# Tensor networks for High Energy Physics: contribution to Snowmass 2021

Yannick Meurice1 James C. Osborn2 Ryo Sakai3 Judah Unmuth-Yockey4 Simon
Catterall3 Rolando D. Somma5 1 Department of Physics and Astronomy, The
University of Iowa, Iowa City, IA 52242 USA  2 Computational Science Division,
Argonne National Laboratory, Argonne, IL 60439  3 Department of Physics,
Syracuse University, Syracuse, NY 13244 USA  4 Fermilab, Batavia, IL 60510  5
Los Alamos National Laboratory, Los Alamos, NM 87545

(August 27, 2024)

###### Abstract

Tensor network methods are becoming increasingly important for high-energy
physics, condensed matter physics and quantum information science (QIS). We
discuss the impact of tensor network methods on lattice field theory, quantum
gravity and QIS in the context of High Energy Physics (HEP). These tools will
target calculations for strongly interacting systems that are made difficult
by sign problems when conventional Monte Carlo and other importance sampling
methods are used. Further development of methods and software will be needed
to make a significant impact in HEP. We discuss the roadmap to perform quantum
chromodynamics (QCD) related calculations in the coming years. The research is
labor intensive and requires state of the art computational science and
computer science input for its development and validation. We briefly discuss
the overlap with other science domains and industry.

###### Contents

  1. I Introduction
  2. II Science goals for HEP
  3. III Relation with Quantum Computing and QIS
  4. IV Selected recent progress
  5. V Technologies and resources needed
  6. VI Overlap with other science domains and industry
  7. VII Conclusions
  8. A Brief background on tensor methods

Executive summary for “CompF6: Quantum computing”

Tensor network methods are playing an increasingly important role in several
branches of physics and in quantum information science. Tensor networks
provide a compact representation of entangled quantum systems and quantum
states. They can be used to reformulate physical theories on a lattice and map
them onto other quantum systems—such as quantum computers or quantum
simulators. These physical theories can then either be studied classically
using the original tensor network formulation, or in the mapped representation
using quantum hardware.

In the original formulation, they present an increasingly attractive
alternative to traditional Monte Carlo methods (used in lattice QCD) which are
currently the only computational methods able to provide first-principle
calculations for a large range of HEP problems, but which also consume large
computational resources. In numerical methods based on coarse-graining, tensor
network methods do not rely on statistical sampling, potentially providing
solutions to problems plagued by _sign problems_ that render Monte Carlo
methods ineffective. In some fields, such as condensed matter physics, their
use is fairly common. However, for problems of interest to HEP, tensor network
methods are still very resource intensive, and are not currently feasible for
complex theories in four space-time dimensions, such as QCD. Further
development of tensor network methods and software is needed for them to make
a significant impact in HEP like Monte Carlo.

In addition to providing an alternative to Monte Carlo methods in HEP, tensor
networks are also useful in connecting HEP to QIS. They can provide a
convenient representation for mapping lattice theories to quantum hardware
(either in terms of digital quantum circuits or analog quantum simulators),
and can also be used to perform classical simulations of quantum circuits.
This is helpful for developing and testing quantum computing algorithms, and
is an important area of research for determining the tipping point for quantum
computers to provide a computational advantage over classical ones. They also
assist in isolating “building blocks” that have limited complexity and can be
optimally approximated for noisy intermediate-scale quantum (NISQ) hardware.

Here we summarize some of the key problems in HEP that could be addressed with
tensor network methods, the current status of applications to HEP, and the
developments in methods, software, and personnel needed to make progress in
these areas.

Executive summary for “TheoryF10: Quantum Information Science”

Tensor network methods are playing an increasingly important role in several
branches of physics and in quantum information science. In the context of HEP,
tensor networks have appeared as ways to reformulate lattice gauge theory
models to obtain a fully discrete formulation that is suitable for quantum
computation and coarse graining methods. Tensor networks also provide tools to
understand entanglement in conformal field theories and their connection to
gravity.

In the context of lattice gauge theory, tensors can be seen as the
translationally invariant, local building blocks of exact discretizations of
the path integral. They encode both the local and global symmetries of the
original model. It is easy to design approximations (truncations) that
preserve these symmetries and to design simplified models that should have the
same correct universal continuum limit as the original model. Developing these
building blocks and optimizing the approximations for NISQ machines and
classical computers are important tasks for the near-term future. Tensor
networks can also be used to perform classical simulations of quantum
circuits. This is useful for developing and testing quantum computing
algorithms and quantum computational advantage.

There is an emerging international and interdisciplinary community developing
new methods in this area, which includes an increasing number of researchers
from the lattice gauge theory community. There is a clear road map to do QCD
related calculations in the coming years. This effort is at the interface of
quantum computing and classical high performance computing (HPC) and requires
state of the art HPC for its development and validation.

Executive summary for “TheoryF5: Lattice Gauge Theory”

Tensor Lattice Field Theory (TLFT) is a new approach for understanding the
non-perturbative structure of lattice theories including QCD. Starting from
the conventional Lagrangian/path-integral formulation, it is possible to
reformulate the models using tensors which are translationally invariant,
local, and can be considered as building blocks of exact discretizations of
the path integral. They encode the local and global symmetries of the original
model. They provide a more general space of theories than the traditional
Lagrangian and Hamiltonian densities. Most models studied by lattice gauge
theorists can be reformulated in the language of TLFT. Truncations need to be
used for practical purposes but can chosen to preserve symmetries in a generic
way. This provides a broad class of models with the same universal continuum
limit as the original model.

TLFT smoothly connects the Lagrangian and Hamiltonian approaches and is very
useful in the context of quantum computing. Moreover, TLFT approaches to the
path integral can also be combined with deterministic coarse-graining methods.
This is very efficient for a broad range of couplings and also allows the
study of complex actions. The truncations performed during coarse-graining are
the only approximations used and, if they can be controlled, exponentially
large volumes can be reached. This is potentially interesting in the study of
models beyond the standard model with slowly running coupling constants.

Using TLFT for models directly relevant for HEP is a feasible goal in the next
decade. Developing software and running codes in TLFT is labor intensive and
requires access to high performance computing.

##  I Introduction

The use of tensor network methods to tackle quantum field theory problems
relevant for HEP is a recent and rapidly developing new area of research. This
is a very interdisciplinary area and workshops involving several communities
have taken place recently. For instance, a recent workshop at the Institute
for Nuclear Theory Catterall  _et al._ had more than 80 participants in High
Energy, Nuclear and Condensed Matter Physics. Before discussing the science
goals, the importance of the approach for quantum computation, recent
progress, and resources needed, we first give a brief overview of the context
in which tensor network methods appear, and their relations with other areas
of research.

An important disclaimer: this white paper does not provide exhaustive lists of
references but rather selected references that are most familiar to tensor
practitioners in the US lattice community. Exhaustive lists of references can
be found in recent reviews Orus (2019); Bañuls  _et al._ (2020); Bañuls and
Cichy (2020); Meurice  _et al._ (2020a); Cirac  _et al._ (2021). Additional
information can also be found in some of the 2020 LOI’s Meurice  _et al._
(2020b); Unmuth-Yockey (2020); Butt  _et al._ (2020a).

  * •

Context
Tensor network methods are playing an increasingly important role in many
branches of physics and data analysis. In the context of HEP and more
specifically lattice gauge theory, tensor network methods appear at the
interface of quantum and classical computing and target questions where sign
problems are present or large volumes are needed. They can also play a role in
understanding conformal and holographic theories. For recent reviews focused
on the use of tensors in lattice gauge theory see Bañuls  _et al._ (2018);
Bañuls and Cichy (2020); Meurice  _et al._ (2020a).

  * •

Terminology
The term “tensor” appears in many other contexts. For instance, energy-
momentum tensors or Kalb-Ramond tensor fields are often used in HEP. In
contrast, “tensor networks” refers to objects with multiple indices that can
be assembled together to represent the states of a Hilbert space or transfer
matrix, compute expectation values of operators, or express partition
functions. Typically, the indices refer to quantum numbers or to a list of
states in a localized Hilbert space.

  * •

Relationship with quantum computing
The reformulations of lattice field theory models using finite sets of tensor
indices provide Hilbert spaces suitable for quantum computing (see Section
III). The space-time tensor networks can be adapted to build quantum circuits.
Additionally, certain quantum states that can be represented using a tensor
network, e.g. matrix product states (MPS), can be prepared efficiently on a
quantum computer. This allows one to address more general problems, e.g.,
quantum simulation, which are still inefficient for classical computers.

  * •

Condensed matter origins
Many of the basic quantum information ideas behind the use of tensor networks
in lattice field theory were originally developed in the context of condensed
matter. White emphasized the need to keep track of the entanglement among
coarse-grained blocks while performing real space renormalization group
transformations and invented the density matrix renormalization group (DMRG)
method to handle this White (1992). DMRG is typically the most accurate method
for computing ground states of one-dimensional lattice Hamiltonians and is
related to tensor networks called matrix product states (MPS). Other tensors
networks, for instance, called projected entangled pair states (PEPS) and the
multi-scale entanglement renormalization ansatz (MERA), play an important role
in dealing with other types of models. The quantum information theory
community has developed a clear understanding of the convergence (or the lack
thereof) of such algorithms in terms of area laws associated with entanglement
entropy. For recent reviews see Haegeman and Verstraete (2017); Ran  _et al._
(2020).

  * •

Tensors for HEP and nuclear physics
There has been a significant interdisciplinary effort starting around 2010,
and highlighted in conferences and workshops at the Aspen Center for Physics,
KITP, the INT and KITPC, to apply some of methods developed in the condensed
matter community to problems in HEP and nuclear physics (NP). The annual
lattice conferences have helped foster interactions among the communities
involved Bañuls  _et al._ (2014); Meurice  _et al._ (2014a); Van Acoleyen  _et
al._ (2014); Shimizu and Kuramashi (2014a); Saito  _et al._ (2014); Unmuth-
Yockey  _et al._ (2014); Meurice  _et al._ (2014b). The number of
contributions has grown steadily with the years. Tensor reformulations of
lattice gauge theory models starting from the standard Lagrangian path-
integral formalism connect smoothly to Hamiltonian approaches developed in
condensed matter physics.

  * •

Tensors as building blocks of computations
The tensors are local objects containing all the universal information about
the model in question such as its dimension and symmetries. Most lattice
models have a tensor reformulation and there is a clear plan to follow the
road map that has been successful for QCD (the “Kogut sequence”, see below).
Tensors can be used to deal with sign problems, strongly interacting systems
and conformal field theory (but efficient numerical methods remain to be fully
developed in 2+1 and 3+1 dimensions). They provide new tools for the lattice
field theory and AdS/CFT communities. Quantum circuits can also be viewed as
tensor networks, providing alternative methods for constructing, transforming
and evaluating quantum computations Markov and Shi (2008); Vidal (2003);
Biamonte and Bergholm (2017).

  * •

The road map
The lattice gauge theory community worked its way over many years to being
able to simulate full QCD with dynamical fermions by starting with simpler
models in lower dimensions. This approximate sequence of models is sometimes
called the “Kogut sequence” or the “Kogut ladder” after review articles Kogut
(1979, 1983). A similar road map is being followed with the tensor
reformulations discussed here Meurice  _et al._ (2020a).

##  II Science goals for HEP

  * •

Quantitatively reliable coarse-graining for strong interactions
Quarks and gluons play an important role for problems with typical energies
ranging from MeV to TeV. Hypothetical models beyond the standard model often
involve effective coupling constants that run very slowly with energy scale.
To tackle such problems it is crucial to develop reliable methods for
performing coarse-graining and renormalization. Tensor renormalization group
(TRG) methods have the potential to meet these expectations because they allow
a clean partition of the degrees of freedom at different scales. Indeed such
methods have been used in simple models to obtain critical exponents to very
high precision. However, the truncation methods that must be employed still
need optimization for use in higher dimensions with both fermions and gauge
fields.

  * •

Dealing with sign problems at finite density
A certain number of problems in astrophysics and nuclear physics require
calculations involving quarks and gluons at finite density. These calculations
are plagued by sign problems which prevent the use of conventional importance
sampling. TRG methods rely on the singular value decomposition (SVD) and are
typically insensitive to sign problems. In 1+1 dimensions, they are
generically very efficient in dealing with problems involving finite density
or complex couplings where the Euclidean Boltzmann weights are not real and
positive. Developing efficient methods in higher dimensions is an important
goal for the tensor community.

  * •

Real-time evolution of strongly interacting particles
The interpretation of hadron collider data relies extensively on event
generation algorithms such as Pythia Sjöstrand  _et al._ (2015). These
algorithms incorporate results from perturbative QCD that are reliable at
short distances and use empirical models to describe the formation of hadrons
at larger distances. Replacing these empirical models by ab-initio
calculations based on lattice QCD is one of the major motivations for quantum
computing in HEP. This is discussed in a separate white paper Bauer  _et al._
. Tensor methods can also in principle be used to handle real-time evolution
and out-of equilibrium situations. This is a well-developed research area in
condensed matter and we hope that this can also become the case in HEP.

##  III Relation with Quantum Computing and QIS

  * •

Discretization of path-integrals
For qubit-based quantum computing a complete discretization of both field
space and spacetime is necessary. Tensor network methods starting from a
lattice path-integral formulation provide a general way to reach this goal.

  * •

Character expansions
For compact groups, character expansions (e.g., Fourier expansion) provide a
natural way to discretize the path integral of models with continuous
variables (e.g., the gluon fields in QCD). They were extensively developed in
the context of strong coupling expansions in the early days of lattice QCD but
they also connect with modern worm algorithms and can be adapted to work well
even at weak coupling.

  * •

Symmetry compatible truncations
For practical purposes, truncations of the character expansions are necessary.
Fortunately, this can be done in a way compatible with symmetries Meurice  _et
al._ (2020a); Meurice (2020, 2019).

  * •

Locality and effective theories
The (quasi) locality of microscopic interactions is a crucial ingredient of
Lloyd’s quantum supremacy argument Lloyd (1996). It is present in tensor
network reformulations and their effective theories obtained by coarse
graining.

  * •

Simulation of the building blocks of quantum algorithms
Quantum algorithms can be interpreted as tensor networks, where each tensor
corresponds to a quantum gate. From this point of view, we can isolate
building blocks of the quantum algorithm and use tensor network methods to
perform the tensor contractions, providing a useful technique to simulate
quantum algorithms on classical computers.

  * •

State preparation
Tensor networks with some local structure allow one to produce state-
preparation quantum circuits of small size or depth. This is beneficial for
quantum simulation. It is also beneficial to study/verify new physical
theories and for observing convergence by changing the bond dimension of the
tensor.

  * •

Quantum state tomography
Tensor networks can also be used in the context of quantum state tomography
Cramer  _et al._ (2010). By imposing a tensor network structure (e.g., a
PEPS), there is a way perform efficient quantum state tomography, which
requires exponential resources in general. Efficient quantum state tomography
is important for many reasons, including the verification of quantum computing
devices and experiments.

  * •

Relation with other approaches
The study of the transfer matrix derived from the tensor formalism leads to
finite algebras Meurice (2019) that can be compared with quantum link
constructions Orland and Rohrlich (1990); Brower  _et al._ (1999). Related
methods Bhattacharya  _et al._ (2021); Berenstein  _et al._ (2022) have been
used in the context of quantum computing. These related formulations should be
compared in the continuum limit. For instance, it has been showed recently
that truncations of the Abelian Higgs model in the charge representation leads
to a phase transition associated with an enhanced symmetry.

##  IV Selected recent progress

  * •

Spin and gauge models with Abelian symmetries
It was found out early Liu  _et al._ (2013) that the character expansions for
Abelian groups leads to simple factorization. The integration of the fields
result in Kronecker deltas which represent the symmetry (modulo $n$ for
$Z_{n}$ versions). The identities representing the symmetry depend only on
these selection rules and not on the specific values taken by the tensor
elements Meurice (2019, 2020). Consequently, truncations preserve symmetries
exactly and it is possible to attempt to reach the continuum limit with highly
simplified microscopic formulations. In addition, for gauge theories, one can
integrate completely the gauge fields without gauge fixing and the procedure
is manifestly gauge invariant.

  * •

Spin and gauge models with non-Abelian symmetries
The mass-gap of the O(3) nonlinear sigma model in 2D was studied in Ref.
Unmuth-Yockey  _et al._ (2015) by fitting the two-point correlation function.
In Refs. Kawauchi and Takeda (2016) the CP(1) model in 2D with and without a
$\theta$-term was analyzed using the higher order tensor renormalization
group, and the loop-tensor network renormalization algorithms. Reference
Bazavov  _et al._ (2019) considered the non-Abelian Higgs model in 2D. This
model is confining across the entire phase diagram, and the string tension was
extracted from Polyakov loop correlators. Extracting spectra and binding
energies from correlations functions is an important step on the path towards
analyzing QCD. Understanding the systematics of the method in the presence of
a $\theta$-term – which has a sign problem – for a theory which is
asymptotically free will help guide the method in higher dimensions with other
non-Abelian groups.

  * •

Scalar theories
The complex scalar $\phi^{4}$ theory with finite chemical potential, a typical
model with the sign problem, is analyzed in 2D and 4D Kadoh  _et al._ (2020);
Akiyama  _et al._ (2020). While the presence of a sign problem is a strong
motivation to use tensor network methods, a notable accuracy of the critical
coupling constant compared to other schemes including Monte Carlo simulations
is reported in the 2D real $\phi^{4}$ theory Kadoh  _et al._ (2019); Delcamp
and Tilloy (2020) following the pioneering work by Shimizu Shimizu (2012). To
discretize the field space a Taylor expansion of the hopping factors combined
with Gaussian quadratures is used.

  * •

Fermions
Application of the Grassmann tensor renormalization group Gu  _et al._ (2010);
Gu (2013) to relativistic fermion models started from a series of
investigation on the 2D Schwinger model with Wilson fermions Shimizu and
Kuramashi (2014b, c, 2018). A key feature of these fermion models is the
presence of Grassmann variables that decorate the tensor networks. In some
special cases that include the 2D Schwinger model with staggered fermions, it
is shown that one can construct a tensor network representation without
Grassmann variables Butt  _et al._ (2020b). A coarse-graining algorithm for
Grassmann variables in higher dimensions that goes well with the higher order
TRG (HOTRG) Xie  _et al._ (2012) and its approximation schemes Adachi  _et
al._ (2020); Kadoh and Nakayama (2019) was proposed in Sakai  _et al._ (2017).

  * •

Supersymmetric models
The 2D $\mathcal{N}=1$ Wess–Zumino model, the simplest supersymmetric model
with a severe sign problem, is analyzed using the Grassmann tensor
renormalization group Kadoh  _et al._ (2018). In this work, a non-interacting
case that is exactly solvable is employed as a numerical test. More
complicated models including the interacting case of the same model and the
$\mathcal{N}=(2,2)$ Wess–Zumino model would be a possible and interesting
direction.

  * •

Quantum gravity
Tensor networks can be useful in the study of quantum gravity as well. One of
the directions where tensor networks have appeared is in the “spin-foam”
formulation of quantum gravity Perez (2013). In Ref. Dittrich  _et al._
(2016), a tensor network formulation, as well as a coarse-graining scheme were
developed for a spin-foam partition function. In Ref. Asaduzzaman  _et al._
(2019) the authors form a tensor network for the partition function of two-
dimensional gravity where the gauge symmetry has been extended to merge the
tetrad and spin-connection variables into a single connection. In this work
the Fisher zeros of the partition function are plotted in the complex-coupling
plane. In a related work the authors in Ref. Bao  _et al._ (2017) model de
Sitter space using the multi-scale entanglement renormalization ansatz tensor
network.

  * •

Other approaches
As explained before, there is a large literature where MPS and PEPS are used
to handle gauge theories Bañuls  _et al._ (2020); Bañuls and Cichy (2020) that
is not reviewed here. For recents progress, see for instance Emonts  _et al._
(2020); Funcke  _et al._ (2020); Frías-Pérez  _et al._ (2021); Williamson  _et
al._ (2021).

##  V Technologies and resources needed

  * •

Method development
A large variety of tensor network algorithms have been developed to
approximately evaluate observables in Hamiltonian and Lagrangian systems (see
section A in the appendix for a summary and references). The methods span a
range of compromises between accuracy, compute time and memory use, with the
most accurate methods generally requiring the most time and memory. For
systems with one spatial dimension, the time and memory requirements of the
most accurate methods can typically be accommodated on modern computers.
However, for three spatial dimensions, the computing demands for the most
accurate methods are well beyond what is presently possible, and a significant
reduction in the size and connectivity of the tensors within the network is
necessary. Several advances have been made to improve the scalability of
methods, which has made initial simulations of some 3+1 dimensional models
possible. However, high precision simulations of simple 3+1 dimensional
systems and initial tests for more complex theories (such as QCD) still remain
a significant challenge. Sustained research into understanding the tradeoffs
among approaches, and developing new methods is necessary in order to reach
the accuracy and scalability needed for many of the problems of interest to
HEP.

  * •

Tensor network software and library requirements
Tensor network calculations rely on efficient tensor (or matrix) math
libraries. The most common operations needed are

    * –

Reshaping the tensor and permuting the order of indices

    * –

Element-wise operations (e.g. scaling elements, adding tensors)

    * –

Contracting indices in a product of two (or more) tensors

    * –

Tensor decompositions (typically SVD or eigen-decomposition)

A list of some tensor software can be found on the web Various  and a more
comprehensive list of packages is available here Psarras  _et al._ (2021).
Many of the tensor libraries were developed and are currently used by the
condensed matter community, and may have MPS or other methods implemented.
Popular programming languages are C++, Python and, more recently, Julia. For
HEP research Numpy is still a popular choice for implementations of TRG and
related algorithms. Since many of the tensor libraries use optimized GEMM
routines for contractions, performance among them for large tensor sizes will
be similar for the same set of operations. The main differences between them
tend to be in the interface (e.g. whether contractions use Einstein notation,
or some other helper functions). This can make a difference in the ease of
use, however, the order of contractions can make a big difference in the
memory and compute requirements, so interfaces that allow specifying a
collection of contractions need to either determine the optimal order or let
the user specify. Determining the optimal contraction order becomes
particularly important for larger networks but is considered to be an NP-hard
problem. Finding efficient methods to determine nearly-optimal orderings is an
active area of research, especially for classical simulations of quantum
circuits.

Tensor decompositions In many cases the exact contraction of tensor networks
is not feasible, and approximations must be made. This is often done by using
an SVD to reduce the size of intermediate tensors by keeping only the largest
modes, and hence having an efficient SVD is important. Since typically only a
small portion of the modes are being kept, a full SVD isn’t necessary. A
partial SVD, such as one using randomized linear algebra Halko  _et al._
(2011), can be more efficient as the tensor sizes grow.

Distributed memory As the tensor sizes grow, the memory available on a single
computer will become a bottleneck. High accuracy calculations of theories such
as QCD will likely require larger amounts of memory found on leadership
computing resources. This will require having distributed memory tensor
libraries that can efficiently perform the necessary contractions and SVD
procedures on distributed tensors.

Symmetries Many problems of interest to HEP have symmetries that can also be
captured in the TN formulation. This leads to sparse structure in the tensors
which can be taken advantage of to make the calculations more efficient and
even more accurate. Some tensor libraries developed for condensed matter
applications can take symmetries into account. Adopting and modifying these
for problems in HEP could be beneficial.

Software ecosystem As noted in Psarras  _et al._ (2021), there are a large
number of libraries available with overlapping capabilities, each developed
for a specific research group or application in mind. Due to the diverse set
of target problems, methods and implementations, consolidating on a single or
small number of libraries may be difficult. For HEP, having a coordinated
research plan to develop tensor network software tailored for HEP problems,
while leveraging libraries actively developed in the condensed matter and
computer science communities may be a viable strategy to avoid further
fragmentation.

  * •

Access to large scale classical computers
Unlike traditional lattice field theory methods which have a cost in compute
time and memory that grows with some power of the system volume, tensor
network methods tend to have more modest resource growth for increasing
volume. Many TN methods have a resource growth that is only logarithmic in the
volume, and some can even work directly in the infinite volume limit. The
major factors influencing the resource requirements for TN methods are

    * –

The number of spatial dimensions in the theory

    * –

The complexity of the theory (requiring a larger tensor on each site)

    * –

The required accuracy of the final result

Increasing the accuracy of simulations requires growing the size of the
intermediate tensors in the corresponding contraction and truncation
procedure, to capture the necessary entanglement in the system.

Improved truncation schemes can help reduce the memory footprint in favor of
more compute cycles (which can be advantageous as the compute power of
computing technology tends to grow faster than memory capacity). However, when
moving to 3+1 dimensions, the memory and compute requirements will both grow
very fast as the accuracy is improved, and high accuracy simulations of
complex systems such as QCD will likely require running across large scale
leadership computing facilities. In this case having efficient distributed
memory versions of the codes will be necessary. Having access to large
machines (with large memory) will also be necessary to develop methods and to
calculate results. Since there is no statistical sampling in TN methods, as
opposed to standard lattice Monte Carlo methods, the total compute time needed
could be less than for traditional lattice methods. The main bottleneck,
however, is likely to be the memory size needed to store tensors required for
the target accuracy.

In some cases, storage of the large tensors for later use may also be desired,
requiring sufficient storage space and I/O bandwidth for tensors of order of
the size of the total available memory.

  * •

Workforce
Progress on these problems will require a collaborative workforce across HEP
and ASCR domains, along with collaborations with other science domains such as
condensed matter physics. This will require skilled domain scientists familiar
with the construction of lattice theories and relating numerical measurements
to physical results. It will also require computational scientists developing
improved tensor network methods that optimize the tradeoff between memory and
compute time for a given science problem and computer hardware. They will also
need to engage computer scientists and applied mathematicians who are
developing libraries and methods for large scale parallelization of tensor
network contractions and decompositions.

##  VI Overlap with other science domains and industry

  * •

Condensed matter physics
The development and application of tensor network methods is an active and
vibrant field in condensed matter physics. More details on the background in
condensed matter physics can be found in previous sections and related
references. Many applications to HEP grew out of the related work by, and in
collaboration with, condensed matter researchers. For example, the
reformulation of lattice gauge theories was developed as part of a
collaboration extending the methods of Tao Xiang’s group Xie  _et al._ (2012).

  * •

Classical simulation of quantum circuits
Tensor network methods can also be used to perform classical simulations of
quantum circuits, used for testing and development of quantum algorithms. This
can be done by using the tensor network to hold a compact approximation of the
full state vector, for example using a PEPS network Pang  _et al._ (2020). One
can also use tensor network methods to simulate the output of quantum
computers without storing the full state vector Villalonga  _et al._ (2020);
Pan and Zhang (2021). The classical simulation of quantum circuits is an
important means of developing and testing quantum algorithms for HEP
applications. Additionally, improvements in methods for simulating quantum
circuits could have an impact in tensor network methods directly applied to
computing HEP problems, and vice versa.

##  VII Conclusions

In summary, tensor network methods are playing an increasingly important role
in several branches of physics and are also becoming important in the context
of HEP. Tensor networks provide a compact representation of entangled quantum
systems and quantum states that can be used to reformulate physical theories
on a lattice, map them onto other quantum systems, such as quantum computers
or quantum simulators, and to classically evaluate simulation results either
in the tensor network formulation, or in the mapped representation onto
quantum hardware.

Tensor Lattice Field Theory (TLFT) is a new approach to models studied in the
context of lattice QCD. It provides reformulations of these models where the
microscopic tensors are the translationally invariant, local, building blocks
of exact discretizations of the path integral. They encode the local and
global symmetries of the original model. They also provide a more general
space of theories than the traditional Lagrangian and Hamiltonian densities.
Most model studied by lattice gauge theorists can be reformulated in TLFT.
Truncations need to be used for practical purposes and preserve symmetries in
a generic way. This provides a broader class of models with the same universal
continuum limit as the original model. TLFT methods connect smoothly between
the Lagrangian and Hamiltonian approaches and can be very useful in the
context of quantum computing.

TLFT presents an increasingly attractive alternative to traditional Monte
Carlo methods, which are currently the only computational methods able to
provide first-principle calculations for a large range of HEP problems but
have limitations. In numerical methods based on coarse-graining, tensor
network methods do not rely on statistical sampling, so can also provide
solutions to problems plagued by _sign problems_ that render Monte Carlo
methods ineffective. In some fields, such as condensed matter physics, their
use is fairly common. However, for problems of interest to HEP, tensor network
methods are still very resource intensive, and are not currently feasible for
complex theories in four space-time dimensions, such as QCD. Further
development in tensor network methods and software is needed for them to make
a significant impact in HEP too.

Tensor networks can also provide a convenient representation for mapping
lattice theories to quantum hardware (either in terms of digital quantum
circuits or analog quantum simulators). They can also be used to efficiently
perform classical simulations of quantum circuits. This is useful for
developing and testing quantum computing algorithms, and is an important area
of research for determining the tipping point for quantum computers to provide
a quantum advantage over classical ones. They help isolate “building blocks”
that have a limited complexity and can be optimally approximated for NISQ
hardware.

We have summarized some of the key problems in HEP that could be addressed
with tensor network methods, the current status of applications to HEP, and
the developments in methods, software and personnel needed to make progress in
these areas.

Using TLFT for models directly relevant for HEP is a feasible goal for the
next decade. Developing software and running codes in TLFT is labor intensive
and requires access to HPC.

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##  Appendix A Brief background on tensor methods

This is a brief overview of some tensor network methods. More complete and
detailed summaries can be found in several review articles Orus (2019); Bañuls
_et al._ (2020); Bañuls and Cichy (2020); Meurice  _et al._ (2020a); Cirac
_et al._ (2021).

The Density Matrix Renormalization Group (DMRG) White (1992) method (and the
similar representation known as Matrix Product States (MPS) Rommer and Östlund
(1997)) was developed for 1D lattice Hamiltonians, and is now the most
accurate method available for many of these types of problems. MPS can also be
applied to 2D Hamiltonians, but this is typically limited to cases with low
entanglement. The Projected Entanglement Pair States (PEPS) Verstraete and
Cirac (2004); Perez-Garcia  _et al._ (2007); Schuch  _et al._ (2010) is an
extension of MPS that can capture 2D entanglement well, but at a greatly
increased cost.

An alternative to MPS is provided by Tree Tensor Networks (TTN). These use a
binary tree of tensors to represent the state, which can be more efficient in
memory, especially for large systems, but generally does not provide as good a
representation as MPS. This can be circumvented by the addition of
_disentanglers_ into the tree network, which provide extra connections within
TTN that allow it to express much greater entanglement (by applying unitary
rotations that remove entanglement without changing the physics). This is the
basis of the Multiscale Entanglement Renormalization Ansatz (MERA) network
Vidal (2007). The disentanglers require extra effort to optimize, but for
Hamiltonian systems, they still allow for an efficient contraction of the
network.

TTNs can be applied to 2D Hamiltonians as well, with and without
disentanglers. An implementation with disentanglers has been shown to work
expecially well Felser  _et al._ (2021), however it comes with a large cost.

For HEP applications, an important problem is the evaluation of partition
functions, and related observables, that would come from Lagrangian systems
(and also classical Hamiltonians). One of the original methods designed for
the evaluation of 2D partition functions was the Tensor Renormalization Group
(TRG) algorithm Levin and Nave (2007). This method explicitly decomposed and
contracted neighboring tensors together to produce a new renormalized tensor
encoding the interactions of several sites. An improved method for performing
the tensor block transformation, based on a higher order SVD for tensors, was
introduced as the Higher Order TRG (HOTRG) method Xie  _et al._ (2012). This
method has a relatively simple form and can easily be extended to higher
dimensions, though the entanglement is limited due to the tree-like structure
of the projections.

Several methods have been introduced to improve on the accuracy of HOTRG. The
Tensor Network Renormalization (TNR) method Evenbly and Vidal (2015) includes
extra disentangling steps, similar to those used in MERA. TNR has only been
applied in 2D systems so far, and generalizing to higher dimensions is non-
trivial. Other methods that attempt to remove unnecessary short-range degrees
of while preserving the long-range ones are Loop-TNR Yang  _et al._ (2017) and
Gilt-TNR Hauru  _et al._ (2018). These are based the idea of removing degrees
of freedom around a loop and are easier to apply in higher dimensions.

One method developed explicitly to scale well in higher dimensions is
Anisotropic TRG Adachi  _et al._ (2020). This is based on an update step that
is similar to the original TRG method, which is relatively cheap, but not as
accurate as some of the above methods at a given bond dimension. Another
variation developed to make the calculations even cheaper in memory and
compute time is to decomposes the network so that it only contains triad
tensors (ones with three indices) Kadoh and Nakayama (2019). This
representation can be combined with other methods such as HOTRG.

Choosing the best method for a given problem can be difficult as it depends on
the details of the observables being measured, the required accuracy and even
the details of the implementations and the computer hardware being used. For
problems in 4D, memory used is a significant constraint. Methods that use less
memory can be run with a larger bond dimension, than those that consume more
memory. In some cases the larger bond dimension can make up for the accuracy
lost (at fixed bond dimension) due to breaking up the tensor, or using a
simpler update method. However this is not always the case, since many of the
methods are based on local update steps which do not necessarily converge on
the correct answer in a consistent manner. In some cases a more expensive, but
more uniformly converging method might give better accuracy for a fixed
hardware budget (memory and/or compute time). Searching for the best
combination of efficiency and convergence for problems of interest to HEP is
an open question.