# Quantum-inspired Techniques in Tensor Networks for Industrial Contexts

Alejandro Mata Ali  i3B, Quantum Development Department, Paseo Mikeletegi 5,
20009 Donostia, Spain<EMAIL_ADDRESS>Iñigo Perez Delgado  i3B, Parque
Tecnológico de Bizkaia, Ibaizabal Bidea, Edif. 501-A, 48160 Derio, Spain
<EMAIL_ADDRESS>Aitor Moreno Fdez. de Leceta  i3B, Unidad de Inteligencia
Artificial, Avenida de los Huetos, Edificio Azucarera, 01010 Vitoria, Spain
<EMAIL_ADDRESS>
###### Abstract

In this paper we present a study of the applicability and feasibility of
quantum-inspired algorithms and techniques in tensor networks for industrial
environments and contexts, with a compilation of the available literature and
an analysis of the use cases that may be affected by such methods. In
addition, we explore the limitations of such techniques in order to determine
their potential scalability.

##  1 Introduction

Quantum computing is a novel field that is receiving considerable interest due
to its particular way of performing operations and processing information,
being able to tackle highly complex computational problems. In the information
and digitalization era in which we find ourselves, this technology has a
remarkable applicability for industrial cases, for example in the optimization
of delivery routes [1, 2], packet storage [3] or quantum machine learning [4]
applied to various contexts [5, 6, 7, 8].

However, despite the great progress in the development of digital quantum (or
‘logic gate’) computers, substantial challenges remain, such as the fragility
of quantum states, errors in quantum operations, or the scalability of the
number of qubits.

In this context, there is a need to explore alternative approaches that can
take advantage of the computational properties of quantum systems, but do not
have to run on digital quantum computers. One such approach may be quantum
annealing [9, 10], which runs on specialized quantum devices dedicated to
combinatorial optimization of particular cost functions. Another one is
digital annealing [11, 12, 13], which draws on properties of quantum annealing
to simulate it classically.

On the other hand, one of the most promising technologies, although more
unknown within the industry, is tensor networks (TN) [14]. Tensor networks are
a class of quantum-inspired algorithms and techniques based on mimicking the
tensor operations performed by a quantum computer, but executing them on
classical computers. By using tensor properties, the execution of such
operations can be optimized, especially in cases where the entire quantum
state vector is not required, but only properties of it.

Another relevant feature of tensor networks is that they allow the efficient
representation of certain families of quantum states, by means of
representations such as the matrix product state (MPS)[15, 16], also called
tensor train (TT), or projected entangled pair states (PEPS) [16]. In this
way, with a reduced amount of memory we can perform quantum calculations and
obtain properties of complex systems. These same representations have been
relevant to the world of machine learning, since they are capable of
compressing models by reducing the memory required without losing a notable
amount of precision [17, 18, 19].

Given all these capabilities, tensor networks are highly susceptible to be
used in industrial contexts, being able to address highly complex and large
problems efficiently. Despite being a relatively mature field at the academic
level (initiated in 1971 by Penrose [20]), its study applied to industry is
recent and always associated with quantum computing. However, given the great
results achieved in recent years by tensor networks, outperforming quantum
algorithms executable on current hardware [21, 22], many companies are
beginning to take an interest in the study and implementation of this
technology.

In this paper we survey the main uses of tensor networks for industrial use
cases, compiling state-of-the-art articles and exposing the strengths and
weaknesses of each technique. In addition, we classify the algorithms
according to their specific use case. The field of tensor networks has an
extensive and complex literature, so the aim of this article is to serve as an
introductory guide to it.

##  2 Introduction to tensor networks

The first thing we will do is to make a brief introduction to tensor networks
in order to understand the subsequent sections. In an effort to stay within
the main objective of the paper, we will assume knowledge of basic linear
algebra concepts and focus on tensor networks as such. Complementary
information on tensor networks can be found in [14, 23].

A tensor network (TN) is a graphical representation of a multilinear equation
between different tensors. We can see an example in Fig. 1, where we have the
tensor network representing the 2-tensor $T_{ip}$ whose elements are obtained
by means of the contraction operation

$T_{ip}=\sum_{j,k,l,m,n,o}A_{ijkl}B_{jm}C_{kmn}D_{lo}E_{nop}.$ (1)

Figure 1: Tensor network representing the 2-tensor $T_{ip}$ of Eq. 1.

As can be seen, this is a very simple representation of a very complex
operation, in whose larger generalizations it is complicated to determine the
relationships between the tensors contracted and the most efficient way to
contract them. In addition, we can further abstract the tensor network by
eliminating the names of the indexes and tensors if we know by some rule what
they should be. An example would be that the tensor indexes are placed
according to the clockwise orientation.

The rules followed by such representations are as follows:

  1. 1.

Each node of the network represents a tensor.

  2. 2.

Each outgoing line of a tensor represents an index of that tensor. Therefore,
a node with $n$ outgoing lines represents an $n$-tensor.

  3. 3.

Each line connecting two tensors represents that both tensors are contracted
with respect to a common dummy index.

  4. 4.

Lines that only come out of one tensor, but do not connect to another,
represent free indexes. Therefore, a tensor network with $n$-free indexes
represents an $n$-tensor.

  5. 5.

Nodes that are not connected by an index are in tensor product with each
other.

A highly interesting and useful type of tensor network is the already
mentioned MPS [15], since it consists of an one-dimensional and decomposed
representation of a tensor by means of the product of matrices. We can see an
example in Fig. 2 a), where we see the MPS representation of a 5-tensor. This
MPS representation is composed of 2 types of indexes: the bond indexes, which
will have dimension $b$, and the physical indexes, of dimension $d$,
corresponding to the dimensions of the tensor to be represented. The physical
indexes are the free indexes, which coincide with the indexes of the tensor to
be represented, while the bond indexes are those that connect the tensors of
the network to each other and are responsible for accounting for the
correlations between the physical indexes. It should be noted that there are
other versions of the MPS representation with different boundary conditions
(the way to link the end nodes), but the simplest is this one. There is also
the possibility that the dimensions of the bond indexes are different, as are
those of the physical indexes, for reasons we will mention at the end of this
section.

For an $N$-tensor MPS representation, with constant bond dimension $b$ and
constant physical dimension $d$, the number of elements to store is
$2db+(N-2)db^{2}$ versus the $d^{N}$ that would be needed in the dense,
contracted version of the tensor, $db$ for each end node and $db^{2}$ for each
node in the interior of the chain.

Any tensor can be represented in MPS representation exactly, although not
always efficiently, since $b$ can be of the order of $d^{N}$ in certain
complex cases and where the correlation is very strong. Therefore, it will be
important to consider how $b$ increases with the size of the tensor to be
represented. Still, many cases of interest can be represented in MPS
representation efficiently. For example, in academic contexts its most
widespread use is that of representing quantum states of $N$ particles, that
is, state vectors of $d^{N}$ components.

An extension of the MPS representation is the Matrix Product Operator (MPO)
[24], consisting in allowing each node to have 2 physical indexes, as we see
in Fig. 2 b). The usefulness of this representation lies in cases where we
want to represent a matrix of $d^{2N}$ components, such as a quantum
interaction. In this case, the number of elements to store is
$2d^{2}b+(N-2)d^{2}b^{2}$.

Figure 2: a) MPS form of a 5-tensor, b) MPO form of a 10-tensor.

Such forms can be obtained mainly in three ways. The first is by logical
construction of the representation, so that the interactions between the
different nodes of the tensor network generate the tensor we want. The second
will be the variational construction, consisting of starting with a test set
of tensors and using techniques such as gradient descent to get it to become
the one we want to represent. The third is by singular value decomposition
(SVD) [25], iterative, where we perform an SVD for each pair of physical
indexes. There is also an optimized method for sparse tensors [26].

Importantly, either representation can be used to represent a vector, matrix,
or general tensor, since we can convert an $m$-tensor to an $n$-tensor by
techniques such as ‘grouping’ and ‘splitting’ [14], a bijective mapping from
one set of indexes to another set of indexes.

Consider an example of obtaining the MPO representation of $n$ nodes of a
$N\times M$ matrix. To do this we will have to start by transforming the
matrix into a $2n$-tensor, by means of a splitting. The resulting $A^{\prime}$
tensor will have dimensions $\vec{D}$ such that

$N=\prod_{i=0}^{n-1}D_{i},\qquad M=\prod_{i=n}^{2n-1}D_{i}.$ (2)

Now we join the pairs of input and output indexes that we want to be part of
the same node of the MPO form and we group them to obtain a tensor
$A^{\prime\prime}$ whose new dimensions $d_{i}=D_{2i}D_{2i+1}$ are
$d_{i}=D_{2i}D_{2i+1}$. Now we perform the iterative process of searching for
the MPS representation of that tensor, to finally split the physical indexes
of each node to obtain the original input and output indexes. The whole
process can be seen in Fig. 3.

Figure 3: Process of compression of a matrix to its MPO representation. 1) We
apply a splitting. 2) We join the indexes of each node in pairs. 3) We apply a
grouping of the index pairs. 4) We perform the iterative SVD. 5) We apply a
splitting to each physical index.

If instead of wanting an exact representation of a tensor, we would like an
approximate representation of it in order to have a smaller $b$ and thus save
memory, we can take advantage of these two representations. In the variational
construction we will simply operate with smaller $b$, while in the iterative
SVD we will perform truncated SVD (TSVD) [27], keeping only the $b$ largest
singular values, which represent the most significant correlations. We can
also approximate a previously obtained MPS representation [28]. The accuracy
of this approximation will depend both on the tensor itself that we want to
represent and on the mapping we use, since if we put together in the same
nodes the most correlated ‘columns’, all their correlations will be within the
node itself. Moreover, if we have the most correlated indexes adjacent to each
other, we will need a smaller bond dimension, since the intensity of the
correlations between two nodes of an MPS representation decays exponentially
with the geodesic distance between them, the minimum distance that separates
them.

##  3 Basic tensor properties and more complex techniques

In this section we will make a connection between the basic properties of
tensor networks and certain more advanced and complex techniques. In this way
we will create a bridge between the abstraction of tensor networks and their
practical applications. To do so, we will approach each case explaining which
properties are determinant in it and which is the advantage it offers with
respect to other methods.

###  3.1 Simulation of quantum systems

As is evident from their historical context, tensor networks are especially
intended for the efficient representation of many-body physical systems. This
is of great interest in fields such as condensed matter physics [29, 30] or
quantum computing [31, 32].

For this purpose an estimated (or ‘ansatz’) solution that is representable as
a tensor network is chosen and worked with, translating into tensor notation
also physical operators and magnitudes. This allows us to work with a limited
amount of memory, unlike the case of working with the entire statevector.
Cases where tensor networks are very successful are in obtaining minimum
energy states in quantum chemistry [33] and in the time evolution of certain
quantum states to obtain certain magnitudes and properties of them [21, 22].

###  3.2 Machine learning model compression

The best known application of tensor networks is the compression of machine
learning models [17, 18, 19]. This can be done either by compressing an
already created model and training it or by creating a compressed model and
training the tensor network that represents it directly. The key to model
compression lies in finding an efficient representation of the model that
preserves its analysis properties. We will explain how it works by compressing
a dense layer in a neural network to its approximate MPS and MPO
representations.

To begin with, we will have to keep in mind that the action of a dense layer
of a neural network, prior to the action of an activation function, can be
written as

$\vec{v}=A\vec{x}+\vec{c},$ (3)

where $A$ is a matrix of dimension $N\times M$, $\vec{v}$ and $\vec{c}$ are
vectors of dimension $N$ and $\vec{x}$ is a vector of dimension $M$. The input
to the layer is the vector $\vec{x}$, while the output will be the vector
$\vec{v}$. The layer will be characterized by the $M$ components of the matrix
$A$ and the $N$ components of the vector $\vec{c}$, which will be the ones to
be trained. Therefore, the layer will have a number of $N(M+1)$ parameters. To
reduce the number of parameters we can represent the matrix $A$ as an
approximate MPO form with a certain bond dimension $b$ so that we have a
smaller number of parameters than in the original dense version. We can do the
same with the vector $\vec{c}$ obtaining its approximate MPS form. In this
way, we have compressed the model layer. This is favorable both to save memory
space and to train the model faster and avoid problems such as overfitting,
depending on the representation used and the original model.

Since the $A$ matrix approximation and the $\vec{c}$ vector approximation can
be made arbitrarily close to the originals by increasing the $b$ bond
dimensions, we can always start from the original model and compress it until
we saturate a preset tolerable error. However, we can also compress the
elements of the model and run it to see how accurate it is with respect to the
original model, increasing the compression until we reach that maximum error.

Another possibility would be to use a compressed version of the untrained
model, with a controlled initialization as in the original, and train the
compressed model directly. This has the advantage that we do not lose accuracy
by approximating our model, but train an already pre-approximated model. We
can also initialize the model by compressing an already trained model and
retraining it.

There are many compression schemes in addition to the MPS and MPO, such as the
PEPS [34, 35] or the MERA [36]. These allow to efficiently perform certain
operations on their compressed representations and to apply various
optimization methods, such as the Density Matrix Renormalization Group (DMRG)
[36, 37].

###  3.3 Application of high dimensionality operations

A problem that we may have in real industrial cases is the need to apply high
dimensional operations, such as creating certain kernels for methods like the
Support Vector Machine [38]. We can take as an example wanting to apply a
kernel in tensor product of the components of an $\vec{x}$ input vector of $N$
components [39]. This kernel is

$\phi(\vec{x})_{i_{1},i_{2},\dots,i_{N}}=\phi_{1}(x_{1})_{i_{1}}\phi_{2}(x_{2})_{i_{2}}\dots\phi_{N}(x_{N})_{i_{N}},$
(4)

where each kernel $\phi_{j}(\cdot)$ affects the $j$-th component of the vector
$\vec{x}$. In this case, we can consider a kernel $j(\cdot)$.

$\phi_{j}(x_{j})=(x_{j},1),$ (5)

so that the global kernel $\phi(\cdot)$ will give us all possible products
between the components of the input vector, as a $2^{N}$ component tensor to
express correlations in the vector. As we can see, in the standard way this
would require an exponential amount of memory and time in the size of the
input vector. Furthermore, if we want to apply a matrix to such a kernel we
would need a matrix $A$ of dimension $M\times 2^{N}$, which scales
prohibitively.

However, by taking advantage of the properties of tensor networks we can use
an MPO representation for such a matrix, of the form depicted in Fig. 4, where
only $2N$ smaller tensors need to be contracted.

Figure 4: Eq. 4 and MPO layer for an input vector of $N=5$ components. Figure
5: Contraction of the first three nodes of the MPO layer with the first three
nodes of the kernel. 1) We contract a kernel node with its corresponding MPO
layer node. 2) We contract the next kernel node with its layer node. 3) We
contract the two new nodes obtained. 4) We repeat the contraction process for
the next pair of nodes. 5) We contract the current nodes, maintaining the
physical index.

With this methodology we can apply an operation in which we weight elements of
type $x_{1}x_{2}x_{3}x_{4}x_{5}$, $x_{1}x_{2}x_{3}x_{4}$, $x_{1}x_{2}x_{3}$,
$x_{1}x_{2}x_{2}$ and $x_{1}$, efficiently and with a reduced number of
elements.

###  3.4 Data compression and denoising

In the field of big data there is a need to work with large volumes of data,
which can be very expensive or unfeasible for certain systems. The tensor
compression property that we have seen in representations such as MPS allow us
to compress [40] and handle [41] large amounts of data. To do this we just
need to convert the data into a tensor and compress it. In addition, if we
allow ourselves to reduce the precision of its elements or are only interested
in its global characteristics, we can use approximate representations of the
tensor to save memory space.

###  3.5 Combinatorial optimization

Solving most combinatorial optimization problems requires runtimes
exponentially increasing with problem size. Given the properties of quantum
computation, many can be tackled through it efficiently. A similar approach to
quantum lies in the use of tensor networks for combinatorial optimization.
There are different types of combinatorial optimization solvers.

The first one is the one developed in [42], running in a black box. This
algorithm starts from a cost tensor, $T$, whose subscripts are the components
of the solution vector. This tensor will have the cost of all possible states,
although it is not necessary to know the tensor explicitly: it is enough to
have a function that gives us the cost as a function of its indexes. With this
tensor, we will make an approximation of its max volume submatrix, the
submatrix containing its largest singular values in product, by means of an
MPS representation. From this representation we will obtain, with high
probability, the lowest cost element.

The problem with this algorithm is that when we work with problems with many
invalid configurations, it tends to fail because it sees that almost all or
all the elements are the same, since it is sampling the tensor. Therefore, we
need an encoding, however complicated it may be, in which we reduce the space
of invalid configurations, so that the algorithm sees a certain functional
map.

The second possibility is based on the use of Generative Artificial
Intelligence (AI) boosted with tensor networks [43].

The third possibility is the use of imaginary time evolution to a quantum
state in uniform superposition of all possible combinations, equivalent to the
application of a Hadamard gate to each qubit of a quantum circuit. If from
this uniform superposition we damp each combination exponentially to its
associated cost, the optimal combination will be the one with the largest
amplitude, resulting in a higher measurement probability.

Due to its non-unitareity, at the quantum level this type of operation cannot
be performed, but at the classical level it is possible to perform [44]. If we
needed to take into account all possible combinations, we would need an
unbearable amount of memory, resulting in a brute-force search. However, by
combining the ability to compress information and simulate quantum states with
the ability to apply non-unitary tensor network operations, we can create such
a state for various problems [45, 46, 47]. This can be done in an exact or
approximate way, by recompressions, depending on how much memory we want to
devote to it.

The main limitation is that for NP-Hard problems, the most complex ones, we
will need an exponential amount of memory and time to solve them exactly. By
making intelligent adjustments in the encoding and expression of the problem,
the creation and contraction of the tensor network and applying various
simplifications in the constraints, we can create efficient approximate
algorithms from the exact ones, avoiding exponential scaling.

The fourth is the simulation of a physical system represented by a variational
tensor network, with a cost function given as a Hamiltonian that we can write
in MPS or MPO representation. In this case we will perform variational updates
of the state tensor network, using techniques such as gradient descent or DMRG
to minimize the energy of the system to find the optimal combination. The
effectiveness of these types of techniques depends very much on the problem to
which they are applied and they can get stuck in local minima, especially for
problems with constraints. In addition, there is the limitation that we must
be able to express the cost function in a tensor network representation.

##  4 Use case studies

In this last section we will present the main use cases in which we can see a
direct and significant potential of tensor networks techniques to industrial
and business contexts.

###  4.1 Finance

In the contemporary financial domain, high dimensionality emerges as a
challenging phenomenon. With rapid technological advancement and the
availability of large-scale data, financial markets have become inherently
multidimensional. Thanks to this, in-depth market analysis is possible.

####  4.1.1 Portfolio optimization

Portfolio optimization consists of determining the best allocation of
financial assets in which to invest in order to maximize a cost function,
which is usually one that maximizes return and minimizes risk. This problem is
already complicated in a static way, becoming extremely complex in a dynamic
way, with a portfolio to be optimized over a series of trading days. The high
dimensionality and constraints of the problem make it generally an intractable
problem in an exact manner.

Tensor networks can be used to explore the space of inversion solutions in
order to search for the optimal combination. In [48] we can see how they use
imaginary time evolution on an MPS representation to solve the optimization
problem. They achieve sharpe ratio results, a measure of excess return per
unit of risk on an investment, superior to those obtained by other methods for
large sizes, although with lower payoff and much longer run time for large
problems than Dwave’s hybrid solver.

####  4.1.2 Interpretable predictions

In [49], a recurrent neural network method is presented for making
interpretable predictions, those in which the underlying reasoning can be
understood, using an MPS representation of the recurrent layer weight matrix.
It can be observed that the accuracy, Sharpe and return results are higher in
the tensor network version than in the other compared versions.

###  4.2 Medicine

The field of medicine is one of the fields with the highest dimensionality,
both for the diagnosis and treatment of diseases, as well as in the design and
production of drugs. Because of this, there has recently been a great interest
in the application of Artificial Intelligence techniques in this field,
obtaining great results.

####  4.2.1 Drug discovery

Drug simulation is a monumental challenge in contemporary science due to the
complexity of molecular interactions and the enormous number of possible
configurations. In this context, quantum computing offers a clear advantage as
it is able to model and treat phenomena such as the electronic structure of
molecules and interactions between atoms more adequately than classical
computing. As explained above, the ability of tensor networks to efficiently
represent quantum states may allow us to simulate certain molecules that are
currently not possible with quantum computers.

Similarly, we can save time and costs of new drug discovery by predicting new
therapeutic properties associated with new drugs. This can be achieved in
several ways, one of them being the tensor decomposition [50]. To this end, we
create a drug-gene-disease tensor, which accounts for the relationships found
between these drug-gene, drug-disease and gene-disease pairs. With this, we
formulate a tensor completion problem. The initial tensor will have only the
direct relationships of the pairs we have found, while the rest of the more
complex relationships will be the ones we have to determine. For this, we will
perform a generalized tensor decomposition (GTD), to model part of these
relationships, and we will apply a multilayer perceptron (MLP) to capture the
rest of the relationships.

The main limitation of this method is that its predictive capacity is inferior
for new diseases, so other techniques must be coupled to it.

####  4.2.2 Medical image analysis

The analysis of images, and especially medical images, is a very complicated
task, either because of the characteristics of these images, the noise in them
or the high dimensionality that the problem can reach. However, as we have
seen, tensor networks are ideal for cases with high dimensionality and noise.

In [51] authors present methods for classifying medical images, using exactly
the scheme we explained in Ssec. 3.3, interpreting the images as an input
vector and a feature map cosenoidal, and the MPO layer as a classifier. Thus,
by contracting the tensor network we will obtain a vector that will tell us
the probabilities of that image belonging to one group or another. This is
done in combination with other more advanced machine learning techniques and
finally we obtain an extremely high level of compression with levels of area
under the ROC (Receiver Operating Characteristic) curve similar or superior to
the methods with which it is compared. All this with much lower GPU
requirements.

Medical image segmentation is vital for disease diagnosis, treatment
monitoring, surgical guidance, experimental drug development and medical
education. A method for medical image segmentation with tensorized transforms,
introduced in the paper, is shown in [52]. In this method, a tensorized
version of the self-attention module, a step of the transformer, is realized.

###  4.3 Simulation of quantum materials and topological materials

Simulation of quantum materials and topological materials is an extremely
complex problem, because they are high-dimensional problems. Quantum many-body
systems require an exponential memory in the number of elements to represent
their wave functions. The utility of these materials is enormous, given their
curious applications, such as high-speed electronics, solar cells, quantum
sensors, telecommunications and quantum computing.

To deal with this, concrete ansatz are proposed [53, 54, 55, 56] which
accurately and efficiently represent the minimum energy states of these
systems. We start with an initial state and perform various transformations,
such as DMRG or imaginary time evolution, to bring it down to its minimum
energy state. Once this state is obtained, we can easily obtain properties of
the state by applying the tensors of the quantities to be measured.

###  4.4 Battery simulation

Battery simulation is a very useful problem, but it also requires the study of
quantum many-body systems. For this we can also use ansatz in tensor networks
[57, 58], with which we can obtain properties of the minimum energy states.

###  4.5 Optimization

Optimization problems in industrial environments are very important, as they
may require the optimization of the order in which processes are performed,
such as the assignment of times and tasks to machines in the Job Shop
Scheduling Problem (JSSP). However, due to the problem size in these cases, it
is very expensive to solve such instances due to the high number of
combinations and the complexity of the cost functions.

####  4.5.1 Route Optimization

The Traveling Salesman Problem (TSP) is a widely studied historical problem,
in which a route must be chosen that traverses all nodes of a certain network
only once and offers the lowest cost associated with it, given by the costs of
going from one node to another. Generalizations and particular cases of this
problem model sharing problems, which are very important in industry.

For this purpose, [45] propose a TN method that can solve generalized TSP
cases, with different variants. This method takes as variable the node in
which one is at each step of the route and thus model the cost function as a
Quadratic Unconstrained Discrete Optimization (QUDO) to the nearest neighbor
with non-repetition constraints. Its basic mechanics resides in the imaginary
time evolution described in [46] and the connection of a set of layers that
impose the constraints, eliminating non-compatible combinations in the
represented state. By modifying such constraint layers and the QUDO modeling,
different generalizations can be addressed, such as being able to repeat up to
$N$ times a certain node, that the start and end can be different or that we
can only pass through each group of nodes once. All these cases are studied in
[45].

The main limitation of this method is the exponential scaling it suffers with
the number of nodes due to the constraints, but iterative methods such as the
one explored in [47] are able to limit this scaling successfully.

####  4.5.2 Post assignment optimization

A particular case of generalized TSP is the assignment of jobs to workers.
Assume a set of workers who are already assigned to a set of jobs and we have
a set of vacancies. The objective will be to redistribute workers to the set
of jobs, including filled and vacant jobs, such that we maximize a certain
cost function and there are only 0 or 1 workers per job. This problem has been
studied and treated at the end of [45], using the formulation of [59], which
seeks to maximize a combination between the productivity of the job and the
compatibility of the worker assigned to it.

####  4.5.3 Manufacturing Sequence Optimization

The optimization of manufacturing sequences is a very useful industrial
problem. We have a set of products to be manufactured sequentially, such that
there is additional manufacturing time when moving from manufacturing one type
of product to manufacturing another. This time can be a time to change parts
of a machine, configuration or materials. Therefore, we want to obtain the
sequence that minimizes this overall additional time.

This case is studied in [45], since it can be seen as a TSP in which we can be
at each node as many times as there are outstanding products of the class
associated to that node in the initial set. With this perspective, a node
represents a class rather than a product to optimize the constraint layers.

####  4.5.4 Machine task assignment optimization

Optimal assignment of tasks to machines can be modeled as the choice of task
execution on a set of machines such that there is a constraint between the
tasks performed by one and another. An example of a constraint would be the
following: ‘if machine 1 does a cutting, machine 2 cannot do a peeling’. These
constraints can be extracted from a historical record or from the business
knowledge of the usual machine manager. In addition, each task will have its
own execution time on its corresponding machine.

This problem is addressed in [47], where a TN method with imaginary time
evolution and constraint layers like those of [45] is combined with a genetic
algorithm and an iterative method to reduce the time and memory scaling of
such algorithm, taking advantage of properties of this type of problems. The
main limitation of the method is that, although it has a high success rate,
for certain cases it cannot obtain satisfactory solutions for a given memory
limit.

###  4.6 Big Data

As its name indicates, in the field of big data large volumes of data are
handled, so we need specific systems to deal with them.

The work [40] presents different methods to deal with big data problems using
tensor network decompositions, as well as other topics. On the other hand, in
[41] a tensor network method for secret sharing in big data is presented,
based on the decomposition into an MPS or a Tucker decomposition [60] of the
original data tensor by means of consecutive TSVDs. Between each TSVD a tensor
perturbation will be performed, by multiplying the new node obtained from the
SVD by a $\Delta$ matrix, and the tensor to be decomposed in the next step by
$\Delta^{-1}$. This achieves that the global tensor is the same, but the nodes
of the representation are different. Finally, the first node of the
representation is randomized. In this way, parts of the dataset can be stored
in a distributed way while maintaining the security of the data, since no one
will be able to obtain any information from the data until all the nodes of
the representation are available. At the same time, it presents a way to
perform distributed computing for big data.

###  4.7 Classification

As we have seen in the case of medical image classification, using methods
such as the one presented, different types of images or general data can be
classified. In addition, there are more general methods. On the one hand, the
inspiration of quantum circuits in tensor networks methods to make
classification circuits [61]. In this case, hierarchical trees are addressed.

On the other hand, we have the use of PEPS for image classification, where
results similar to those of GoogLeNet, VGG-16 and AlexNet [62] are achieved by
means of simple coupled tensor network systems.

###  4.8 Artificial Intelligence

As we saw in Ssec. 3.2, various models dedicated to artificial intelligence
can be compressed using tensor network representations. Combining that with
what was shown in Ssec. 3.3, we can deal with much larger and more complex
problems. In addition, we can add various quantum machine learning techniques
[4] to design new models with the features we need.

On the other hand, creating more compact models that can be inserted and
trained directly on more modest devices, such as mobile devices, can allow
experimentation with more models simultaneously and new applications.

We can also realize privacy-preserving machine learning techniques using
tensor networks [63] techniques. This is of vital importance when working with
sensitive data, such as medical records.

###  4.9 Cybersecurity

Cybersecurity is a very complex field, in which extraordinary measures are
needed to protect users’ privacy and detect possible attacks [64]. Here we can
encompass the techniques already seen above applied to this context.

###  4.10 Anomaly detection

Anomaly detection is a problem consisting of identifying unusual cases in a
data set. This problem is very useful in detecting computer attacks. Because
anomalous data can present many structures, having a large space of
possibilities, it is required to model in a very effective way the normal data
so that they can be identified with respect to the anomalous ones. This
usually requires analyzing high dimensional data or applying very complex
operations to it.

The work [39] discuss a supervised method precisely to be able to deal with
this problem. By applying an ansatz product like the one we explained in Ssec.
3.3, and an operation in variational MPO that we will train, we can project
the normal data to a hypersphere. In this way, the normal points will remain
on the surface of the sphere, while the anomalous ones will go to the center
or to the outside of the sphere. Moreover, the process can be performed very
efficiently due to the structure of the tensor network to be contracted,
scaling remarkably well with the size of the problem.

It has been tested with different datasets against different algorithms,
giving good results. Its main limitation is that it is a supervised algorithm,
in which we require a well-performed labeling prior for it to work properly.
Even so, it can be extended to be part of an unsupervised algorithm by adding
automatic pre-labeling.

##  5 Conclusions

We have seen that tensor networks are a set of techniques with great
applicability to the industrial context in an immediate way, with proven
advantages and an extensive literature on the subject. We have explained the
main features that make them especially attractive for high dimensionality
problems and machine learning. We have also highlighted several applied cases
and some of the techniques that have been used in the state of the art to
address them, as well as their advantages and disadvantages.

## Acknowledgement

The research leading to this paper has received funding from the Q4Real
project (Quantum Computing for Real Industries), HAZITEK 2022, no.
ZE-2022/00033.

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