# Neural-Network Quantum States: A Systematic Review

David R. Vivas  Centre for Bioinformatics and Photonics (CIBioFi), Universidad
del Valle, Edificio E20 No. 1069, 760032 Cali, Colombia  Departamento de
Física, Universidad del Valle, 760032 Cali, Colombia  Javier Madroñero  Centre
for Bioinformatics and Photonics (CIBioFi), Universidad del Valle, Edificio
E20 No. 1069, 760032 Cali, Colombia  Departamento de Física, Universidad del
Valle, 760032 Cali, Colombia  Víctor Bucheli  Escuela de Ingeniería de
Sistemas y Computación, Universidad del Valle, 760032 Cali, Colombia  Luis O.
González  Departamento de Ingeniería, Universidad Nacional de Colombia, Sede
Palmira, Colombia  John H. Reina  Centre for Bioinformatics and Photonics
(CIBioFi), Universidad del Valle, Edificio E20 No. 1069, 760032 Cali, Colombia
Departamento de Física, Universidad del Valle, 760032 Cali, Colombia

###### Abstract

The so-called contemporary AI revolution has reached every corner of the
social, human and natural sciences—physics included. In the context of quantum
many-body physics, its intersection with machine learning has configured a
high-impact interdisciplinary field of study; with the arise of recent seminal
contributions that have derived in a large number of publications. One
particular research line of such field of study is the so-called Neural-
Network Quantum States, a powerful variational computational methodology for
the solution of quantum many-body systems that has proven to compete with
well-established, traditional formalisms. Here, a systematic review of
literature regarding Neural-Network Quantum States is presented.

##  Introduction

The exponential growth on the dimensionality of a quantum system in regard to
its number of bodies impedes its classical, direct computational solution for
anything more than small systems: in order to solve a 300-quantum bit (qubit)
system with non-trivial interactions via diagonalization of its Hamiltonian it
is required to solve a system of $2^{300}$ linear equations, a number larger
than the estimated number of atoms in the universe[1]. Due to this curse of
dimensionality, through the history of computational physics several
computational methodologies such as the Variational ansatz Montecarlo[2]
(VMC), the Density-Matrix Renormalization Group (DMRG) [3, 4, 5] or the Path-
Integral Montecarlo formalism[6] have been developed to approach this problem.
Nevertheless, the aforementioned methods are not exempt of limitations such as
the lack of accurate ansatzs, the sign-problem or the specialization to one-
dimensional configurations.

In 2017, Carleo & Troyer[7] developed a novel methodology based on the use of
artificial neural networks (NNs) for the simulation of quantum many-body
systems. This novel approach is based on the Variational Montecarlo technique,
but uses as ansatz an adjustable parameter blackbox represented by a neural-
network (Restricted Boltzmann Machine, RBM), thus eliminating the need of
fine-tuning a hypothesis function modeled in terms of the properties of the
ground-state, which are generally unknown in the lack of an analytical
approximation or solution.

This approach, regarded as Neural-Network Quantum States (NQS) in the
literature, has achieved state of the art results both in approximating the
ground-state of large chains and lattices of quantum spins [8, 9] with an
amount of variational parameters significantly lower than the used by methods
such as DMRG [7, 9] and with the capability of executing unitary time
evolution [7, 10, 11, 12] and of reconstructing physical wavefunctions parting
from experimental data [13].

Several recent reviews[14, 15, 16, 17, 18, 19, 20] encompass neural-network
quantum states as a discussion topic: In [14] several machine learning
algorithms and their quantum physics applications are extensively discussed.
In [15] a review of NQS representational power, relation to tensor networks
and applications to condensed matter and quantum computing is presented. In
[16] the authors describe the grand landscape of Machine Learning applications
to several areas of the physical sciences. In [17], the authors review and
discuss the impact of Restricted-Boltzmann machines in quantum physics, with
special emphasis on NQS and condensed matter. In [18], the author analyzes the
grand landscape of the intersection between machine learning and quantum
matter, with a section devoted to discussion on NQS. In [19] the authors
review NQS from a technical-focused point of view. Lastly, in [20] the authors
provide a practical guide for supervised phase transition learning and
unsupervised quantum tomography and ground-state approximation by using
different neural-network architectures. Regarding such works, this review
focuses on providing the reader with a broad landscape of both the
computational problem and the vast existing literature related to NQS since
its development in 2017[7].

Although a relatively novel methodology, the rapid growth of published works
related to NQS research raises several questions: Which are the advancements
and what is the current state of this methodology? What are the main
challenges and perspectives in this line of research? In this work, we address
such questions by means of a systematic review of literature that provides
with a state-of-the art on Neural-Network Quantum States.

This article is structured as follows:

  * •

Section 1 contextualizes the reader on the computational problem, the method
of interest and the open research questions up to date.

  * •

Section 2 describes the methodology executed for the literature search and
presents the network of researchers obtained from this collection.

  * •

Section 3 describes and discusses the identified literature.

  * •

Section 4 presents the conclusions and perspectives reached during this
review.

##  Methodology and systematic search of literature

The search and compilation of the relevant literature was achieved via
specialized search equations applied both in Scopus and Google Scholar. The
used search equations are listed in table 1. An author map (see figure 1) was
generated from the compiled bibliography using VOSViewer.

Figure 1: Author map generated from the compiled bibliography. A cluster
Between Carleo, Carrasquilla, Torlai, Cirac and Melko is appreciated,
indicating collaboration between the Universities of Waterloo, Toronto,
Complutense de Madrid and the Flatiron institute of New York. Mapping was
limited to the biggest connected author cluster for visualization purposes.

Initially, the generic search equations ("machine learning" AND "quantum
physics") and ("Neural-Network Quantum States" OR "Neural network quantum
states" OR "Neural quantum states") were considered as a naive starting point.
By parting from this search equation, taking [7] as a reference publication
and iteratively refining the obtained results via keyword mapping, the
remaining search equations (table 1: 3-11) were subsequently derived. The
study of the collected literature was achieved by means of an annotated
literature scheme.

We remark the particular importance of Google Scholar in this work, given that
this platform provides references both in indexed journals and in pre-print
platforms such as the arXiv; with the latter being of key importance in the
finding of recent literature on novel, rapidly-growing lines of research such
as the one subject of this work.

# |  Search Equation
---|---
1 |  "Machine Learning" AND "Quantum Physics"
2 |  "Neural-Network Quantum States" OR "Neural network quantum states" OR "Neural quantum states"
3 |  (("Neural Networks" AND ("Autoregressive" OR "Recurrent")) AND ("wavefunctions" OR "wave functions" OR "quantum many body" OR "condensed matter OR "quantum phase transition" OR "quantum dynamics" OR "spin")
4 |  ("Machine Learning" OR "Deep Learning" "Neural Networks" OR "Restricted Boltzmann Machine") AND ("wavefunctions" OR "wave functions" OR "quantum many body" OR "condensed matter" OR "quantum phase transition" OR "quantum dynamics" OR "spin")
5 |  ("Machine Learning" OR "Deep Learning" "Neural Networks" OR "Restricted Boltzmann Machine") AND (("spin" OR "fermion" OR "boson") AND ("chain" OR "lattice"))
6 |  ("Machine Learning" OR "Deep Learning" "Neural Networks" OR "Restricted Boltzmann Machine") AND ("Ising" OR "Heisenberg" OR "Bose" OR "Hubbard")
7 |  ("Machine Learning" OR "Deep Learning" "Neural Networks" OR "Restricted Boltzmann Machine") AND (("Ising" OR "Heisenberg" OR "Bose" OR "Hubbard") AND (Localization OR Delocalization))
8 |  ("Machine Learning" OR "Deep Learning" "Neural Networks" OR "Restricted Boltzmann Machine") AND ( ("quantum information" OR "quantum state tomography") OR ("spin" AND ("Chain" or "Lattice")) )
9 |  ("Machine Learning" OR "Deep Learning" "Neural Networks" OR "Restricted Boltzmann Machine") AND ("spin" AND ("chain" OR "lattice") AND "dynamics")
10 |  ("Machine Learning" OR "Deep Learning" "Neural Networks" OR "Restricted Boltzmann Machine") AND (("Ising" OR "Heisenberg" OR "Bose" OR "Hubbard") AND "dynamics")
11 |  ("Machine Learning" OR "Deep Learning" "Neural Networks" OR "Restricted Boltzmann Machine") AND ("Variational Montecarlo" OR "Variational Monte Carlo")
Table 1: Identified search equations for literature finding.

##  State of the art

Since 2017, several lines of research focused on the study of the intersection
between machine learning and quantum mechanics have surfaced in the form of
several scientific publications that report significant advancements in
computational physics. These contributions have enriched the current landscape
of computational physics by taking advantage of both modern computational
power and the rich theoretical and empirical advancements that have been
developed during the past two decades and culminated in the so-called
contemporary AI revolution.

Of the aforementioned contributions, the work by G. Carleo and M. Troyer [7]
can be regarded as one of the first modern publications on NQS. In this
pioneering work, that coined the term ‘Neural-Network Quantum State’, a
paradigmatic generative model termed Restricted Boltzmann Machine (RBM) is
used as a generic neural-network variational wavefunction ansatz (neural-
network quantum state) in order to approximate the ground-state wavefunction
of a quantum many-body system. The computational complexity of this RBM-based
method scales quadratically with respect to the number of bodies, carrying
this a significantly lower computational cost than a direct diagonalization
scheme for the case of large many-body systems. Restricted Boltzmann Machines
were initially conceived under the name Harmonium by P. Smolensky as part of
his Harmony Theory of cognitive science[21]. They were later popularized by G.
Hinton, who conceived an efficient algorithm for their training[22]. RBMs
consist of an input, visible unit layer and a latent, hidden unit layer, and
previous to their application in quantum physics they were widely applied and
studied for their dimensionality-reduction capabilities, and were usually
trained by direct or indirect minimization of the Kullback-Leibler (KL)
divergence between the probability distribution of the RBM and the one
associated to training data. RBMs are physically equivalent to a system of two
fully-connected layers of classical spins described by the canonical
ensemble[23, 24].

The NQS approximation to the ground state wavefunction is reached via an
iterative optimization scheme of the energy functional based on a complex-
valued generalization of the natural gradient method, known in the literature
as Stochastic Reconfiguration (SR)[25]. The SR method accelerates the
convergence of variational Montecarlo methods compared to first order
optimization algorithms such as stochastic gradient descent. Stochastic
reconfiguration is, effectively, an imaginary time evolution over the
variational subspace of the NQS wavefunction. Each step in the optimization
process involves sampling from the RBM via Markov-Chain Metropolis Montecarlo
(MCMMC), where the obtained set of samples effectively acts as a step-wise
training set. For large enough systems, sampling cannot be performed in a
exact way due to the intractable nature of the RBM partition function, the
computation of which can be circumvented via the construction of the samples
using the Metropolis algorithm. Recently, the linear method has also been
considered as a feasible approach for optimizing NQS[26].

Following the sampling process, an stochastic estimate of the energy
functional gradients are computed from the samples. The final step of each
training epoch involves the solution of a linear equation system, as required
by SR. The dimensionality of such system grows linearly with the number of
variational parameters of the RBM, and its solution can be obtained via direct
inversion (or pseudoinversion) of the SR Matrix or via iterative Krylov
subspace methods such as conjugate gradient, with the latter being already
successfully connected with SR in [27]. The process described above is
repeated until reaching a specified convergence threshold. The work by Carleo
and Troyer also demonstrates the capability of executing unitary time
evolution over the approximated ground-state, thus supporting time-dependent
variational Montecarlo. Although SR offers faster convergence, Adam optimizer
[28] has also proven capable of optimizing NQS [29, 8]. The described scheme
is illustrated in figure 2. In reference [7] it is shown that, after training,
each neuron of the RBM acts as an effective filter over a region of the spin
chain or lattice, in a similar fashion to convolutional neural networks; a
reproduction of this result is illustrated in figure 3.

Figure 2: General training scheme for the ground-state approximation of a NQS.

Another high-impact result, published in parallel to the aforementioned,
introduces a Quantum State Tomography (QST) protocol base on neural-network
quantum states[13]. This novel approach, regarded as Neural-Network Quantum
State Tomography (NQST), uses a neural-network to reconstruct the wavefunction
of a quantum system given a set of configuration measurements or samples. The
involved computational methodology minimizes the KL divergence between the
probability distribution associated to the samples and the probability
distribution defined by the squared norm of the NQS wavefunction, with the
added difficulty that such wavefunction is a complex-valued probability
amplitude. The reported optimization scheme uses a complex-valued RBM trained
via ordinary stochastic gradient descent for its radial part and natural
gradient descent for its complex-phase. The reconstructed wavefunction was
also proven capable of executing unitary dynamics. Following this work,
subsequent contributions have improved the precision of the method [30, 31];
extended it to the density-matrix formalism[32]; used recurrent neural
networks as an alternative NQST architecture[33, 34, 35]; extended the
protocol to local-measurement-based QST[36]; and even proved the capabilities
of powerful generative models such as variational auto-encoders[37] to
reconstruct complex many-body wavefunctions[38, 39, 40].

Figure 3: Evolution of the norm of the weights connecting the visible layer
with an arbitrary hidden neuron for a 1D transverse field Ising model ground-
state wavefunction described by an RBM NQS. At the end of the training,
weights connecting the chain with individual hidden neurons effectively act as
local filters over different regions of the chain, reflecting the short-range
interactions of this particular Hamiltonian. Given that fully-connected NNs
such as RBMs are not aware of the relative position of its inputs (in
comparison to CNNs), this result shows that the model naturally learns the
relative position of the spin configurations during training. The results are
reproduced following [7].

Despite sampling methods based on the Metropolis algorithm allow the
construction of samples from otherwise intractable probability distributions,
these methods are prone to exhibit both high correlation between samples, and
also the inability to provide adequate samples when approaching energy
barriers, such as the ones usually present in certain regimes of quantum
matter.

Neural Autoregressive Density Estimators (NADEs)[41], popularized by Google
Deepmind[42, 43], are generative neural-networks that define a probability
distribution from which sampling can be performed exactly. Given that the
connections between layers of a NADE are causal, sampling can be performed via
the chain rule of conditional probabilities, thus eliminating the need of
using construction methods such as MCMMC. A recent contribution [8] has
extended the probabilistic framework that supports NADEs to the notion of a
complex-valued probability amplitude, from which a physical ground-state
wavefunction can be obtained. This approach completely removes the need of
using MCMMC for NQS representation and provides uncorrelated samples with
higher computational efficiency: in figure 4 we illustrate via principal
component analysis how uncorrelated autoregressive samples naturally fall into
a representative region of the configuration space, while the MCMMC samples
are rather spread along a correlated path. Another recent contribution by
Hibat et al.[9] reports the use of a recurrent NQS architecture, which also
supports exact sampling via the autoregressive property, to solve prototypical
systems of up to 1000 spins with a remarkably low amount of variational
parameters and with the capability of estimating Renyi Entropies[44].

Further works related to NQS ground-state research include a construction
method for the classical representation of quantum states via Deep Boltzmann
Machines[45], an error[46], learnability scaling[47] and representability[19]
analysis of NQS under different tensor operations, an expressive power
analysis on different spins, Hamiltonians and architectures [48], a study of
the expressive power of deep NQS [49] compared to variational tensor networks,
an application to the ground-state simulation of few-body bosonic systems[50],
an analysis of the limitations of shallow NQS architectures[51] and RBMS[52]
as universal ground-state approximators, a detailed analysis of the
transverse-field Ising model[53], a random-sampling approach to the solution
of the eigenvalue problem[54], a pair-product approach to compliment the
expressive power of RBM-based NQS[55], an study of the connection between RBMs
and tensor-network from a representability perspective[56], a deep study of
the entanglement representation capabilities of convolutional neural-
networks[57], the ground state approximation[58, 59], phase boundary
exploration[60] of paradigmatic bosonic systems such as the Bose Hubbard
Model, the use of group-equivariant convolutional neural-networks to optimize
NQS ground-state approximation[61], the construction of gauge-invariant
autoregressive NQS[62], the parametrization of electronic wavefunctions by
means of autoregressive NQS[63], a super-resolution approach to extrapolate
spin configurations of the transverse-field Ising model to bigger
lattices[64], the iterative retraining of Recurrent NQS to progressively scale
to larger systems[65], the study of disordered quantum Ising chains via dense
and sparse RBMs[66], the approximation of the ground-state of the frustrated
$J1-J2$ Heisenberg model by means of a dual NQS approach[67], the smmulation
of spin liquids on a Honey lattice via RBMs[68] and, lastly, the use of graph
neural-networks to learn a parametrized structured variational manifold NQS
that achieves state-of-the-art results on quantum many-body benchmarks[69].

Figure 4: Principal component analysis of samples obtained both by
autoregressive sampling and a Hamiltonian MCMC chain for the ground-state of a
100-long transverse-field Ising chain in the disordered phase, represented by
an autoregressive transformer. The uncorrelated samples of the autoregressive
sampler fall inside what can be regarded as a representative region of the
ground state of the system, while the Hamiltonian MCMC samples are confined to
a less-diverse chain on a distant starting region of the configuration space.

The NQS unitary dynamics algorithm introduced in [7] is, computationally
speaking, a problem similar to SR, but demands a higher number of samples and
limits the possibility of regularizing the involved linear equation system,
implying this a higher computational cost and the surfacing of new challenging
numerical instabilities. The dynamics simulation capabilities of NQS have been
explored and extended in several subsequent works. In [11], dynamics over up
to $8\times 8$ transverse-field Ising lattices are explored and compared to
iPEPS[5] results, with the authors reporting the numerous computational
techniques and workarounds used in order to ensure the stability and accuracy
of the time evolution. Another contribution [12] proposes the use of an
alternative algorithm based on the solution of the time-dependent Schrodinger
equation with an NQS wavefunction via the midpoint-method; this novel approach
involves the training of a new NQS at each time step $t+1$ by minimizing a
cost function with respect to the $t$ time step wavefunction over a set of
samples, thus allowing the possibility of changing and adapting the NQS
architecture on the fly, with this being specially useful in cases where
certain physical regimes reached during the time evolution require a higher
expressive power NQS. In [70] it is shown that a simple recurrent neural-
network is able to predict the time evolution of a one-dimensional transverse-
field Ising model, and also scale to larger systems after training. In [10]
NQS unitary dynamics simulation over strongly-interacting Bose gases in
continuous space is executed. Reference[71] introduces a novel quantum
algorithm based on the use of a modified RBM for the variational simulation of
closed and open many-body dynamics, while circumventing the vanishing gradient
problem. In [72], a method for the many-body dynamical simulation on the
frequency domain is devised. NQS have also proven capable of simulating the
dynamics of open quantum systems, with [73] introducing a first approach for
small systems and [74] further improving the previous results with the use of
autoregressive NQS. In [75], a quantum renormalization group approach is
combined with neural-networks to compute the long-time coherent dynamics of
large many-body localized systems. In [76], stationary states of open quantum
many-body systems have been approximated using NQS and several physical
observables have been computed. Finally, in [77] a simulation of up to 40
spins in a dissipative Heisenberg lattice in one and two dimensions has been
reported.

NQS have also been successfully applied to the analysis, exploration and
discovery of physical phases; including the finding of quantum critical
points[78], phase-diagram reconstruction of the Bose-Hubbard model[79],
approximation of states on chiral topological phases[80, 81], the study of
entanglement transitions by means of a RBM[82] and representation capability
analysis near quantum criticality[83, 84]. Although state configurations (i.e.
spin configurations) can be regarded as relatively simple computational data,
they are a reflection of highly-complex phenomena that takes place inside the
Hilbert space of a quantum many-body system. Being fed by such configurations,
fedforward neural-networks have also proven capable of identifying phases of
matter, phases transitions and order parameters from raw-spin configurations
on a wide variety of Hamiltonians following a supervised learning approach[85,
86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101]. Alternative
machine learning algorithms have also proven highly capable of handling
physical phases of matter[102, 103, 104, 105, 106, 107, 108, 109, 110, 111,
112, 98].

##  Final Remarks and Perspectives

Although recent contributions have reported the optimization of NQS up to
$10^{3}$ bodies[9], successful NQS dynamics are still in the order of $10^{1}$
bodies. A main perspective is the development of novel methodologies and/or
computational techniques that make possible the study of the dynamics of
larger systems. Current advances in NN architectures such as self-attention
mechanisms[113], Hamiltonian NNs and graph NNs open interesting perspectives
for the further development of NQS architecture research.

Neural-networks have also proven capable of approximating ground-states of
arbitrary sparse Hamiltonians in continuous spaces[114]. Current state of this
approach provides the advantages of a universal ansatz but is, however, still
penalized by the quantum curse of dimensionality. A more efficient approach is
yet to be devised.

With the current development of hardware specifically designed for deep
learning purposes, a new perspective for the representation of NQS on such
specialized hardware arises. This proposal was already analyzed in [115, 116],
which identifies both advantages and difficulties. A practical implementation
remains to be executed.

Despite not being capable of providing explicit probability distributions as
needed for NQS ground-state approximation and NQST, Generative Adversarial
Networks (GANs) have found applications in many-body physics, by introducing
an approach for modelling physical phases of quantum spin systems with these
kind of models[117]. A wider spectrum of applications for these powerful
generative models remains to be seen.

As illustrated in this work, the diverse applications of Neural-network
Quantum States in many-body quantum physics have configured a rapidly-growing,
high-impact field of research that has vastly expanded the current landscape
of computational physics, condensed matter, and quantum information science.
The recent birth of this formalism together with its rapid advancements opens
promising perspectives both for new computational techniques to come and the
maturity of the already present ones.

##  Acknowledgments

This work was supported by Vicerrectoría de Investigaciones at Universidad del
Valle (Grant CI 71212).

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