⃗ k as a number state, denoted | Z k 〉 , defined in a vector space of total dimension d ^N . Specifically, we associate each integer z ∈ Z d with a number state |z⟩ in a d-dimensional Hilbert space, represented as per Eq. 1, such that the full state vector | Z k 〉 is given as the tensor product of the single site states, Z k = z k 0 z k 1 z k 2 … z k N − 1 . (5)

Similarly the data labels y _k are recast as number states |y _k ⟩ in a c-dimensional space. The diagrammatic tensor notation for these states is presented in Figure 3. Given this embedding of our training data, a classifier can be represented as tensor network T that maps states 〈 Z k in | from the lattice of N sites of dimension d to states 〈 Z k out | on a single site of dimension c, 〈 Z k in | T = 〈 Z k out | , (6) see also Figure 2B for an explicit example.

In general, the accuracy of T as a classifier could be quantified by evaluating the scalar products of the output states with the label states, ⟨ Z k out | y k ⟩ , where a large scalar product would indicate good classification. However, in the particular case of unital number-preserving networks T , the norm of states is preserved such that all scalar products ⟨ Z k out | y k ⟩ either evaluate to unity (indicating correct classification of the data sample with label y _k ) or to zero (indicating incorrect classification of the data sample). Thus, the number of correctly classified samples N _correct simply evaluates as the sum over all the scalar products, N correct = ∑ k Z k in T y k . (7)

The diagrammatic tensor notation for Eq. 7, in the particular case that T is a binary MERA, is presented in Figure 4. It follows we should use Eq. 7 as the cost function for training the tensor network T for the supervised learning problem: the tensors contained within T should be optimized as to maximize N _correct. Methods for achieving this are discussed in the following section of this manuscript.

Before moving on, we remark that the formalism we described (or similar formalisms consider previously [21, 23, 26–28, 31]) for addressing supervised learning problems using tensor networks could, in principle, employ arbitrary tensor networks T as classifiers (not only those built from number-state preserving tensor networks). However, it is only for certain types of network, such as MPS and TTN, that scalar products of the form Z k in T y k can be efficiently evaluated. The cost of (exactly) evaluating the overlap of a product state with a more sophisticated tensor network state, such as a MERA, typically does not scale efficiently with system size. Thus, one would expect that a general MERA network would only be computationally feasible as a classifier for problems with a small number of sites (or variables). In contrast the output state 〈 Z k out | of Eq. 6 can be efficiently evaluated for any number-state preserving tensor network, with cost that scales only linearly in the number of tensors in T . Nonetheless, the result that a scalar product Z k in T y k is efficient to evaluate does not in itself imply that the network T can be efficiently trained. In Section 5 we formulate additional requirements for network T that are sufficient to allow for efficient training.

In this section we propose a method to optimize the tensors of a network T to maximize the number N _correct of correctly identified training samples in a supervised learning problem, as formulated in Eq. 7. We follow the same strategy of single tensor updates developed in the context optimizing MERA [48], where only a single tensor in the network is changed at any time while all other tensors in the network are held fixed. These single tensor updates can then be organized into “sweeps,” in which all tensors in the network are optimized in turn, and the sweeps iterated until the entire network is sufficiently converged.

Key to this optimization strategy is the notion of a tensor environment, which can be understood as the derivative of the network with respect to a single tensor. Specifically, given a network that evaluates to a scalar such as that from Figure 4B, the environment Γ _u of a tensor u results from contracting the entire network sans the particular tensor u under consideration. It follows that the number of correctly classified samples N _correct from Eq. 7 can always be expressed as the scalar product of a tensor u ∈ T with its environment Γ _u , N correct = t r u ⋅ Γ u † , (8) where, for notational simplicity, we have recast u and Γ _u into input-output matrices, see Figure 4C. We relegate a description of the general method for computing environments Γ _u to Section 5 of the manuscript, and proceed here assuming Γ _u is already known.

Let us now turn to the problem of finding the optimal number-state preserving tensor u _opt., u opt. ≡ argmax u t r u ⋅ Γ u † (9) which maximizes the number of correctly identified samples N _correct of Eq. 8, given a known environment Γ _u . Here it is easy to see that u _opt. can be built by simply identifying the location of the maximal element in each row of Γ _u and then placing the unit element at the corresponding location in each row of u _opt., with all other entries zero. Note that if the maximal element in a row of Γ _u is degenerate then u _opt. is not uniquely defined; one can still obtain an optimal solution by simply selecting one of the maximal elements in that row of Γ _u . Let us consider a concrete example: imagine we are updating a tensor u with a 4 × 4 input-output matrix of the form given in Figure 1b-iii, and assume that the environment has been evaluated as Γ u = 10 12 9 8 5 6 9 2 21 18 7 22 12 15 13 14 . (10)

Then the (unital and number-state preserving) 4 × 4 matrix u _opt. that maximizes Eq. 8 is given as u opt. = 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 . (11) and the number of correctly classified training samples after this optimal update is given as N _correct = (12 + 9 + 22 + 15) = 58. Some remarks are in order regarding this optimization strategy. Firstly, we notice that unlike many commonly used algorithms for training neural networks, our approach is not based upon a gradient descent. Instead we can directly “hop” to the true maximum for any single tensor (given that the other tensors in the network are held remain fixed), provided the environment is exactly known. While this strategy has some advantages over gradient based methods with respect to avoiding local maxima, getting stuck in a solution that is not globally optimal can still remain a possibility depending on the problem until consideration.

We now discuss methods to introduce some randomness into the optimization, in order to reduce the possibility of getting trapped in a local maxima. One approach could be to employ a similar strategy as used in the stochastic gradient descent methods [49], where randomness is introduced by using only select “batch” of training samples for each update. Instead, here we advocate a different strategy inspired by Monte Carlo methods [50] used in sampling many-body systems. Rather than updating to the optimal tensor u _opt. at each step, we propose to allow updates to sub-optimal solutions of Eq. 8, with a probability diminishes exponentially in relation to how far the solution is from the optimal solution. For this purpose we first introduce the difference matrix Ω, given by subtracting from each row of Γ the maximal element within the row, Ω i j = Γ i j − max j Γ i j . (12)

For the example environment Γ _u given in Eq. 10 the corresponding difference matrix is Ω = − 2 0 3 4 4 3 0 7 1 4 15 0 3 0 2 1 . (13)

We then use the difference matrix to generate a matrix p ^trans. of transition probabilities, defined element-wise as p i j trans. = exp Ω i j / α ∑ j exp Ω i j / α (14) where α is a tunable parameter that sets the amount of randomness. For the example difference matrix Ω of Eq. 13 and setting α = 2 we get the transition matrix p trans. = 0.21 0.58 0.13 0.08 0.10 0.16 0.72 0.02 0.35 0.08 0.00 0.57 0.10 0.45 0.17 0.28 . (15)

The transition matrix is then used to perform a stochastic update of the tensor u under consideration: values in each row of p ^trans. set the probability for the unit element in the equivalent row of the updated u to be placed at that particular location (note that Eq. 14 has been defined such that each row of p ^trans sums to unit probability). Notice that in the limit α → 0 the matrix p ^trans. tends to u _opt. (provided Γ had no degeneracies in its maximal row values), since all non-optimal transitions are fully suppressed. Conversely, in limit α → ∞ all probabilities in p ^trans. tend to the same value, representing completely random transition probabilities.

Here we describe evaluation of tensor environments, crucial to the optimization algorithm discussed in the previous section. For simplicity, we describe this evaluation assuming the tensor network T under consideration is a binary MERA, although the same methodology can be employed for arbitrary (number-state preserving) tensor networks.

Rather than tackling the problem of computing tensor environments Γ directly, we first introduce the concept of configuration spaces |ϕ⟩. Proper use of configuration spaces |ϕ⟩, which play an analogous role to the local reduced density matrices ρ used to optimize tensor networks in the context of quantum many-body systems, will greatly simplify the subsequent evaluation of environments. Let us assume that the output index of the tensor network T under consideration has been fixed in some specified label state |y⟩, and that the lattice on which it is defined has been partitioned into a region A and its compliment B. Then, given a number state | Z A 〉 on region A, we define the configuration space |ϕ ^B ⟩ as ϕ B = ∑ c o n fi g s : σ Z σ B , (16) where the sum runs over all valid configurations σ of number states | Z σ B 〉 defined on region B such that the combined number state Z A Z σ B is classified by T into the correct category |y⟩, i.e., such that Z A Z σ B T y = 1 . (17)

An example of a network that could be contracted to evaluate a configuration space |ϕ ^B ⟩ is depicted in Figure 5A. It is seen that this network can be simplified, as shown Figure 5B, by lifting the input number state | Z A 〉 through tensors in T where-ever possible (i.e., where-ever a tensor has a number state available on all of its incoming indices), using the number-state preserving tensor properties as outlined in Figure 1. It is convenient to define the configuration causal cone C ( B ) associated to region B as the set of tensors remaining in the network T after this simplification; equivalently C ( B ) can be defined as the set of tensors C ∈ T whose output state can be affected by the choice of input state on region B.

Notice that this configuration causal cone C ( B ) is precisely equivalent to the (standard) causal cone [36, 51] that would emerge from an isometric MERA for the same region B, defined as the set of tensors that can affect the local reduced density matrix ρ _B . However, the origins of these causal cones are drastically different: the causal cones in isometric MERA result arise due to the isometric constraints imposed on tensors, whereas the number-state preserving tensors proposed in this manuscript are not required to be isometric. Similarly, configuration causal cones arise only in networks that preserve number states, and are thus ill-defined for generic MERA [Note that it is, however, possible to have networks with tensors that are both simultaneously isometric and number-state preserving, see Supplementary Appendix SA for further discussion]. Despite the difference in the origins of these two forms of causal cone, it is not a fluke that they were exactly equivalent in the previous example. It can be understood that the configuration causal cones in any number-state preserving tensor network are always equivalent to the causal cones found in an isometric tensor network of the same geometry, provided that the index orientations (specifying incoming and outgoing indices) match between the networks. Given this equivalence, we will henceforth drop the distinction between the two definitions, such that the term “causal cone” can refer to either definition.

The process of evaluating the configuration space for a region B of three sites from a binary MERA is depicted in Figure 6A. This evaluation can be formulated as a sequence of contractions that each “lower” the configuration space through the causal cone, … | ϕ 2 B 〉 → | ϕ 1 B 〉 → | ϕ 0 B 〉 (18) where bracketed subscripts denote configuration spaces at different depths within the network. Each of the lowering contractions is implemented by one of two geometrically different lowering operators, depicted in Figure 6B, which are the direct analogues to the descending superoperators [48] used in the evaluation of density matrices from isometric MERA.

In our example using a binary MERA, the cost of evaluating | ϕ B [ 0 ] 〉 for a region B of three contiguous sites scales at most linearly with the network depth, since the form of the lowering operators are self-similar at all depths. In a general (number-state) preserving network the computational cost of evaluating configuration spaces will be related to the causal structure of the network: the leading order cost will scale exponentially with maximum width of the causal cones. Thus it is apparent that not all number-state preserving tensor networks can be efficiently evaluated for local information (characterized by the configuration space |ϕ _B ⟩); only those for which the maximum causal width is not too large. However, since MERA are precisely designed to have bounded causal width (i.e., the causal width never spreads beyond some small number of sites), it follows that number-state preserving versions of MERA networks precisely fall within the class of networks that can be efficiently evaluated.

Given that the evaluation of configuration spaces has been understood, we now turn to the task of building the environment Γ _u associated to tensor u, as depicted in Figure 7, which is accomplished as follows. First we lift the initial number state | Z k in 〉 to a new number state | Z ̃ k 〉 that lives on the boundary of causal cone C ( u ) associated to tensor u, as depicted in Figure 7A. Then we compute the configuration space 〈 ϕ ̃ k u | defined on the output indices of tensor u, as depicted in Figure 7B. Then the environment Γ _u is given by taking the outer product of the configuration space 〈 ϕ ̃ k u | with the piece of the state | Z ̃ k 〉 supported on the input indices of u, denoted | Z ̃ k u 〉 , while summing over all training samples k, Γ u = ∑ k Z ̃ k u ϕ ̃ k u , (19) see also Figure 7C.

In this section we present benchmark results for how number-state preserving tensor networks perform as classifiers in some simple problems. The goal here is to establish the feasibility of our proposal, rather than to establish performance for challenging real-world tasks, which will be considered in future work. In particular we demonstrate 1) that the proposed optimization algorithms can efficiently and reliably train the networks under consideration, and 2) that number-preserving networks perform comparably well to unrestricted networks for classification tasks.

We have proposed the class of number-state preserving tensor networks for use as classifiers in supervised learning tasks and have shown that a large class of these networks, specifically those with bounded causal structure, are efficiently trainable for large problems. In particular we have described a training algorithm that, for any chosen tensor in the network under consideration, exactly identifies the optimal tensor for that location (i.e., that which maximizes the number of correctly classified training samples), all with cost that scales only linearly in number of training samples. Importantly, the class of efficiently trainable number-state preserving networks includes realizations of sophisticated networks such as MERA, which would otherwise be computationally intractable. As such, we believe this could be the first computationally viable proposal which would allow MERA, close tensor network analogues to convolutional neural networks, to be applied as classifiers for challenging tasks such as image recognition. This remains an interesting direction for future research.

Although number-state preserving tensors represent a highly restricted class of tensor, the preliminary results of Section 6 are encouraging that this class is sufficient when applying tensor networks as classifiers for learning problems as outlined in Section 3. It still remains to be seen whether number-state preserving tensor networks are as powerful as generic tensors networks for these tasks; this question requires further theoretical and numerical investigation. However it is relatively easy to understand that, in the limit of large bond dimension, a number-state preserving tensor network could in principle achieve 100% accuracy on any training problem outlined in Section 3. The reasoning follows similarly to the argument that a generic tensor network can represent an arbitrary quantum state in the limit of large bond dimension. Consider, for instance, the MERA depicted in Figure 2B. One could increase the bond dimension of indices within the network until the output index of each w tensor matches the product of its input dimensions, in which case each w could be fixed as a trivial identity tensor when viewed as an input-output matrix. In this scenario, the top tensor w _top could implement an arbitrary classifier that would perfectly map every training sample to its designated label, regardless of the training data given.

A major difficulty with the use of MERA in D = 2 or higher spatial dimensions [37, 38] is their high scaling of computational cost with bond dimension χ. However, there is reason to be more optimistic for their application as classifiers. The cost of contracting an isometric metric MERA for a density matrix, necessary for its optimization towards the ground state of a local Hamiltonian, is related to the size of the maximum causal width of the network. For instance, the most efficient known 2D isometric MERA [38] has a causal width of 2 × 2 sites, such that the density matrices within the causal cone have 8 indices. The cost of computing these density matrices can be shown to scale at most as O (χ ¹⁶). However, while a number-state preserving version of this 2D MERA would also have a causal width of 2 × 2 sites, the relevant configuration space |ψ⟩ within the causal cone would only have 4 indices (which follows as the density matrix involves both the bra and the ket state, whereas the configuration space only involves the ket). Thus the cost of optimizing a number-state preserving version of this 2D MERA, where the key step is the evaluation of configuration spaces, will scale roughly as O (χ ⁸) (i.e., the square-root of the cost of optimizing an isometric MERA for a quantum ground state). This square-root reduction in cost scaling as a function of bond dimension χ from isometric to number-state preserving networks will hold in general, such that number-state preserving networks could realize much larger bond dimensions given a fixed computational budget. This advantage is somewhat mitigated by the fact that the cost of optimizing a number-state preserving network comes with a factor n _samp related to the size of the training set, which could be very large. However, it would also be straight-forward to parallelize the evaluation of environments over the samples.

Although the main text of this manuscript focused on number-state preserving versions of MERA, many other forms of hierarchical network could also be of useful as classifiers as discussed further in Supplementary Appendix SB1. In particular the network of Supplementary Figure SB2, which does not have an isometric counterpart, seems to be the closest tensor network analogue to a convolutional neural network. Rather than disentanglers, this network uses δ-function tensors to effectively allow neighboring w tensors to “read” from the same boundary sites, mirroring the overlap of feature maps arising in a convolution (and similar to the generalized networks recently proposed in Ref. [31]). It would be interesting to compare the effectiveness of this structure versus a traditional MERA, which will be considered in future work.