#  Standard Model Physics and the Digital Quantum Revolution: Thoughts about
the Interface

Natalie Klco<EMAIL_ADDRESS>Institute for Quantum Information and Matter
and Walter Burke Institute for Theoretical Physics, California Institute of
Technology, Pasadena CA 91125, USA.  Alessandro Roggero<EMAIL_ADDRESS>InQubator for Quantum Simulation (IQuS), Department of Physics, University of
Washington, Seattle, WA 98195, USA.  Address after July 2021: Dipartimento di
Fisica, University of Trento, via Sommarive 14, I–38123, Povo, Trento, Italy.
Martin J. Savage<EMAIL_ADDRESS>InQubator for Quantum Simulation (IQuS),
Department of Physics, University of Washington, Seattle, WA 98195, USA.

(07 July 2021)

###### Abstract

Advances in isolating, controlling and entangling quantum systems are
transforming what was once a curious feature of quantum mechanics into a
vehicle for disruptive scientific and technological progress. Pursuing the
vision articulated by Feynman, a concerted effort across many areas of
research and development is introducing prototypical digital quantum devices
into the computing ecosystem available to domain scientists. Through
interactions with these early quantum devices, the abstract vision of
exploring classically-intractable quantum systems is evolving toward becoming
a tangible reality. Beyond catalyzing these technological advances,
entanglement is enabling parallel progress as a diagnostic for quantum
correlations and as an organizational tool, both guiding improved
understanding of quantum many-body systems and quantum field theories defining
and emerging from the Standard Model. From the perspective of three domain
science theorists, this article compiles _thoughts about the interface_ on
entanglement, complexity, and quantum simulation in an effort to contextualize
recent NISQ-era progress with the scientific objectives of nuclear and high-
energy physics.

††preprint: IQuS@UW-21-007

###### Contents

  1. I Introduction
  2. II Entanglement in Quantum Few-Body, Many-Body, and Field Theories
    1. II.1 Identifying and Characterizing Entanglement
    2. II.2 The Role of Entanglement
      1. II.2.1 …as an organizational principle:
      2. II.2.2 …as an order parameter for symmetry breaking:
      3. II.2.3 …as insight into the structure of hadrons:
      4. II.2.4 …as a source of local thermalization:
      5. II.2.5 …as an indicator of geometry:
      6. II.2.6 …as guidance to quantum simulation design:
      7. II.2.7 …as a harbinger of computational complexity:
  3. III The Vision of Quantum Simulation
    1. III.1 “Gedanken” Scaling of Quantum Resources
    2. III.2 Quantum Algorithm Development: “Gedanken” to Reality
  4. IV Complexity Classes in Context: Asymptopia Versus Reality
    1. IV.1 Asymptopia: The Utopia of Infinite Resources
    2. IV.2 Bounded Error: Uncertainties and Approximate Simulations
    3. IV.3 Reality: Hacking Bounded-Error Quantum Simulations
    4. IV.4 Bottom Line: Enhancing Leading Order Complexity
  5. V Quantum Techniques and the Standard Model
    1. V.1 Mapping Quantum Fields to Quantum Degrees of Freedom
      1. V.1.1 Fermions (quarks, neutrinos, electrons, nucleons)
      2. V.1.2 Scalar Fields (The BQP-Complete “Gold Standard”)
      3. V.1.3 Abelian and Non-Abelian Gauge Fields: Toward Lattice QCD
    2. V.2 Quantum Fields for Quantum Information
    3. V.3 Preparing Wavefunctions: Ground States and Finite-Density
    4. V.4 Probing Quantum Systems in Real Time: Scattering and Inelasticities
  6. VI Standard Model Applications: Select Implementations on Digital Quantum Hardware
    1. VI.1 Lattice Gauge Field Theory Dynamics
    2. VI.2 Structure and Reactions of Nuclei
    3. VI.3 Collective Neutrino Oscillations
  7. VII Closing
  8. VIII Acknowledgements

##  I Introduction

Quantum computation, simulation, and communication utilize the coherence and
inherent non-locality of entanglement in quantum mechanics to store, transmit,
encrypt and process quantum information with the fundamental particles and
forces of nature described by the Standard Model (SM). Advances in
understanding, creating and manipulating entanglement and coherence in the
laboratory, through broad-based collaborative research programs at
universities, national laboratories and technology companies, have recently
led to first demonstrations of a quantum advantage 111 A quantum advantage has
been achieved for a particular calculation when the measured time to solution
(with a given precision) on a quantum device is less than that achievable with
the most capable classical computers. using superconducting quantum devices
Arute  _et al._ (2019); Zhong  _et al._ (2020); Wu  _et al._ (2021). This
remarkable milestone in computing, heralds a paradigm-shifting change in how
we manipulate and engage with information. Its accomplishment was a tour de
force in the design and fabrication of materials, in quantum algorithms,
software and circuitry, in engineering systems controls and device
fabrication, and in the integration and isolation of classical and quantum
mechanical systems. It ushers in the second impact era of quantum
mechanics—after revolutionizing our understanding of the subatomic world, now
the controlled and coherent manipulation of entangled quantum states offers to
revolutionize broad scientific applications from computing, simulation and
theoretical developments to sensing and experiment.

First highlighted in the famous EPR paper in 1935 Einstein  _et al._ (1935),
Bell’s 1964 work Bell (1964) provided inequalities to quantitatively probe
entanglement, a distinctive attribute of quantum mechanics. The first
experimental confirmation of violations of classical predictions in
correlations between observables, in a way consistent with quantum mechanics
and the nonlocality of entanglement, was performed in 1972 by Clauser and
Freedman Freedman and Clauser (1972) using atomic transitions. Since that
time, remarkable experiments probing entanglement, and more generally quantum
mechanics, have been performed e.g., Refs. Aspect  _et al._ (1982a, b). This
growing literature has closed loop holes in the interpretation of the early
experiments and continues to inspire remarkable theoretical progress. The
meticulous design and formidable experimental prowess applied to these works
have firmly established that the subatomic rules governing our universe are at
least as complex as quantum mechanics, and have been instrumental in reaching
the present state of quantum information and quantum computing.

In the early 1980’s, Feynman and others Landauer (1961); Bennett (1973);
Benioff (1980); Manin (1980); Feynman (1982); Fredkin and Toffoli (1982);
Feynman (1986); Landauer (1991) recognized that simulations of all but the
simplest features of complex quantum systems lie beyond the capabilities of
classical computation, and ultimately require simulations using quantum
systems themselves i.e., quantum computers, sparking extensive activity in the
area of quantum information Williams (1998) 222See Ref. Preskill (2021) for a
recent discussion of the origins of quantum computing.. Precisely controlled
systems of cold-atoms, trapped-ions, annealers, optical, SRF and
superconducting qubit devices, and related programming languages, are becoming
available to scientists to begin exploring analog, digital and hybrid
simulations of simple quantum many-body (QMB) systems and quantum field
theories (QFTs). These devices fall into the category of Noisy Intermediate
Scale Quantum (NISQ) devices Preskill (2018). They are of modest size and of
“OK” fidelity. Their quantum registers are not well isolated from the
environment, operations applied to the registers are imperfect, and they
operate more like quantum experiments with associated systematic errors to be
quantified than, say, a laptop. There are limited (or no) protocols or
redundancies included in the hardware or software to correct for errors
introduced into the wavefunction due to imperfect isolation and operation,
such as phase-flips or spin-flips, i.e. there is no intrinsic error-correction
and the devices are not fault tolerant e.g., Ref. Gottesman (2009). While
quantum annealing devices 333 Quantum annealers, such as the D-Wave systems
Dwa , utilize quantum tunneling to adiabatically relax to the ground state, or
nearby states, of systems mapped onto a spin model Farhi  _et al._ (2000)
e.g., Refs. King  _et al._ (2018); Ajagekar  _et al._ (2020). are now
operating with thousands of qubits Dwa , systems that implement a universal
quantum gate set e.g., Ref. Lamata  _et al._ (2018); Alexeev  _et al._ (2021),
such as trapped ion, superconducting qubit, Rydberg atom systems and more, are
currently in the tens of qubits range, and are expected to scale to hundreds
in the next few years. Rather than relying upon software error correction with
large physical-to-logical qubit ratios, there are also major research efforts
focused on non-Abelian anyons to provide topological robustness Kitaev (2003),
including by Microsoft Microsoft  and the Quantum Science Center at Oak Ridge
National Laboratory Oak Ridge National Laboratory . While it is understood
that high-precision calculations at scale of direct comparison to experiment
are not expected during the NISQ era, which is expected to extend over the
next several years, there are good motivations to pursue a quantum advantage
for observables that scale poorly with classical computation, such as real-
time evolution, the structure of finite density systems and key aspects of
entangled systems Nuclear Physics and Quantum Information Science, A Report
from the NSAC Quantum Information Science Subcommittee (2019). Activities are
starting toward better understanding the complexity (in the _formal computer
science_ meaning of this word) of important problems in the SM, to determine
those that may be efficiently addressed with quantum simulation.

The potential of quantum computing for scientific applications is becoming
widely appreciated. For an ideal quantum computer with an $n$-qubit quantum
register, the dimensionality of its Hilbert-space is $d=2^{n}$. For $n=299$,
$d\sim 10^{90}$, a number larger than the number of atoms in our universe,
$\sim 10^{86}$. The exponential growth of the number of states with the number
of qubits parallels, for obvious reasons, the growth of the number of states
required to describe $A$-body systems. States in a Hilbert space are accessed
through unitary operations among the qubits, and while exponential time is
required to access all states Poulin  _et al._ (2011), physically relevant
states for a local Hamiltonian are generally accessible through time evolution
in polynomial time from other such states. Though a quantum advantage remains
to be obtained for a scientific application, there is a threshold in $A$
beyond which quantum simulations become the only known avenue with the
potential to achieve the requisite precision. This threshold depends upon the
system, the process to be computed, the desired precision, and the efficiency
of classical algorithms and implementations on high-performance computing
(HPC) systems. As such, one expects to see different quantum advantage
thresholds for e.g., lattice gauge theory (LGT), nuclear structure and
reactions, and coherent neutrino evolution.

Nuclear and high-energy physics research is focused on the nature of a diverse
array of fundamental and emergent QFTs and QMB systems, defined by unique
interactions and properties. The SM Glashow (1961); Higgs (1964); Weinberg
(1967); Salam (1968); Politzer (1973); Gross and Wilczek (1973) and beyond,
constructed around global and local symmetries and associated particle
content, underlies the properties and dynamics of these complex systems.
Analytic and computational techniques in QMB systems and QFTs developed during
the $20^{\rm th}$-Century, proved central to establishing the SM, in providing
predictions for processes beyond experimental reach, and in defining and using
low-energy effective field theories (EFTs) and phenomenological models to
describe an array of phenomena in strongly interacting, correlated systems of
mesons, nucleons and hyperons. Classes of computations projected to require
beyond exascale classical resources have been identified Habib  _et al._
(2016); Carlson  _et al._ (2017), with common attributes including finite
density, real-time non-equilibrium dynamics, inelasticities, the need for
large Hilbert spaces, quantum coherence and other intrinsically quantum
features. This circa 2016 recognition is daunting if only a classical HPC path
forward were possible, but exciting in light of the coinciding events of early
quantum devices becoming more widely accessible and the first QFT simulation
performed on a quantum device Martinez  _et al._ (2016). Soon after
superconducting quantum devices became available to scientists via the cloud
IBM ; Karalekas  _et al._ (2020), the first calculation of the deuteron
binding energy Dumitrescu  _et al._ (2018) was accomplished by a multi-
disciplinary team at Oak Ridge National Laboratory, and similar computations
of light nuclei with EFT interactions soon followed Lu  _et al._ (2019);
Shehab  _et al._ (2019); Roggero  _et al._ (2020a); Yeter-Aydeniz  _et al._
(2020); Di Matteo  _et al._ (2021).

Our article is framed from the perspective of quantum simulation of SM
quantities, though is relevant for quantum systems more broadly through
communications protocols, sensing, error correction, and generic large-scale
quantum computation. Beginning with the acknowledgement that nonlocality of
information is fueling the quantum revolution, Section II illustrates the
polycephalic form of entanglement at the interface—including weaving an
organizational fabric between QMBs and QFTs and the quantum devices that will
simulate them, and illuminating hierarchies in low-energy EFTs and their
phases. Predictive capabilities for the SM in the first era of quantum
mechanics advanced, in part, through the construction of effective
interactions and EFTs, providing _classically tractable_ leading order
calculations that can be systematically improved with perturbation theory
e.g., perturbative quantum chromodynamics (QCD). While most EFTs have been
built around local interactions and vanishing entanglement, the prospect of
capable quantum computers motivates applying this mindset toward devising
_quantumly tractable_ leading order calculations that naturally incorporate
entanglement for quantum simulation at-scale. Where possible, the practical
utility of such organizations is to extend the scientific reach of bounded-
error quantum simulation, mitigating naïve asymptotic complexity
classifications, as discussed in Section IV. Progress over the last few
decades has established an extensive toolbox of quantum protocols that can be
used, adapted and improved for quantum simulations of SM quantities. In
Sections V and VI, we present an overview of relevant algorithms and
techniques, progress that has been made in their application to SM
simulations, and three SM areas that are being pursued with available quantum
devices.

##  II Entanglement in Quantum Few-Body, Many-Body, and Field Theories

Beyond purist motivations of understanding the non-local correlations within
our fundamental descriptions of nature, studying the structure of entanglement
within the SM is expected to provide guidance both in the formulation of
theoretical frameworks and in the design of associated quantum or classical
simulations. Because entanglement is a basis-dependent property, sensitive to
the way in which a physical system is mapped onto quantum degrees of freedom,
a detailed understanding of the entanglement properties expressed in subatomic
interactions with natural bases will inform every level of quantum simulation,
from initial decisions of the quantum description to the specifications for
architectural co-design.

###  II.1 Identifying and Characterizing Entanglement

A necessary and sufficient condition for identifying quantum entanglement can
be expressed in terms of the separability of the density matrix describing the
union of two or more Hilbert spaces $\mathcal{H}=\cup_{i}\mathcal{H}_{i}$. For
the specific case of $\mathcal{H}=\mathcal{H}_{A}\cup\mathcal{H}_{B}$, the two
Hilbert spaces may be e.g., spatial regions of a field, separate momentum
scales, valence and sea quantum number contributions, or a spatial region of a
nucleon and its complement with a size set by the probing momentum transfer.
If the density matrix can be written as a classical mixture of separable
density matrices, $\rho_{A\cup B}=\sum_{i}p_{i}\ \rho_{A,i}\otimes\rho_{B,i}$
with $p_{i}\geq 0$, then the system is separable and not entangled. If the
density matrix cannot be written in this way, the system is entangled. The
existence of entangled states, flouting attempts to describe their
correlations classically, is culpable for the exponential growth of Hilbert
spaces (and thus classical computational resources) required to describe QMB
systems. To date, every theoretical attempt to evade this exponential state
space required by non-classical correlations has been ruled out by experiment,
the most famous example being that of nature’s consistent response, through
the violation of Bell’s inequalities, that a local hidden variable description
is inconsistent with observation.

While experiment indicates that a complete evasion of the exponential growth
of Hilbert space is not likely to provide a successful description of nature,
it is possible to systematically delay the onset of this burdensome scaling.
The language of one concrete and powerful example Bañuls  _et al._ (2018)
contains at its heart the pure-state Schmidt decomposition,

$|\Psi\rangle=\sum_{k=1}^{\chi_{m}}\lambda_{k}|\Psi_{A,k}\rangle\otimes|\Psi_{B,k}\rangle\
\ \ ,$ (1)

the expansion of a pure quantum state in terms of the separate eigenvectors of
its $A$\- and $B$-space reduced density matrices. For pure states, the two
density matrix spectra, $\vec{\lambda}$, will have equal non-zero eigenvalues
of number limited by the dimensionality of the smaller Hilbert space,
$\chi_{m}$. Only for $\chi_{m}$ = 1, in which local pure states remain, is the
state separable across the chosen bipartition. Entanglement-motivated
approximations to the state in Eq. (1) can be achieved by introducing a
truncation $\chi\leq\chi_{m}$, with a viable and affable entanglement measure
identified as $\chi$’s logarithm. Despite the opacity of entanglement
structure when an $n$-qubit state is written in the computational basis,
$|\Psi\rangle=\sum\psi_{\vec{b}}\
|b_{1}\rangle\otimes\cdots\otimes|b_{n}\rangle$ with $b_{i}\in\\{0,1\\}$, an
entanglement-driven hierarchical structure of the wavefunction can be exposed
through a series of sequential Schmidt decompositions Vidal (2003, 2004),

$\psi_{\vec{b}}=\Gamma_{\alpha_{1}}^{(b_{1})}\lambda_{\alpha_{1}}\Gamma_{\alpha_{1}\alpha_{2}}^{(b_{2})}\lambda_{\alpha_{2}}\cdots\lambda_{\alpha_{n-1}}\Gamma_{\alpha_{n-1}}^{(b_{n})}\
\ \ .$ (2)

Here, the $\Gamma$’s are tensors of maximal dimension $\chi$ characterizing
the single-qubit Hilbert space and the $\lambda$’s are locally-contracted
vectors with a dimensionality of $\chi$ at the bipartition. These
$\lambda$-vectors, serving as entanglement conduits between local qubit
Hilbert spaces, may be truncated, leading to a family of systematically
improvable approximations to $|\psi\rangle$ with reduced entanglement. Rather
than the $2^{n}$ complex numbers capturing the exact wavefunction, the
truncated description requires a memory that scales $\sim\chi^{2}n$, linearly
in the number of qubits but exponentially in the entanglement measure
mentioned above. Efficiently representing low-entanglement quantum states by
truncating their local Schmidt decompositions underpins the robust technology
of tensor networks Bañuls  _et al._ (2018), continuing to provide leading
capabilities in low spatial dimensions and for short time evolutions where
local bipartite entanglement is naturally constrained.

Though the clear separability criterion above seems at first simple and
innocuous, determining whether a particular density matrix can be transformed
into a separable mixture is, in general, a difficult and open area of
research. In particular, there is no known necessary and sufficient criterion
for separability that is also efficiently computable as the dimensionality
and/or number of Hilbert spaces becomes large. Unfortunately, _large_ in the
previous sentence happens to be around three qubits, a pair of three-state
qutrits, or the presence of a four-state qudit. It may be no surprise that
describing the complex non-local correlations that quantum systems are capable
of expressing is not easily achieved through a single observable or criterion.
Rather, analogous to the Bayesian approach to data analysis through classical
probabilities as a “dialogue with the data” Sivia , the exploration and
characterization of entanglement in quantum mechanical systems often requires
a collection of entanglement criteria, each one perhaps being necessary _or_
sufficient, whose combination of fragmented information may be synthesized
into more complete logical conclusions.

For example, a quantum generalization of the Shannon entropy leads to the
entanglement entropy, $\mathcal{S}=-Tr[\rho_{A}\log\rho_{A}]$, used to explore
the entanglement of bipartite states that are globally pure. The entanglement
entropy is calculated in terms of the reduced density matrix,
$\rho_{A}=\text{Tr}_{B}\rho_{A\cup B}$, and is symmetric with respect to the
system $A,B$ that is chosen to be traced over. Because entanglement entropy
heralds the presence of entanglement by alerting us to mixedness in reduced
density matrices (pure states exhibit $\mathcal{S}=0$), the interpretation of
this entropy becomes ineffectual when the global state is already mixed. For
the language of mixed states, one may turn to conditional entropy and mutual
information that combine otherwise-ambiguous marginal distributions into more
informative relative entropies. The important caveats continue as such mutual
information criteria are sensitive to classical correlations in addition to
quantum correlations; it is possible to find a non-zero mutual information for
a classically mixed unentangled state, and thus non-zero mutual information is
necessary though insufficient for detecting entanglement. To address this, one
may look yet further to the family of positive transformations, those that
retain positive eigenvalues of the density matrix. By acting locally
(subsystem) with such a transformation, any emergence of a negative eigenvalue
must herald inseparability. A common positive transformation to consider, due
to its relatively amiable computability, is the transpose Horodecki  _et al._
(1996); Simon (2000); Vidal and Werner (2002); Plenio (2005); Horodecki
(1997); Horodecki  _et al._ (1998). Physically, the _partial transpose_
implements a local momentum inversion that will produce a valid, positive
density matrix if the two regions distinguished by the transpose are
unentangled. Though the presence of negative eigenvalues, non-zero
_negativity_ , is unperturbed by classical correlations, negativity is
sufficient for detecting inseparability, but it is not necessary Horodecki
(1997); Horodecki  _et al._ (1998); Horodecki and Lewenstein (2000); Werner,
R. F. and Wolf, M. M. (2001). Aptly named, the multitude of entanglement or
separability _witnesses_ Terhal (2000), will individually fail to identify the
properties they are designed to observe in a subset of Hilbert space; their
strength, however, often lies in collaboration among multiple witnesses and
deductive reasoning. Following Murphy’s law, and in the spirit of EFTs, it is
usually a safe default to assume: if states of particular entanglement
properties can exist (e.g., positive under partial transposition but
inseparable), they do.

In addition to entanglement witnesses that conclusively herald but
inconclusively exclude entanglement or separability, is a category of measures
regarding entanglement more fundamentally as a resource for quantum
information protocols. Examples of such measures are the entanglement of
formation Wootters (1998); Hill and Wootters (1997) or the distillable
entanglement Bennett  _et al._ (1996a, b), quantifying the asymptotic ratio of
maximally entangled resource states necessary to produce an ensemble of the
state or the ratio capable of being distilled back from a given ensemble
through local operations, respectively. While operational measures provide
welcomed clear physical interpretations, they are commonly computationally
prohibitive. It is thus common to combine heralding measures with operational
measures, e.g., the logarithmic negativity is an upper bound to the
distillable entanglement Vidal and Werner (2002); Audenaert  _et al._ (2003).
Consistent with the above creation of entanglement measures in a collaborative
array connected to e.g., operational communications protocols, it is likely
that developing the dialogue for exploring the entanglement of QMB and QFT
systems will entail the creation of an extended array of entanglement
measures, inspired by the natural symmetries and entanglement fluctuating
interactions of the subatomic world.

###  II.2 The Role of Entanglement

Though the landscape for quantifying the structure of entanglement in QMB and
QFT systems can seem daunting and precarious at times, this connection has
already begun and will continue to provide novel perspectives of entanglement…

####  II.2.1 …as an organizational principle:

The successes of tensor network methods in the simulation of low dimensional
quantum lattice models Bañuls  _et al._ (2018) arise from the use of an
entanglement-centric computational framework. By forming a basis of
variational states truncated by the entanglement entropy crossing any
bipartition of the space, the exponential computational demands associated
with the inclusion of entangled states can be systematically mitigated and
included only as necessary e.g., at long times in dynamical scattering
simulations Milsted  _et al._ (2020). These successes motivate the formation
of other structural hierarchies guided by entanglement.

The success of nuclear shell-models is due, in part, to the wavefunction of
many nuclei being close to a tensor product of a valence model-space
wavefunction with that of an inert closed-shell core. Inspired by the advances
arising upon incorporation of non-local correlations into classical
computational methods, modern nuclear structure calculations, using
phenomenological and QCD-based chiral nuclear forces combined with
sophisticated nuclear many-body numerical algorithms and codes, are now
investigating the role of entanglement as an organizational principle. Around
2015, the role of two-orbital mutual information for organizing bases of
nucleon orbitals in medium mass nuclei Legeza  _et al._ (2015) was examined.
Extensive work by Gorton and Johnson in 2018 Gorton (2018) examined the
entanglement entropy in light, medium and heavy nuclei using the BIGSTICK code
Johnson  _et al._ (2018). Inspired by this work and the multidimensional
structure of entanglement, a detailed exploration of Helium nuclei using
configuration-interaction calculations for a range of single particle bases of
various complexities, studied the single-orbital entanglement entropy, mutual
information and two-orbital negativity Robin  _et al._ (2021). Starting from
matrix elements in a harmonic oscillator basis derived from chiral
interactions, and self-consistently evolved to include two-body correlations
in identifying a single particle basis, the localized entanglement within a
nucleus was observed to become manifest in such a natural basis e.g.,
decoupling the two-neutron halo of 6He from its 4He core at the level of
fundamental quantum correlations Robin  _et al._ (2021). These works show that
entanglement observables are capable of providing insight into otherwise-
obscure structures that may be leveraged for a computational advantage in
exploiting future hybrid quantum-classical architectures. Subspaces with large
entanglement are amenable to quantum computation, and integrated with other
such subspaces using classical computation (as the quantum correlations
between subspaces is small, by construction).

Complementary to the role of entanglement as an organizational principle is
the role of entanglement fluctuations in constraining the forms of dynamical
interactions or in illuminating a hierarchy among operators. For example, in
high energy scattering processes at tree-level in QED, a ubiquity of maximally
entangled helicity states at particular final scattering angles was observed
Cervera-Lierta  _et al._ (2017). Promoting this observation to an
organizational principle led to the intriguing realization that demanding
maximally entangled final states to be achievable in QED and weak interactions
produces, in the space of theoretical deformations to these fundamental
vertices, a potential landscape that appears to be extremized by the SM. The
capability of an entanglement extremization principle to constrain the
structure of interactions, similar to more familiar rotational symmetries and
gauge invariance, inspires further investigation into the role of quantum
correlations in the theoretical description of fundamental interactions.

An exploration of low-energy elastic, s-wave scattering of 2- and 3-flavor
nuclei through the strong interaction has demonstrated another connection
between emergent symmetries and entanglement structure Beane  _et al._ (2019).
Rather than the capability to produce maximally entangled final states, this
work quantified the operational entanglement power Zanardi  _et al._ (2000);
Ballard and Wu (2011) of the $S$-matrix, finding that the presence of spin-
flavor symmetries coincide with parameter regimes of diminished entanglement
_fluctuations_. When symmetry in this system is heightened e.g., through
Wigner’s SU(4) symmetry, the singlet/triplet phase shifts vanishing/residing
at unitarity, or at a kinematical $\pi/2$ phase shift difference where spin
spaces are exchanged (SWAP operator Beane  _et al._ (2019); Low and Mehen
(2021)), the entanglement power of the $S$ matrix becomes suppressed.
Promoting this observation to an organizational principle led to an
entanglement motivation for the emergent SU(16) symmetry calculated in Lattice
QCD at heavy pion mass Wagman  _et al._ (2017) by establishing a conjecture
Beane  _et al._ (2019) that the dynamical suppression of entanglement
fluctuations may drive emergent symmetries. This observation has the potential
to inform the hierarchy of local operators used to design EFTs of nuclei and
hypernuclei, extending beyond those from large-$N_{c}$ alone.

While central to quantum communication, studies of entanglement and
separability in Gaussian systems also describe the nonlocality in harmonic
chains Srednicki (1993); Audenaert  _et al._ (2002); Botero and Reznik (2004);
Audenaert  _et al._ (2007); Marcovitch  _et al._ (2009); Coser  _et al._
(2017); Di Giulio and Tonni (2020) or in the free lattice field theories
directly relevant to high-energy and nuclear physics. More generally,
entanglement studies in field theories have spanned the areas of conformal
field theories (CFTs) Calabrese and Cardy (2004, 2009); Calabrese  _et al._
(2009, 2012, 2013); Ruggiero  _et al._ (2018) and near CFTs such as pQCD Ho
and Hsu (2016); Baker and Kharzeev (2018); Tu  _et al._ (2020); Kharzeev and
Levin (2021), AdS/CFT and gravity-dualities Maldacena (1998, 2003); Ryu and
Takayanagi (2006a, b), lattice simulations Klco and Savage (2021a, b), the SM
Cervera-Lierta  _et al._ (2017) and low-energy EFTs Beane  _et al._ (2019);
Beane and Ehlers (2019); Beane and Farrell (2020). Aiming to understand the
microscopic mechanism that incorporates entanglement into EFT construction,
through perhaps a small expansion parameter or generation of a mass scale, the
exponential decay of distillable entanglement between disjoint regions in the
massless non-interacting scalar field vacuum can be studied Marcovitch  _et
al._ (2009); Klco and Savage (2021a, b). In doing so, a dimension-independent
UV-IR connection indicating that the entanglement or inseparability at long
distances is governed by the high momentum modes of the field is found Botero
and Reznik (2004); Klco and Savage (2021b). Extending a result of quantum
information to the realm of field theories, the observation that Gaussian
states provide minimal-entanglement approximations Wolf  _et al._ (2006) opens
the possibility that such features will persist in the presence of
interactions, though further exploration is required. Being a framework built
upon an expansion in local operators with inherent decoupling upon the
appearance of a separation of scales, EFTs may require novel methods in order
to incorporate the fundamental features of non-localities.

####  II.2.2 …as an order parameter for symmetry breaking:

A beneficial interaction has developed in recent years between quantum
information, condensed matter, and SM physics. One influential direction of
progress has been in the utilization of entanglement structures to identify
phase transitions, provide sensitivity to probe symmetries or non-perturbative
phenomena such confinement, and classify states of matter, particularly those
exhibiting non-trivial topology Klebanov  _et al._ (2008); Wen (2019). Because
separability reflects correlations between Hilbert spaces on all length
scales, entanglement measures offer valuable non-local probes of quantum
mechanical structure complementary to the local observables commonly leveraged
in EFTs.

As an example in the nucleon, the scale-dependent breaking of chiral symmetry,
as the valence sector interacts with the parton sea, can be tracked by
observing entanglement entropy Beane and Ehlers (2019). Specifically, by
analyzing contributions to the nucleon state vector on a null-plane, where the
internal helicity degrees of freedom can be separated from kinematical
contributions, quantum correlations arising in the presence of collective
excitations perturbing the separable large-$N_{c}$ descriptions are an
indicator of chiral symmetry breaking. As high-energy sea partons are
incorporated into the nucleon description, the entanglement between the
valence sector and the sea increases. This entanglement saturates near its
maximum value, blurring the distinction of the valence spin that dominates
only for a low-energy parton sea and suggesting a fundamental understanding of
the small valence contributions to the nucleon spin. In the chiral multiplet
basis, it was shown explicitly that chiral symmetry is broken only when the
nucleon state is entangled, and therefore entanglement is an order parameter
for chiral symmetry breaking.

In the SM beyond the nucleon, early quantum simulations of neutrino flavor
dynamics are beginning to highlight the role played by dynamical phase
transitions in collective neutrino oscillations Roggero (2021a, b), coinciding
with a modification in the structure of entanglement. Notably, these first
explorations were made possible by exploiting the capabilities of tensor
networks to efficiently compress many-body states with low levels of bipartite
entanglement Vidal (2003). These systems will be discussed further in Section
VI.

####  II.2.3 …as insight into the structure of hadrons:

From heavy quark effective theory (HQET) Isgur and Wise (1989), to the
large-$N_{c}$ expansion, to parton distribution functions (PDFs), coherent
quantum effects are often successfully incorporated by expanding around a
basis that is effectively classical. The successes of nuclear structure and
reaction calculations, with basis states of (color-singlet) nucleon degrees of
freedom and a hierarchy of multi-nucleon interactions, have been a testament
to the nucleon’s role as a fundamental building block in the finely-tuned
transition from quarks and gluons to the predominantly classical world of
macroscopic physics, i.e., nuclei have more structure than would be expected
of a quark-gluon droplet. This perspective supports the sensibility of
beginning with factorizable or tensor-product structures at leading order and
systematically incorporating entanglement effects at sub-leading orders e.g.,
suppressed by inverse powers of a heavy quark mass, number of colors, or
parton momenta. It is the experimental accessibility and control of sub-
leading quantum coherent effects that is likely to become emblematic of 21st
century explorations.

In the last decade, increased attention has been placed on the factorization
treatment of PDFs in non-Abelian gauge theories Collins  _et al._ (1989);
Collins and Qiu (2007); Rogers and Mulders (2010); Cherednikov and Stefanis
(2012); Buffing and Mulders (2014); Schäfer and Zhou (2014); Buffing and
Mulders (2015); Adare  _et al._ (2017); Zhou (2017). This investigation has
been guided by observed correlations and asymmetries that elude description
when transverse momentum PDFs are assumed independent (factorizable) for each
final state hadron. In a phenomenon that has become known as _color
entanglement_ , two-gluon process provide the opportunity for a coherent
exchange of color degrees of freedom. This coherent exchange produces quantum
mechanical correlations between hadronic Hilbert spaces, and thus an
inseparability in the PDFs that characterize them.

From another perspective, entanglement can be considered inherent to hadronic
structure produced either dynamically or through the enforcement of symmetries
Kovner and Lublinsky (2015); Kharzeev and Levin (2017); Kovner  _et al._
(2019); Tu  _et al._ (2020); Peschanski and Seki (2019); Castorina  _et al._
(2021); Ramos and Machado (2020); Duan  _et al._ (2020); Gotsman and Levin
(2020); Andreev  _et al._ (2021); Baty  _et al._ (2021); Kharzeev and Levin
(2021). For example, spatial entanglement between regions of a proton
delineated by the volume probed in deep inelastic scattering Kharzeev and
Levin (2017) can be considered. In small Bjorken $x$ regimes expected to be
accessible with the Electron Ion Collider (EIC), entanglement entropy is
predicted to become maximal and naturally governed by a simple logarithm of
the gluon distribution. Further evidence for partonic entanglement arising
from the color-singlet structure of hadrons is expected to be accessible in
LHC proton-proton collisions Tu  _et al._ (2020). A deeper understanding of
the role of quantum correlations in hadronic structure is expected as quantum
coherent effects become more accessible to experiment.

####  II.2.4 …as a source of local thermalization:

Connected to its natural role of delocalizing quantum information, the
generation of entanglement has been linked to the local emergence of
statistical mechanics in unitarily evolving closed quantum systems (see e.g.,
Refs. Deutsch (1991); Srednicki (1994, 1999); Eisert  _et al._ (2015);
D’Alessio  _et al._ (2016); Anza and Vedral (2017); Anza  _et al._ (2018)).
Recent experimental demonstration in a Bose-Einstein condensate of Rubidium
atoms in a 2D optical lattice has supported this connection as local
observables became statistically governed while the global quantum state
remained pure Kaufman  _et al._ (2016). More generally, the role of coherent
evolutions has been linked to the achievement of statistical distributions
more rapidly than would be expected through semi-classical approaches alone
Bell  _et al._ (2003); Giovannetti  _et al._ (2003a, b, c); Sawyer (2004). In
the sense that non-local correlations provide a mechanism of quantum
mechanical mixing in addition to scattering events, the timescale of
collisions need not limit the time scale of locally emergent thermalization. A
natural application of this phenomenon is in complex, non-equilibrium systems,
such as heavy-ion collisions, where thermalization timescales appear to be
much shorter than naïvely expected by cross section analysis. Early studies in
this direction suggest that entanglement may be providing mechanisms of
thermalization necessary to describe experimental observations of deep
inelastic scattering and heavy-ion collisions Muller and Schafer (2011);
Akkelin and Sinyukov (2014); Ho and Hsu (2016); Kharzeev and Levin (2017);
Berges  _et al._ (2018); Müller and Schäfer (2017); Baker and Kharzeev (2018);
Berges  _et al._ (2019); Feal  _et al._ (2019); Iskander  _et al._ (2020),
probing mixed-state subsystems of globally entangled states.

####  II.2.5 …as an indicator of geometry:

Connections between geometry and entanglement are plentiful: from area laws in
low-energy states of 1-dim gapped Hamiltonians Hastings (2007); Eisert  _et
al._ (2010) and the role of geodesics guiding complexity in the design of
unitary quantum circuits Dowling and Nielsen (2008), to classifications of
correlations through entanglement polytopes Tóth  _et al._ (2007); Walter  _et
al._ (2013) and potentially the form of spacetime itself Swingle (2018).
Holographic perspectives on the over-completeness of degrees of freedom in
field theory descriptions ’t Hooft (1993); Susskind (1995) have led to
concrete examples of bulk theories of quantum gravity in hyperbolically curved
space being exactly dual to a boundary CFT in one dimension lower, known
broadly as the AdS/CFT correspondence Maldacena (1998). Extending the
Bekenstein-Hawking formula Bekenstein (1973); Hawking (1975), connecting the
entropy of a black hole to the surface area of its event horizon, to the
framework of holographic dualities has produced an intriguing conjecture, the
Ryu-Takayanagi formula Maldacena (2003); Ryu and Takayanagi (2006a, b),
connecting the entanglement entropy of a region in CFT to the surface area of
its associated geodesic in the dual gravitational bulk. Beyond opportunities
for tractable explorations in quantum gravity, such connections illuminate the
dispersion of information into higher-dimensional embeddings—a common strategy
in subatomic physics, though not usually described in this language.

One notable recent exploration connecting holographic dualities to geometry
and entanglement for nuclei Beane and Farrell (2020) has done so through the
$S$-matrix formulation of scattering, with explicit examples in the $s$-wave
nucleon-nucleon system. Rather than the EFT action, this approach focuses upon
trajectories of the $S$-matrix on an abstract surface defined by unitarity.
The entanglement power Zanardi  _et al._ (2000); Ballard and Wu (2011),
providing a quantification of operator entanglement, serves as a measure of
distance to non-entangling subspaces and as a source of non-trivial curvature
to $S$-matrix trajectories. Regarding the role of entanglement and symmetry
complementary to the hierarchy of local operators allows this formulation to
address non-local interactions, corresponding to deviations into the bulk from
inelastic effects, with a novel perspective reminiscent of holographic
dualities.

####  II.2.6 …as guidance to quantum simulation design:

While the impact of the entanglement-guided style of thinking is expected to
apply to quantum simulation design for QMB systems broadly, most concrete
examples have been demonstrated in the simulation of quantum fields. In the
theoretical planning for the quantum simulation of the scalar field, it is
straightforward to realize that the entanglement structure of the vacuum state
of the free field is trivial if represented in a basis of momentum modes—a
collection of tensor product oscillators and the starting point for
perturbative calculations in the language of Feynman diagrams. Considering
further the entanglement structure that will dynamically grow upon the
incorporation of local self-interactions indicates the production of
entanglement between any momentum-conserving collections of mode oscillators.
For any representation of momentum oscillators mapped onto physical quantum
degrees of freedom for computational implementation, the self interactions
necessarily produce momentum-space entanglement Balasubramanian  _et al._
(2012); Jordan  _et al._ (2014) that requires interactions in the Hamiltonian
that are beyond nearest neighbor 444Note that while genuine $n$-point Green’s
functions cannot be reduced into 2-point contributions e.g., 3- and 4-nucleon
forces cannot be represented by 2-nucleon forces, any unitary operator can be
decomposed into nearest-neighbor two-qubit entangling operations.. Motivated
by a desire to mitigate the entanglement structure demands on quantum
hardware, a variety of bases for digitizing the scalar field upon a spatial
lattice or with single particle states have also been explored Jordan  _et
al._ (2012, 2014); Somma (2016); Macridin  _et al._ (2018a, b); Klco and
Savage (2019); Barata  _et al._ (2021), focusing on the necessary entangling
operators for the preparation of low-energy states and the time evolution of
the field. It is expected that developing intuition in the entanglement
structure of dynamical fields will guide the design of quantum simulations of
more complex fields, including those with non-Abelian gauge symmetries (see
Section V.1.3).

Classical tensor network methods Bañuls  _et al._ (2018) or strategies for
reorganizing unitary quantum circuits Klco and Savage (2020a) provide viable
paths for incorporating simple area-law or symmetry-embedded entanglement at
lower computational cost, incorporating additional entanglement perturbatively
for the approximate initialization of entangled quantum states. Distinct from
the conduit-style treatment of entanglement in classical tensor networks,
truncating in entanglement traversing all possible bipartitions of the space,
one can imagine leveraging correlation structures to truncate instead in the
size or non-locality of entangling operators utilized in quantum simulation.
For example, guided by the knowledge of exponentially localized spatial
correlations in the ground states of massive fields, preparation circuits can
be constructed in which the relevance of long-distance quantum gates are
exponentially suppressed Klco and Savage (2020b, c). This approach is
analogous to that used in the Approximate Quantum Fourier Transform (AQFT)
Barenco  _et al._ (1996). When mapping such fields to quantum architectures
through a position-space distribution of qubits, the locality of correlations
in the simulated field becomes manifest in the hardware
implementation—interactions performed experimentally also enjoy exponential
localization. The above examples provide inspiration for the synergy expected
between fundamental studies of the structure of entanglement in QFT or QMB
systems and the design of their efficient simulation using quantum devices.

####  II.2.7 …as a harbinger of computational complexity:

Absent in the above discussions is the role of entanglement in distinguishing
computational complexity or in allowing for a parametric reduction in the
temporal and/or spatial resources demanded by a calculation. While quantum
speed-ups have a dependence upon the presence of sufficiently scaling
entanglement that follows practically by definition (see e.g., Refs.
Braunstein and Pati (2002); Jozsa and Linden (2003); Vidal (2003); Schuch  _et
al._ (2008) for more precise elaboration), the desired statement that
entanglement is the source of quantum advantage is perhaps misleadingly
broad—some categories of highly entangled states can be efficiently
represented and simulated by classical computational architectures,
occasionally challenging the succinct identification of quantum advantages.

The canonical counterexample to the desired simplified understanding equating
entanglement and computational complexity is the stabilizer formalism
Gottesman (1997, 1998), $S_{i}|\psi_{S}\rangle=|\psi_{S}\rangle$, describing
quantum states $|\psi_{S}\rangle$ that are stabilized in the $+1$ eigensector
of a commuting set of stabilizers, $\vec{S}$. The $2^{n}$ elements of
$\vec{S}$ can be parameterized in the basis of tensor product (extended) Pauli
operators, $P=\pm\sigma_{1}\otimes\sigma_{2}\otimes\cdots\otimes\sigma_{n}$
with $\sigma_{i}\in\\{\mathbb{I},X,Y,Z\\}$, and uniquely identify the vector
$|\psi_{S}\rangle$. For example, the three-qubit ($n=3$) state
$|GHZ\rangle=\frac{|000\rangle+|111\rangle}{\sqrt{2}}$ can be described by a
set of 8 stabilizers,
$\vec{S}=\\{\mathbb{I}_{3},XXX,Z\mathbb{I}Z,ZZ\mathbb{I},\mathbb{I}ZZ,-XYY,-YXY,-YYX\\}$.
At first, this appears to be an exacerbation of memory requirements. However,
leveraging the group structure of the stabilizers, $S$ can be captured by its
$n$ generators e.g., $G_{S}=\\{XXX,ZZ\mathbb{I},Z\mathbb{I}Z\\}$. Constructing
a classical binary basis assigning two classical bits (with values 0,1,2,3) to
each qubit Hilbert space to distinguish the four possible Pauli operators,
along with an additional bit for the sign of each generator leads to the
classical storage of $G_{S}$ in $(2n+1)n$ classical bits. Applying a quantum
circuit, $U$, to a stabilizer state can be achieved by simply transforming
each of the generators in $G_{S}$ as $g_{i}\rightarrow
g_{i}^{\prime}=Ug_{i}U^{\dagger}$. The crucial feature of the stabilizer
formalism that enables its classical efficiency throughout computation is that
each transformed generator $g_{i}^{\prime}$ remains a single tensor product of
Pauli operators, $UPU^{\dagger}=P^{\prime}$, _if_ the circuit is in the
Clifford subgroup $U\in\\{H,S,{\rm CNOT}\\}$ (which also generates the Pauli
operators) 555$\\{H,S,{\rm CNOT}\\}$ are the Hadamard, S-phase, and
controlled-NOT gates, as discussed in Ref. Nielsen and Chuang (2010).. Thus,
any circuit acting on a stabilizer state that is comprised of $\\{H,S,{\rm
CNOT},X,Y,Z\\}$ gates is efficiently classically simulatable Gottesman (1997,
1998). This includes the creation of highly entangled states e.g., GHZ states
and their $n$-qubit extensions, as well as non-trivial quantum protocols e.g.,
quantum teleportation. Of course, to produce universal quantum computation one
requires additionally the quarter phase $T$-gate Kitaev (1997a); Dawson and
Nielsen (2005), understood in this context to proliferate tensor product Pauli
operators, $TPT^{\dagger}=\sum P^{\prime}$, and thus forces one to abandon the
classically efficient representation. It is for this reason that $T$-gate
count can be a meaningful quantity characterizing simulation complexity,
providing a source of departure from a classically efficient computational
strategy. Furthermore, though the state space of the stabilizer formalism is
highly restricted (e.g., any non-zero amplitude must have the same value and
all measurements must be either deterministic or exhibit a random 50%
probability) and will not capture the elaborate wavefunctions of SM systems,
the formalism is an important and accessible language for the exploration of
quantum error correction codes Gottesman (1997). The high degrees of structure
and symmetry presented by stabilizer states e.g., measurement constraints or
restrictions to a finite octahedron of points on the Bloch sphere, further
support the above discussion that symmetry is important when pursuing a
connection between entanglement and computational complexity. As a
constructive example of the Gottesman-Knill theorem (see Ref. Aaronson and
Gottesman (2004)), this formalism demonstrates that there exists a subset of
quantum computation that is encapsulated within the complexity of classical
computation and yet includes the production of states highly entangled in the
computational basis. From a complementary perspective, as mentioned above, the
organizational structure of tensor networks suggests a necessary condition
that entanglement grows more rapidly than logarithmically with the system size
in order to support exponential computational advantages Vidal (2003); Schuch
_et al._ (2008). Thus, to truly understand the role of experimentally
controlled entanglement in expanding the accessibility of non-equilibrium
properties and dynamics in QMB and QFT systems, it will be necessary, as
demonstrated in the growing literature on entanglement measures, to push well
beyond binary classifications of separability to understand how to work with,
rather than against, the complex and subtle structure of entanglement.

One of the challenges we face, as theorists _at the interface_ , comes from
the fact that much of our QFT tool-set developed in the $20^{\rm th}$-Century
is based around the locality of interactions and the enormous success of the
operator product expansion in understanding high-momentum processes used to
establish the SM. Our experience with systematically improvable calculation
frameworks for non-local quantities is limited, one example of which is the
inclusion of QED in finite-volume lattice QCD+QED calculations, see e.g.,
Refs. Uno and Hayakawa (2008); Borsanyi  _et al._ (2015); Fodor  _et al._
(2016); Hansen  _et al._ (2018); Davoudi  _et al._ (2019). This points to a
necessity of learning new ways to think about our calculations, including
developing intuition about the inclusion and organization of non-localities
and entanglement. Einstein is quoted to have said “To raise new questions, new
possibilities, to regard old problems from a new angle, requires creative
imagination and marks real advance in science.” From this new angle of quantum
information, the necessary creative imagination can be sourced naturally from
the reality of nature at the microscopic scale. Celebrating the versatile
impact that such non-perturbative inspiration can provide, it is likely that
the new perspectives developed in the process of exploring quantum simulation
will provide valuable insights throughout existing techniques and algorithms,
both quantum and classical e.g., Ref. Osborne and Nielsen (2002).

##  III The Vision of Quantum Simulation

As the rules of quantum mechanics remain unchanged over vast energy scales,
the precise control of atomic-scale quantum devices is expected to provide
naturally matched complexity for expressing quantum systems from QCD to
chemistry. This notion of intersimulatability is fundamentally connected with
the concept of universality of a computational device: a universal computer is
capable of efficiently simulating the behavior of a different computer
belonging to the same class, or one contained therein. As such, important
aspects of QFT and QMB systems are anticipated to become accessible when
incorporating quantum degrees of freedom directly into computational
architectures.

###  III.1 “Gedanken” Scaling of Quantum Resources

Feynman introduced the concept of a universal quantum simulator as a
computational device constructed with quantum mechanical degrees of freedom
(such as spins) that would be capable of simulating the behavior of compatible
different quantum systems Feynman (1982, 1986). The question that remained
open was how many classes of universal quantum simulators are there or, in
other words, what controls the intersimulatability of different quantum
systems? To make this question more concrete, a physical quantum system can be
approximately encoded in a finite number of discrete degrees of freedom to be
representable with a finite number of computing elements. The efficiency of a
simulation is then determined by the scaling of required resources with
increasing system size. In particular, efficient computational strategies are
those with resources increasing at most as a polynomial in the size of the
system. The physical intuition followed by Feynman was that a quantum
simulator constructed with locally-interacting quantum degrees of freedom
should be able to efficiently simulate the behavior of a large class of other
quantum systems also described by locally-interacting quantum degrees of
freedom. This intuition was finally proved correct by Lloyd Lloyd (1996), who
was able to show that, indeed, mutual locality allows mutual simulation. The
key insight was to configure the quantum computer to approximately simulate
the dynamics over short time-intervals while controlling the total error
accumulated during the entire evolution. The simulation strategy can
accommodate fermion statistics, and requires a compute time that is
appealingly proportional to the total time of the internal simulation. Prior
to this milestone result, Shor’s algorithm for factoring Shor (1994) was the
main candidate shown to provide a future quantum advantage, naïvely quite
distant from Feynman’s initial vision.

Efficient simulation strategies on universal quantum computers have been
formulated for a variety of SM systems from quantum chemistry Wecker  _et al._
(2014); Reiher  _et al._ (2017) to fermion-lattice models Wecker  _et al._
(2015); Kivlichan  _et al._ (2018, 2020), including pionless EFT Kaplan  _et
al._ (1998a, b); van Kolck (1999) on a lattice Roggero  _et al._ (2020a), and
some relativistic QFTs Byrnes and Yamamoto (2006); Jordan  _et al._ (2012,
2014); Shaw  _et al._ (2020); Mathis  _et al._ (2020). Once again, physical
intuition suggests that to simulate a quantum system with $n$ degrees of
freedom interacting locally for a time $T$, one would need a quantum device
with $\mathcal{O}(n)$ degrees of freedom and perform $\mathcal{O}(T)$ time
steps, for a total of $\mathcal{O}(nT)$ operations. For a system on a lattice,
this is equivalent to the expectation that the time needed for simulation
would be independent of the volume of the system. First suggested in Ref.
Jordan  _et al._ (2014), this idea was ultimately shown to be correct with a
further proof that the strategy has optimal asymptotic scaling with the number
of operations Haah  _et al._ (2021). The main caveat is that, in order for the
simulation time to be independent of the volume, the degrees of freedom in the
quantum simulator should be organized on a lattice with the same
dimensionality as the simulated system, e.g., a 2-dim array of spins
interacting locally will not in general be sufficient to simulate a 3-dim
system with local interactions in time $\mathcal{O}(T)$ that is independent of
$n$. The introduction of fermions usually leads to geometrically non-local
interactions and, in general, simulation run times for quantum algorithms are
expected to scale as $\mathcal{O}(poly(n)T)$, still an exponential speed-up
with respect to classical simulation strategies.

###  III.2 Quantum Algorithm Development: “Gedanken” to Reality

The first proposal for a universal quantum simulation algorithm in Ref. Lloyd
(1996) used the Lie-Trotter Trotter (1959) approximation for the time
evolution operator, which requires a number of operations scaling as
$\mathcal{O}(T^{2}/\epsilon)$ (per measurement) in order to ensure a maximum
error $\epsilon$ in a simulation of time $T$. Higher order Suzuki-type
integrators Suzuki (1991) enjoy a better scaling
$\mathcal{O}(5^{k}T^{1+\frac{1}{k}}/\epsilon^{\frac{1}{k}})$ with
$k\in\mathbb{N}$ the approximation order. Due to the exponentially increasing
prefactor, these approximations are typically used only with $k=1,2$ (e.g.,
Ref. Childs  _et al._ (2018) for a more detailed discussion). More recently,
alternative approaches to simulating real-time evolution with a reduced
dependence on $\epsilon$ have been proposed, enabled by increased qubit
overhead. Important examples are techniques based on the linear combination of
unitaries (LCU) Childs and Wiebe (2012) and the Taylor expansion Berry  _et
al._ (2015a, b), which achieve an exponential improvement in the error scaling
as $\mathcal{O}(T\frac{\log^{2}(1/\epsilon)}{\log\log(1/\epsilon)})$, and also
techniques based on Qubitization and Quantum Signal Processing (QSP) Low and
Chuang (2017, 2019), which are able to achieve the optimal scaling
$\mathcal{O}(c_{T}T+c_{\epsilon}\log(1/\epsilon))$. Better than
$\mathcal{O}(T)$ scaling is forbidden by the no-fast-forward theorem unless
some of the structural properties of the Hamiltonian are used in the design of
the simulation scheme Berry  _et al._ (2007); Atia and Aharonov (2017), i.e.
codesign. It is important to point out that we do not expect to be able to
implement generic unitary transformations Knill (1995) and that being able to
construct a qubit or qudit Hamiltonian describing a physical system is a
necessary but not sufficient condition for the existence of an efficient
simulation strategy (see, e.g., Ref. Childs and Kothari (2010)).

Besides second quantization schemes, an alternative mapping can be obtained
using first quantization language where the particle’s statistics are included
by explicit symmetrization Abrams and Lloyd (1997); Aspuru-Guzik  _et al._
(2005). The main advantage of working in this basis presents itself in
situations where the number of particles $\eta$ in the system is conserved and
discretization over a large number of fermionic modes $N$ is required—in
second quantization the required number of qubits is at least $n=N$ (using
e.g., the JW or BK mappings described in Sec. V.1.1 below), while in first
quantization this can be reduced exponentially to $n=\mathcal{O}(\eta\log(N))$
instead Babbush  _et al._ (2019) when $N\gg\eta\gg 1$. This reduction in
Hilbert space complexity is typically accompanied by an increase in time
complexity and, for some important situations, the two descriptions have
comparable simulation performance (see, e.g., Ref. Su  _et al._ (2020)).
Analogous symmetry or conserved quantity projections are also under
development in simulations of gauge field theories (see Section V.1.3), and
are found to increase classical preprocessing requirements.

More recently, growing interest in reducing the number of quantum gates
required for realistic simulations has led to an increasing appreciation of
the potential benefits of including stochastic components to simulation
algorithms. Notable ideas in this direction are, for example, randomizing the
order of exponentials in the Trotter expansion of the evolution operator,
resulting in effective higher-order integrators Childs  _et al._ (2019), and
utilizing the fully stochastic compilation of Trotter evolution with the
quantum stochastic drift protocol (QDRIFT) Campbell (2019) and its extensions
Chen  _et al._ (2020); Faehrmann  _et al._ (2021); Berry  _et al._ (2020).
Further developments toward realistic simulations have leveraged EFT
renormalization group (RG) ideas to prioritize the low energy subspace in
dynamical simulations Şahinoğlu and Somma (2020).

Due to the finite gate fidelities of real quantum devices, an inverse
relationship exists between the number of qubits in a time-evolved system and
the number of Trotter steps feasible within the “gate fidelity coherence
time”. As discussed by Martonosi and Roetteler Martonosi and Roetteler (2019),
different algorithms for simulations with different target objectives will
perform optimally on different architectures. In particular, the scaling of
the number of gates with increasing system size for a given algorithm will
determine performance on near-term devices with given error rates. The
important connection between gate errors and number of qubits for pursuing
computation has led to the exploration of more holistic metrics than qubit
number or error rate alone e.g., the quantum volume Moll  _et al._ (2018).
This metric characterizes the performance of “square” circuits e.g., if a 5
qubit depth 5 circuit provides reliable results (so defined from a random
quantum circuit), but a 6-qubit, depth 6 circuit does not, then the quantum
volume is $2^{5}=32$.

Figure 1:  Schematic representation of simulation configurations defined in
terms of their logical (right panel) and physical (left panel) resource
requirements on the device used for implementation. See the text for further
discussions of this diagram and example problems belonging to each region. For
this diagram, NISQ devices are considered without error correction. This
figure is inspired by those of Refs. Martonosi and Roetteler (2019); Kelly
(2018).

In light of the possibility of using error correction schemes to encode
quantum information in a redundant way allowing for a direct reduction of the
effective error rate afflicting a given simulation, it’s important to realize
that quantum algorithms requiring seemingly very different logical quantum
resources could be implemented successfully on the same general purpose
quantum machine. In order to illustrate this point, we show in the right hand
panel of Figure 1 three indicative configurations of computational problems
(denoted with labels $A$,$B$ and $C$) requiring different logical resources.
The gray region indicates instead the regime in logical resources we expect to
be available in the near term NISQ era: a few hundred to a thousand logical
qubits and a few thousand logical operations. By means of encoding, the
problem classes defined at the logical level can be mapped to wide regions in
terms of physical resources. Provided the error rate in the physical device is
smaller than the threshold for error correction (indicated with a dashed blue
line on the left panel of Figure 1), it is possible to trade a larger number
of physical qubits for lower error rates. As a concrete example, the
configurations of problems denoted as $A$ in Figure 1 could be implemented
successfully on both a small quantum device with $\mathcal{O}(10^{2})$
physical qubits and very small error rates $\mathcal{O}(10^{-12})$ or
equivalently on a large quantum device with $\mathcal{O}(10^{5})$ physical
qubits and much larger error rates $\mathcal{O}(10^{-4})$. Due to the
formidable engineering challenges in reducing the limiting noise level to
arbitrarily low levels, a commonly adopted strategy in the pursuit of
universal quantum computing devices e.g., as discussed by technology companies
Kelly (2018), is to first achieve error rates a few order of magnitude below
the error correction threshold, and then develop larger qubit arrays that will
enable scalable fault tolerance. A representative path for this hardware
development is shown as the dotted black line in Figure 1.

While estimates of classical computing resources are well established in many
areas, such activities for quantum computing are just beginning. Research
teams have undertaken extended periods of resource estimation and coordinated
algorithm development tied to present-day simulation capabilities e.g., Refs.
Wecker  _et al._ (2014); Reiher  _et al._ (2017); Babbush  _et al._ (2018);
Lee  _et al._ (2020a), to reduce naive quantum resource requirements from, in
some cases, billions of device-years to a handful of device-days. One of the
common standards for performance measures and resource estimation is the
Hubbard model and other spin systems with “simple” interactions, e.g., Ref.
Childs  _et al._ (2018). At this point is important to highlight the fact that
problem instances depicted in both panels of Figure 1 are necessarily narrowly
defined problems with clearly stated observables and target precision goals,
examples of problems belonging to the 3 configurations depicted are:

  * •

configuration A: computing the ground-state fermion condensate in the Lattice
Schwinger model with $N=16-64$ lattice sites and a electric field cutoff
$\Lambda=2-8$ using an optimized scheme without amplitude estimation Shaw  _et
al._ (2020)

  * •

configuration B: estimating the frequency dependent, inclusive response
function with resolutions $\Delta\omega=10-100$ MeV, at one value of the
momentum transfer for a system of $A=\mathcal{O}(50)$ nucleons described using
Lattice EFT Lee (2009) with a minimal basis defined on a $10^{3}$ lattice
Roggero  _et al._ (2020a)

  * •

configuration C: minimal instance of a problem in financial derivative pricing
showing a quantum advantage Chakrabarti  _et al._ (2021). The estimate is that
a gate depth of $54\times 10^{6}$ on 8K logical qubits will be required.

With rapidly increasing focus on developing quantum algorithms to address
scientific applications, and on dedicated and general purpose hardware,
benchmarks and estimates of quantum resources requirements are expected to
evolve rapidly. In some instances, the estimated resources will increase as
our knowledge improves and naïvety recedes, but reductions in resource
requirements for quantum advantages in scientific applications are anticipated
as methodologies become more sophisticated and devices advance.

The flexibility enabled by error correction mappings inspires the viability of
engineering universal quantum devices able to tackle a variety of problems
with widely different computational requirements at the logical level. For our
three problem configurations, for example, a quantum device characterized by
the yellow pentagon in the left panel on Figure 1 would be able to execute
instances from all three configurations by using various levels of error
correction. On the other hand, we expect near term applications in the NISQ
era to use little or no error correction and the gray region of logical
resources in the right panel of Figure 1 will be mapped, almost unaltered,
onto the physical resources. One complementary route to early success in
obtaining a quantum advantage for a problem of scientific interest is the
adoption of ideas of codesign to develop special purpose architectures
tailored to specific problems 666 In the development of high-performance
classical computing in the US, the QCDSP and QCDOC systems Christ (2011) were
specifically codesigned for lattice QCD, with on-chip memory and low-latency
communication fabric. These systems constituted the precursor to IBM’s
successful Blue-Gene series of HPC systems. There were parallel developments
in Europe, with the APE machines Marinari (1986). . Codesign of special
purpose machines has the potential of drastically impacting the mapping
between logical computational resources and physical ones displayed in Figure
1. For instance, it could be possible to design a quantum device with the
physical properties denoted by the yellow star in Figure 1, which would be
able to perform simulations of problems belonging to configuration $A$ more
efficiently but unable to tackle problems belonging to other configurations.

The arrival of heterogeneous classical architectures led to an extended period
of time during which scientific applications were ported from CPU-only to CPU-
GPU hybrids, and now, most of the performance resides on the GPUs and not
CPUs. We anticipate the same sort of transition in the introduction of QPUs
into heterogeneous computing Pogorelov  _et al._ (2021). Even assuming that
there is a polynomial speed up to parts of the calculations, there is benefit
in identifying parts of calculations that would be accelerated by a QPU. In
particular, computations of highly entangled subsystems could be performed on
QPUs, while the classically efficient elements of the computational workflow,
and re-assembly of the results from the QPUs, could be performed with
classical computers. Analogous to the widespread adoption of GPU frameworks in
scientific computing aided by the standardization of the hardware interface,
standardization at both the low-level API and abstracted hardware-specific
compilers will provide portability and high-level approaches to quantum
algorithmic design. A variety of different framework and standards are
currently being explored e.g., Cirq team and collaborators (2020), Qiskit
Aleksandrowicz  _et al._ (2019), Qiskit Pulse Alexander  _et al._ (2020),
OpenQASM Cross  _et al._ (2017), xacc McCaskey  _et al._ (2020), Q# Microsoft
Q# Language Design Team .

Systematic uncertainty quantification generally requires an ensemble of
simulations with strategically chosen input parameters such that extrapolation
to asymptotic configurations beyond the capability of a single simulation
configuration (a hero run) can be achieved. This ensemble maps out a surface
in parameter space that enables extrapolations and interpolations to reliably
estimate observables of interest and their uncertainties e.g., Ref. Li and
Benjamin (2017); Temme  _et al._ (2017); Kandala  _et al._ (2019), injecting
another layer of analysis to resource estimation. As is the case for classical
calculations and experimental design, designing an optimal manifold of
simulations to achieve the desired precision in observables requires planning
and simulations of the simulations 777 While hero runs are not the pure focus
of simulation, they may be the relevant mode of operation for other quantum
applications..

Part of designing the simulation workflow involves identifying and optimally
interleaving such calibrations with the physics simulations. While it is
obvious that the nature of the calibration is hardware dependent, they will
also depend upon the structure of the quantum circuits being executed.
Simultaneous adoption of a suite of error mitigation protocols Li and Benjamin
(2017); Temme  _et al._ (2017); Endo  _et al._ (2018); Kandala  _et al._
(2019); Dumitrescu  _et al._ (2018) has been used to assess the reliability of
the extraction of physical quantities and their sensitivity to the specific
noise properties of the device being utilized, e.g. Ref. Roggero  _et al._
(2020a). As NISQ-devices currently resemble experiments more than classical
computers, device calibrations that are temporally correlated (integrated)
with physics calculations are valuable to in vivo monitor and subsequently
identify physics data accumulated during periods in which the device is
performing within specified parameters. An example of such “cuts” is provided
in Ref. Klco and Savage (2020a), where calibration circuits that measured the
in-medium fidelity of a Hadamard gate were included to provide contextualized
performance information in initializing symmetric wavefunctions.

Another important aspect of workflow depends upon the (user defined)
partitioning of the simulation into classical and quantum components,
dependent on the performance of the QPU relative to the CPU+GPU, and the
efficiency of communication between them. The current state of the NISQ
ecosystem mandates that scientific applications must include a diverse range
of integrated controls and diagnostics of the quantum hardware, both
accessible and inaccessible to the users. In addition to making available the
first cloud-accessible quantum computers, IBM Q Experience IBM  provides
regular calibration data for each of their superconducting quantum systems,
built-in functionality for some error mitigation protocols, and access to
pulse-shaping for gate optimization through their Qiskit Aleksandrowicz  _et
al._ (2019) programming environment—demonstrating a successful path (and
potential template) for co-design collaborations with universities and
national laboratories toward scientific applications.

##  IV Complexity Classes in Context: Asymptopia Versus Reality

###  IV.1 Asymptopia: The Utopia of Infinite Resources

No discussion of quantum resources would be complete without considering
complexity classes in the formal sense of computer science, see e.g., Refs.
Watrous (2008); Bernstein and Vazirani (1993); Cleve ; com (2013). The scaling
of computational resources required to determine a given observable with
increasing system size for asymptotically large systems and resources defines
the complexity class within which the observable resides. For instance, the P
class (PTIME) contains all problems that require computational time scaling as
a polynomial of systems size (on a deterministic Turing machine), and is
loosely defined by the set of problems that can be solved efficiently. In some
cases, the assignment of problems to complexity classes depends on the
performance of known algorithms, though not all classifications provide
constructive guidance. Because classical complexity classes, defined with
respect to Turing machines (deterministic or non-deterministic), do not
capture the scaling of problems addressed with quantum devices, additional
_quantum_ complexity classes have been introduced. For example, BQP is the
complexity class containing problems with solutions of bounded-error requiring
polynomial-scaling time using quantum resources. Since a quantum circuit can
simulate a classical one, both P, and it’s probabilistic generalization BPP,
are contained in BQP. Jordan, Krovi, Lee and Preskill Jordan  _et al._ (2018)
have shown that scattering within interacting scalar QFT with external
classical sources lies within BQP-complete, indicating that all problems in
BQP can be mapped to scattering in scalar QFT with polynomial-scaling time to
solution, and further that the scattering problem itself is in BQP and thus
efficiently simulatable quantum mechanically.

While it would have been convenient if all scientifically interesting problems
were in BPP or BQP, the unfortunate fact 888This fact may be rather fortunate
if this complexity is essential to support life. is that they are likely not.
Many important problems lie in NP (nondeterministic polynomial time),
requiring beyond polynomial resources to solve (assuming P $\neq$ NP) but only
P resources to verify solutions. The quantum generalization of NP with
polynomially intractable problems is denoted by QMA, and problems belonging to
this class are not expected to be solved efficiently even with a quantum
computer (unless BQP$=$QMA). Similar to the NP-completeness of finding the
ground state energy of a classical Ising spin glass due to an exponentially
vanishing gap to the first excited state (see e.g., Ref. Barahona (1982);
Istrail (2000); Troyer and Wiese (2005)), the problem of determining the
ground state energy of a $k$-body Hamiltonian is QMA-complete for $k\geq 2$
Kempe  _et al._ (2005). A $k$-body Hamiltonian (often referred to as a
$k$-local Hamiltonian in the computer science literature, though no sense of
spatial locality is intended 999 The most interesting aspects of $k$-localcs
(as defined by computer scientists) systems to physicists are likely the
$k$-nonlocalphys attributes induced by entanglement.) is defined as an
operator that admits a decomposition into a polynomial number of terms acting
individually at most on $k$-qubits. Interestingly, the $2$-body Hamiltonian
problem was shown to be QMA-complete Kempe  _et al._ (2005) also when
restricted to spatially local interactions on a 2D grid. These results show
that it will not be possible to solve efficiently for ground-state energies of
even simple Hamiltonians in general, similar to the NP-hardness of a general
solution to the sign problem in Quantum Monte Carlo methods Troyer and Wiese
(2005).

It is important to point out that membership to a particular complexity class
is a statement about the worst-case instance of a particular problem, and does
not necessarily reflect the average- or best-case. A great example of this
dichotomy between average case complexity and worst-case complexity is the
Minesweeper game: in this game, players are given a 2D lattice with $L^{2}$
sites containing $M$ mines. When a site without a mine is uncovered, the site
is assigned a numerical value corresponding to the number of neighboring
mines. The goal is to uncover all sites not containing mines using inference
from an initial set of uncovered sites. This problem has been shown to be NP-
complete Kaye (2000) (or more precisely co-NP-complete Scott  _et al._ (2011))
and therefore extremely hard to win in the worst case scenario. For those who
have played Minesweeper, this result may clash with practical experience, in
which the game can often be solved very quickly. An important structural
property of the Minesweeper game is the mine density $\rho=M/L^{2}$, and it
was recently argued that a complexity phase transition occurs once the mine
density exceeds a critical value $\rho_{c}$ Dempsey and Guinn (2020). At small
mine density, the game is typically easy and inference can be carried out
locally on the lattice. As the critical mine density is approached, successful
inference of the mine’s location requires consideration of lattice patches
approaching the full volume of the lattice, causing solutions to become
expensive to find. This “frustration effect” shows up in certain regimes of a
class of NP-complete problems, for which a prototypical example is the phase
transition in random k-SAT problems Monasson  _et al._ (1999). This discussion
draws obvious analogies with chiral symmetry breaking, confinement, and the
QCD phase diagram, where transitions are accompanied by the delocalization of
natural degrees of freedom capturing the microscopic properties of quarks and
gluons.

###  IV.2 Bounded Error: Uncertainties and Approximate Simulations

To effectively connect quantum simulations of SM observables to experiment, a
complete quantification of uncertainties is required. Lattice QCD calculations
have established well-defined uncertainty quantification methodologies (for a
review, see Ref. Beane  _et al._ (2015)), in addition to demonstrating the
benefits of independent teams of researchers and code developments, and
comparable access to independent computing environments. Lattice QCD is a low-
energy EFT of QCD that can faithfully reproduce low-energy QCD observables.
Such simulations involve discretizing volumes of spacetime, introducing a
lattice spacing, and bounding a spatial extent in each of the four spacetime
directions. Therefore, observables will necessarily deviate from their QCD
value. In addition, as simulations only involve dimensionless quantities, an
overall length scale is required to be determined, along with input values for
the quark masses. This is typically achieved by making comparisons with a
small number of precisely known experimental quantities, such as hadron
masses. Such tunings are imperfect and iterative, and introduce further
deviations from QCD values. EFTs, such as $p$-regime chiral perturbation
theory ($\chi$PT), play an essential role in post-computational processing of
simulation results to make reliable predictions of QCD with fully quantified
uncertainties (see, e.g., Ref. Golterman  for mesons and Ref. Savage (2006)
for nuclei and multi-nucleon scattering systems). For small enough lattice
spacings, compared with the QCD scale and external kinematics, the Symanzik
action Symanzik (1983a, b) includes the structure of the gauge-invariant
operators that may be introduced into the simulation (dependent upon the
nature of the field discretization) to compensate for the finite lattice
spacing in each direction, or alternately dictate the nature of the power-
series in the lattice-spacing(s). While the quark-mass dependence of a number
of low-energy observables is known from $\chi$PT, or HQET, calculations are
typically performed in the neighbourhood of the physical point (or the mass
points of interest) and interpolations are performed to mitigate tuning
errors. On top of systematic errors, there are statistical errors and the
range of behaviors of observables with sampling. The extraction of energies
and observables requires sufficient statistics (sampling of gauge
configurations) to recover correlation functions with sufficient precision to
be able to fit relevant quantities with controlled estimates of mean values,
statistical uncertainties and systematic errors.

For quantum simulations, there are further systematic errors associated with
the quantum hardware and the qubit representation of the simulated system.
Mapping bosons onto a finite Hilbert space introduces digitization errors and
field truncation errors, to be quantified and/or mitigated. There are errors
associated with state-preparation (similar to source structure in Euclidean
space, which may inform quantum simulations), and measurement, and both are at
early stages of being understood and quantified. Other sources of error that
must be quantified are device noise and field pixelation (operator smearing)
on quantum correlations (i.e., the structure of entanglement between separated
regions of a latticized field, e.g. Ref. Klco and Savage (2021a, b)). The
systematic and statistical uncertainties associated with a quantum simulation
(determined by the device, algorithms and theoretical framework) can be
written as,

$\epsilon^{(t)}\sim\epsilon_{\rm
th.}+\epsilon_{a}+\epsilon_{L}+\epsilon_{\theta}+\epsilon_{M}+\epsilon_{\rm
Meas.}+\epsilon_{\rm ops.}+\epsilon_{\rm
digital}+\epsilon_{\Lambda}+\epsilon_{\rm Trotter}+\epsilon_{\rm Noise}\ ,$
(3)

where $\\{\epsilon_{\rm
th.},\epsilon_{a},\epsilon_{L},\epsilon_{\theta},\epsilon_{M},\epsilon_{\rm
Meas.},\epsilon_{\rm ops.}\\}$, associated with the theory, lattice spacing,
volumes, parameter tuning, scale setting, measurements and operator
structures, are familiar (but some will be handled differently) from classical
simulations. The $\epsilon_{\rm digital},\epsilon_{\Lambda},\epsilon_{\rm
Trotter},\epsilon_{\rm Noise}$ associated with field digitization and
mappings, truncations in field space, Trotterized time evolution, and device
noise, are further sources of uncertainties that need to be quantified.

The statistical behavior of an observable, and how it is constructed from
ensemble measurements, is another important feature of algorithmic design.
With direct access to the wavefunction of a system not practical (tomography
scaling exponentially with system size), optimizing observable estimators is
essential for efficient use of quantum simulation (as it is classically).
Cancellations between contributions to expectation values can create sign
problems at the measurement stage of a calculation, exacerbating the number of
circuit executions required to achieve a given precision e.g., Ref. Roggero
and Baroni (2020). Because the statistical structure of the measurements is
mapping (basis) dependent, it is possible that circuit design and algorithms
with less favorable asymptotic scaling may provide compensating enhanced
statistical convergence.

###  IV.3 Reality: Hacking Bounded-Error Quantum Simulations

The complexity classes of SM observables will determine the observables that
can be computed with arbitrary precision using combinations of classical and
quantum resources. However, if they lie outside P or BQP, it does not mean
that significant progress, and potentially all of the required progress,
cannot be made in addressing important problems, gaining significant physical
insights and even proposing new experiments. Explicitly, some observables
formally in P may scale with a sufficiently high-order polynomial that they
are unreachable with the requisite precision for foreseeable computers, while
some in NP or QMA may scale benignly for small or modest system sizes,
enabling computations with “sufficient” precision to have near-term or long-
term impact. The impact of any given simulation is limited by its precision,
not only of the computation, but also of the theory input into the simulation.
Scientific simulations have progressed organically with simulation providing
results of precision $\epsilon^{(t)}$ (where the superscript $t$ denotes total
uncertainty at the time of the simulation). Once $\epsilon^{(t)}$ is
specified, there is freedom to define approximate (perturbatively-close)
theoretical inputs and tolerances of the computation, to furnish a result that
lies within $\epsilon^{(t)}$ of the true result. So, for bounded-error
simulations, with a target tolerance $\epsilon^{(t)}$, as is the case with
quantum devices, the input theory (model or EFT) can deviate from the target
theory, provided that the induced errors are $\leq\epsilon^{(t)}$. This builds
upon the recognition that only a subset of possible quantum observables are
required to be simulated e.g., low-energy scattering amplitudes and not
amplitudes up to the Planck scale, which leads to the employment of EFTs and
phenomenological models 101010 An implicit distinction in motivations for
developing simulations of arbitrary Hamiltonians and domain science
applications. .

Leveraging lower-complexity leading order interactions has enabled classical
simulations to make significant progress in addressing problems formally
beyond the reach of classical computing. The SM has many examples of QMB and
QFT systems that are amenable to such perturbative expansions centered on
lower-complexity leading orders:

  * •

Density Functional Theory (DFT) is a highly successful framework for nuclear
structure and reactions (see e.g., Ref. Colò (2020)). The foundational result
upon which DFT is based is the Hoenenberg-Kohn theorem Hohenberg and Kohn
(1964), which states that the problem of determining the ground state energy
of QMB or QFT systems can be replaced by an equivalent optimization problem
over one-body density matrices $\rho$ using an appropriate functional
$F[\rho]$ as a cost function. Central to the accuracy of the method is the
availability of accurate approximations to the functional $F$ since, once this
is available, the optimization can be usually carried out efficiently in P. At
first glance, this seems at odds with the observation that estimating ground
state energies of even simple $2$-body Hamiltonians is QMA-complete. Indeed,
it is possible to show that, even though a DFT calculation is usually
tractable once the functional $F$ is known, finding an accurate functional is
a QMA-complete problem Schuch and Verstraete (2009). However, with the
guidance of physical understanding and experiment, approximate functionals of
sufficient quality to reproduce observables like binding energies within a
given accuracy can often be found—nature hacking complexity. The QMA
classification in this case is thus a statement about the intrinsic difficulty
of designing functionals capable of arbitrary precision in general.

  * •

Low-energy nuclear forces approximately manifest Wigner’s emergent SU(4) spin-
flavor symmetry Wigner (1937a, b) (which also emerges from large-Nc
considerations Kaplan and Savage (1996), vanishing entanglement power
considerations Beane  _et al._ (2019), and from lattice QCD calculations
Wagman  _et al._ (2017)). Quantum Monte Carlo simulations of multi-nucleon
system ground states, e.g., using the pionless EFT, generally exhibit a sign
problem. Because the sign problem vanishes in the limit of SU(4) symmetry,
successful hybrid schemes exist involving “pre-conditioning” systems with the
SU(4) symmetric interactions followed by implementation of the full
interaction, mitigate the impact of the sign problem Lee (2016). This process
identifies a leading order part of the QMA calculation that is in BPP, and
perturbatively estimates the full result to fixed precision.

  * •

Nuclear structure has been tremendously impacted by Green’s Function Monte
Carlo (GFMC) calculations of light nuclei (see Ref. Carlson  _et al._ (2015)
for a review). Despite requiring exponentially increasing resources with
baryon number, for an extended period of time, GFMC was able to provide
accurate results for system sizes not reachable with polynomially-scaling
algorithms, such as Coupled Cluster. This balance of practical computational
power has almost entirely reversed in recent years due to the adoption of low-
resolution interactions (derived from chiral-EFT Weinberg (1990, 1991);
Ordonez  _et al._ (1996); Kaplan  _et al._ (1996, 1998a, 1998b); Epelbaum  _et
al._ (1998); Machleidt and Entem (2011)) and RG techniques Kaplan  _et al._
(1998a, b); Bogner  _et al._ (2003) (see, e.g., Ref. Hergert (2020) for a
recent review).

  * •

Current lattice QCD calculations indicate that:

    * –

Computing the mass of the pion to a given precision can be accomplished
efficiently at scale, consistent with residing in BPP

    * –

The exponentially degrading signal-to-noise at late times in nucleon and
multi-baryon correlation functions Parisi (1984); Lepage (1989); Beane  _et
al._ (2008) (a sign problem) currently suggests Troyer and Wiese (2005) that
the Quantum Monte Carlo computation of the binding energy of nuclei lies
outside of P and BPP.

Through its connection to the Hubbard model, finding the ground state of the
pionless EFT is among the hardest problems in QMA Schuch and Verstraete
(2009); O’Gorman  _et al._ (2021), and therefore, it is likely that lattice
QCD calculations of nuclei lie in QMA or beyond in PSPACE. However,
calculations of energies and properties of smaller nuclei, which have been
demonstrated Beane  _et al._ (2013), permit the bounded-error extraction of
counterterms in nuclear EFTs Barnea  _et al._ (2015); Kirscher  _et al._
(2015); Contessi  _et al._ (2017); Bansal  _et al._ (2018). These results may
be then used to compute the energies, properties, and low-energy dynamics of
somewhat larger nuclei Savage (2006) (for reviews on this subject, e.g., Ref.
Beane  _et al._ (2008, 2011); Savage (2016); Drischler  _et al._ (2019);
Davoudi  _et al._ (2021a); Aoki and Doi (2020)).

  * •

Heavy-quark systems (exhibiting heavy-quark symmetry for
$m_{Q}\rightarrow\infty$) are comprised of one or more heavy quarks, $Q$, and
light degrees of freedom. Leading order wavefunctions of
$Q\overline{l}$-mesons are a tensor product,
$|Q\overline{l}\rangle\sim|Q\rangle\otimes|\overline{l}\rangle$. The higher-
order $1/m_{Q}$ contributions to observables can subsequently be included in
perturbation theory from this wavefunction. Powerful theorems, such as the
Adellmolo-Gatto theorem Ademollo and Gatto (1964) and Luke’s theorem Luke
(1990), ensure that leading order results in the symmetry breaking parameters
are known without model dependence, and that symmetry breaking in certain
matrix elements is suppressed by two powers of the breaking parameter.
Further, any model dependence or parameters that need to be computed elsewhere
enter at higher orders in the perturbative expansion(s), and associated
uncertainties are thus parametrically smaller than leading order.

  * •

The parton model characterizes hadronic structure and processes at high
momentum transfer. The scale invariance of QCD in this regime, due to
asymptotic freedom at short distances, allows many QCD processes to factorize
at leading order in pQCD, providing a foundation for the systematic inclusion
of higher order corrections. In such situations, classical computing has been
proven effective in computing higher-order matrix elements with the required
precision to advance experimental programs.

For quantum simulations, this hacking of bounded-error complexity classes
through perturbative changes to the theory is only now beginning. For example,
precision first principles calculations of fragmentation lie beyond the reach
of classical computing. It is also expected to lie outside of BQP, due to
final state complexities, but perturbatively addressable with simulations that
lie within BQP. The use of Soft-Collinear EFT (SCET) Bauer  _et al._ (2000,
2001); Bauer and Stewart (2001) may enable simulations within desired error
bounds (though this remains to be demonstrated) Bauer  _et al._ (2021a). For
such hacks to be successful, problems should be broken up into a large part
that resides within BQP and can be performed efficiently on a quantum device,
and a perturbative part (containing the terms that place the problem outside
of BQP).

###  IV.4 Bottom Line: Enhancing Leading Order Complexity

Remarkable progress has been made toward understanding nature through the SM
even for problems that reside in complexity classes formally beyond the
capability of classical or quantum computers. This progress has been enabled,
to a large degree, by _points of enhanced symmetry exhibiting reduced
complexity_. In such instances, nature has been “kind” to us by providing
Hamiltonians that can be separated into “large” and “small” parts for the
observables of interest. This separation has been enabled by the
identification of approximate symmetries of the systems, with the “large”
part(s) invariant under the symmetries, and the “small” part(s) breaking the
symmetries. Generally, and in hindsight, these symmetries are identified by
eliminating entanglement and identifying perturbatively close tensor-product
states. However, examples of expanding about leading-order entangled states
are known Kaplan  _et al._ (1998a, b); van Kolck (1999), resulting from a non-
trivial fixed-point in RG flow. For classical computing, the challenge has
been to find symmetries that place leading order approximations into P or BPP
with a controlled perturbative expansion that sufficiently converges. For
quantum computing, the leading order symmetries need not eliminate
entanglement. Instead, incorporation of quantum devices will allow the
advantage of addressing important QMA-complete SM applications from the
standpoint of leading order approximations residing in BQP. A nice example of
this strategy can be found in low-energy nuclear structure where nucleon-
nucleon correlations can be self-consistently included into a more effective
single nucleon basis Robin  _et al._ (2021), perhaps distilling the leading-
order consequences of entanglement from a QMA-complete problem into one that
is in BQP. Broadly, identification of the leading order entangled systems of
enhanced symmetry presents an exciting challenge.

At this early stage of development of quantum simulation, asymptotic scaling
does not conclusively determine the reach of present day or near-term
calculations, including beyond the NISQ-era into the period where error-
correction becomes routine. While a better understanding of the formal
asymptotic scaling of classical and quantum resources required to address
challenges facing the SM is important, it will not be the sole consideration
in identifying goals for near-term quantum simulation, as important progress
may be achievable, possibly even with demonstrations of quantum advantages,
for problems that lie outside BQP.

##  V Quantum Techniques and the Standard Model

###  V.1 Mapping Quantum Fields to Quantum Degrees of Freedom

The mapping one chooses to represent a quantum system of interest with quantum
computational degrees of freedom will dramatically impact the quantum
resources required. In particular, the representation will affect the cost of
all steps in a quantum simulation from the entanglement structures created in
initial state preparation, to the interactions needed to perform time
evolutions, to the distribution of information desired to be probed in the
final measurement procedure. It is expected that the basis-dependent property
of entanglement may guide the development of intuition in mapping fields onto
quantum degrees of freedom (see Section II.2.6). Furthermore, exploring the
repercussions of various representations informs theoretical flexibility to
codesign simulations.

A basic question to consider in quantum simulation design (that has begun to
be addressed in the application to fields) is _Where is the Line_ , referring
to the flexible partitioning of computational responsibility between the
quantum device and the large-scale classical resources to which it will be
symbiotically married. One category of explorations shifting this line has
been in state preparation e.g., through the use of quantum devices to optimize
the design of interpolating operators for classical LGTs Avkhadiev  _et al._
(2020), through the use of classical snapshots of a volume set by the
correlation length scale to inform state preparation on quantum volumes
capturing many correlation length scales Klco and Savage (2020b, c), or
through purely quantum techniques inspired by the structure of classical
tensor network design Moosavian  _et al._ (2019). A second category of
explorations shifting _the line_ is in the basic representation of the field.
The influential papers of Jordan, Lee, and Preskill (JLP) illustrated an
efficient approach for expressing real-time dynamics of scattering events
quantum mechanically Jordan  _et al._ (2012, 2014). With polynomial resources
in particle number, energy, precision, and volume, a microscopic description
of interacting scalar field theory represented by local qubit degrees of
freedom was shown to produce final state probability distributions that
reflect, to systematically controlled precision, the exact probability
distributions that would be experimentally observed. Further inspiring the use
of quantum devices for this purpose, recent quantification of computational
complexity in tensor network simulation of inelastic false-vacuum bubble
scattering illustrated a growth of entanglement at high energies and multiple
scatterings Milsted  _et al._ (2020), clearly connecting quantum effects to
the voracious consumption of classical computational resources in scattering
processes. As algorithms for the quantum implementation of microscopic time
evolution operators continue to progress, a complementary perspective from EFT
descriptions is under development Provasoli  _et al._ (2019); Bauer  _et al._
(2021b, a), utilizing quantum devices to address non-perturbative physics
isolated in EFT descriptions of gauge field dynamics e.g., of parton showers,
high multiplicity final states, or soft functions of SCET. As a reliable
methodology for systematically isolating energy regimes, the demonstrated
synergy between EFTs and microscopic quantum simulations of fields is expected
to reduce quantum resource requirements by multiple orders of magnitude and
offer a clear route for integration of quantum devices into existing
computational frameworks of experimental impact. Conveniently, the dynamical
techniques developing for the microscopic simulation of gauge theories are
closely related (often identical) to those leveraged for the calculation of
real-time non-perturbative matrix elements within EFT descriptions. As such,
explorations considering different quantum-classical partition lines may be
explored in parallel, to mutual benefit, without concern of obsolescence
dependent on the trajectory of quantum hardware development.

####  V.1.1 Fermions (quarks, neutrinos, electrons, nucleons)

A standard approach for the mapping of fermion degrees of freedom onto qubits
is the Jordan-Wigner (JW) transformation Jordan and Wigner (1928), which
associates one qubit with each fermionic mode in the system. Fermi statistics
are enforced by mapping the creation/annihilation mode operators of the
fermions into products of Pauli matrices acting on multiple spins Ortiz  _et
al._ (2001); Somma  _et al._ (2002),

$c^{\dagger}_{i}=\prod_{k=0}^{i-1}Z_{k}\sigma^{+}_{i}\quad
c_{i}=\prod_{k=0}^{i-1}Z_{k}\sigma^{-}_{i}\ \ ,$ (4)

where the labelling underscores a global ordering of the fermionic modes. In
this notation, $Z_{j}$ is the Pauli-$Z$ matrix acting on the qubit
representing the $j^{\rm th}$ fermionic mode, while
$\sigma^{\pm}_{j}=(X_{j}\pm iY_{j})/2$ act as bosonic creation/annihilation
operators.

Due to the underlying 1-dim structure in the JW transformation, even for
locally interacting systems in more than 1-dim, the resulting qubit
Hamiltonian will typically contain long range interactions acting non-
trivially on $\mathcal{O}(n)$ qubits, which prevents the use of algorithms
that are optimal only for local theories e.g., Ref. Haah  _et al._ (2021).
This problem of induced non-locality caused by the mapping of fermions has
received a considerable attention. An alternative approach is the Bravyi-
Kitaev (BK) mapping Bravyi and Kitaev (2002); Seeley  _et al._ (2012) which,
similar to JW, requires the same number of qubits as fermion modes, but
reduces the induced non-locality exponentially to $\mathcal{O}(\log(n))$.
There are also various mappings employing additional auxiliary-qubits with the
goal of reducing this non-locality by exploiting the connectivity of the
underlying Hamiltonian Verstraete and Cirac (2005); Ball (2005); Havlíček  _et
al._ (2017); Steudtner and Wehner (2018); Derby and Klassen (2020). Notably,
some fermion mappings might present different advantages than the locality of
the resulting spin interactions e.g., the ternary tree construction from Ref.
Jiang  _et al._ (2020) allows for an efficient extraction of $k$-qubit
fermionic density matrices, while the BK-superfast transformation Bravyi and
Kitaev (2002); Setia and Whitfield (2018) has useful error-correcting
properties Jiang  _et al._ (2019) but requires additional auxiliary qubits.
Finally, it is important to point out that the presence of long strings of
Pauli operators in JW mappings do not necessarily increase the computational
cost in some situations e.g., fermionic swap networks Kivlichan  _et al._
(2018).

####  V.1.2 Scalar Fields (The BQP-Complete “Gold Standard”)

Encoding continuous quantum fields, such as scalar and gauge fields, into
quantum simulations presents different challenges and opportunities than the
inclusion of fermions. Wavefunctions can be encoded in the amplitudes of
states in a Hilbert space i.e., the states provide a binary representation of
the digitized argument of the wavefunction while their amplitudes provide the
corresponding value of the wavefunction. To be concrete, let us consider
scalar fields as their mapping onto qubits is the best understood scenario in
the context of QFT.

The objective of any such mapping is that observables of the continuum field
theory are recovered in the limit of vanishing discretization of spacetime and
of the field. The pioneering work of JLP Jordan  _et al._ (2012, 2014)
established a mapping that mirrors that of classical lattice scalar field
theory, but with the continuous field values at each spatial lattice site
replaced by a quantum register of $n_{Q}$, describing $2^{n_{Q}}$ values of
the field. For a lattice of spatial extent $L$ in each direction, the total
number of qubits for such a simulation is $n=n_{Q}L^{3}$. In the JLP basis,
states in the computational Hilbert space correspond to eigenstates of the
field operator $\hat{\phi}|\phi\rangle=\phi|\phi\rangle$ at each lattice site.
After rescaling the Hamiltonian, a (symmetric Klco and Savage (2019)) mapping
can be defined in terms of the maximum value of the field, $\phi(k)=\phi_{\rm
max}\left(-1+2k/(2^{n_{Q}}-1)\right)$, where $k=0,1,...2^{n_{Q}}-1$. A
simulation of high-fidelity requires that, at some level of precision, the
support of the field wavefunction at each site, $\psi(\phi({\bf x}))$, is
contained within site registers, requiring an iterative tuning of the actual
mapping based upon low-resource simulations for the interacting QFT, or direct
evaluation for the non-interacting theory. At this point, a connection can be
made with the Nyquist-Shannon sampling theorem. With uniform sampling of
$\phi({\bf x})$ in both field and field conjugate momentum spaces over the
regions of support, the digitized representation of $\phi$ and
static/dynamical observables can be computed with precision improving
exponentially with increasing number of states at each site Jordan  _et al._
(2012, 2014); Somma (2016); Macridin  _et al._ (2018a, b); Klco and Savage
(2019). This translates into a qubit requirement scaling as
$n_{Q}\sim\log\log\left(1/\epsilon\right)$, where $\epsilon$ is the upper
limit of tolerable systematic error. The systematic errors remaining from the
spatial discretization that are encountered in classical lattice field theory
simulations are also present, requiring $n\sim\log\left(1/\epsilon\right)$.
One of the enjoyable features of this position-space mapping is that on-site
and nearest-neighbor interactions are required for simulation, and only a
small number of “layers” of unitary operations are required for a complete
step in Trotterized time-evolution Jordan  _et al._ (2012, 2014). Instead of
eigenstates of $\hat{\phi}$ to define the field at each site, states
digitizing Hermite polynomials Somma (2016); Macridin  _et al._ (2018a, b) or
directly associated with eigenstates of a harmonic oscillator could be used
Klco and Savage (2019). The oscillators could be tuned to optimize support of
$\psi(\phi({\bf x}))$ with low-resource simulations, and the dimensionality of
the space could be used to define the lowest $2^{n_{Q}}$ levels of the
truncated oscillator.

The direct use of spin systems for mapping fields onto quantum registers
leverages the fact that the minimal requirement for a legitimate mapping is
recovering the correct continuum physics in the limit of vanishing
discretization and digitization. In particular, the approach to the continuum
limit is generally mapping dependent. For scalar field theory (with on-site
wavefunctions that have bounded support in field and conjugate-momentum space,
up to exponentially small corrections), the Nyquist-Shannon sampling theorem
provides double-exponential convergence, with respect to an increasing number
of qubits per site, of field digitization. This supplements the single-
exponential convergence of simulating the field with conventional lattice
systematic uncertainties. An alternate spin-system motivated mapping with one
or two qubits per site makes implicit use of a space-spin RG-blocking to
recover the space and field continuum limits. By mixing space-space and field-
space, the convergence of this latter mapping is expected to be single-
exponential. As discussed previously, this anticipated RG-flow is at the heart
of QLMs Chandrasekharan and Wiese (1997a); Brower  _et al._ (1999).
Identifying efficient mappings for NISQ and beyond quantum simulations is an
important line of investigation, and is already starting to provide some
interesting results, e.g., Ref. Zache  _et al._ (2021). Classical simulations
of a 1-dim Heisenberg comb with two qubits at each lattice site suggest that
the continuum limit of the asymptotically free nonlinear O(3) sigma model
could emerge, and that NISQ devices may have utility in further demonstrations
Singh and Chandrasekharan (2019); Bhattacharya  _et al._ (2021).

####  V.1.3 Abelian and Non-Abelian Gauge Fields: Toward Lattice QCD

A main target of quantum simulation is QCD, and in particular the observables
that cannot be accessed with sufficient precision using classical computing
Nuclear Physics and Quantum Information Science, A Report from the NSAC
Quantum Information Science Subcommittee (2019). The simulation of gauge
theories requires the incorporation of abstract spaces and local symmetries at
lattice sites and the links between them (a thorough review of quantum
simulation of LGTs can be found in Ref. Bañuls  _et al._ (2020)). It is well
understood how to address the internal symmetries and different forms of
inter-site couplings for classical simulations, and also for quantum
simulation using lattice Hamiltonians, but conclusive determinations of
optimal field representations and algorithms for implementation on quantum
devices continue to be developed (for a recent review, see e.g., Ref. Zohar
(2021)), in parallel with ongoing developments in the devices themselves. The
wisdom gained and algorithms explored in the process of simulating scalar
field theory are expected to be valuable in developing simulations of, e.g.,
SU(N) Yang-Mills gauge theories, QCD and O(N) models.

In the case of gauge-field theories, the action-angle variables are one
convenient path for mapping the Hamiltonian to a discrete Hilbert space. While
one can digitize directly the angle variables in a way analogous to the scalar
field in a curved higher-dimensional space, for instance, using discrete
subgroups of SU(N) Petcher and Weingarten (1980); Bhanot and Rebbi (1981);
Grosse and Kühnelt (1981); Alexandru  _et al._ (2019), making a somewhat
direct connection to path integral Monte Carlo methods in Euclidean space, the
action variables alternatively correspond to irreducible representations of
the gauge group and provide a convenient means to define the Hamiltonian and
states associated with the link operator, i.e. eigenstates of the electric
operator. The Kogut-Susskind Hamiltonian Kogut and Susskind (1975); Kogut
(1979), formulated in terms of the Casimir and plaquette operators acting on a
lattice of links defined in the electric basis, is such a construction for
SU(N) gauge field theories, and has been the focus of much recent work. First
examined in the context of SU(3) quantum simulation by Byrnes and Yamamoto
Byrnes and Yamamoto (2006), tensors defining the irreps of each local SU(N)
can be mapped to the states in local Hilbert spaces i.e., tensor indices
mapped to quantum registers localized on each link. For a given lattice
spacing (coupling), the dimensionality of the local Hilbert space can be
increased to achieve the desired precision in calculating observables of
interest, with decreasing lattice spacing demanding an increasing number of
irreps of SU(3) on each link in the electric basis. After exploiting the local
gauge symmetry to integrate over the abstract space at each site Bañuls  _et
al._ (2020); Klco  _et al._ (2020); Ciavarella  _et al._ (2021), Yang-Mills
theory can be written entirely in terms of wavefunctions of link variables
defined by a basis of integers, $(p,q)$, that define the number of upper and
lower indices of each irrep of SU(3). Truncation of each index register at
some maximum value, $\Lambda_{p,q}$, causes the SU(3) lattice to become
defined by connected pairs of $(\Lambda_{p,q}+1)$-dimensional Hilbert spaces
with nearest neighbor connectivity Ciavarella  _et al._ (2021). Such
requirements point naturally to considering qudits, as opposed to qubits,
assigned to each link. Most architectures currently being pursued for qubit
design aim to isolate two-level quantum systems from higher-dimensional
spaces, allowing natural access to qudit structures in e.g., trapped-ions
Klimov  _et al._ (2003); Low  _et al._ (2020), photonics Lu  _et al._ (2020),
ultra-cold atoms and molecules Sawant  _et al._ (2020), and SRF cavities
Holland  _et al._ (2020); SRF . Whether providing intermediate theoretical
organization or asymptotic scaling advantages Gokhale  _et al._ (2019), it
appears that codesign of such systems for application to QCD, using a mapping
of the Kogut-Susskind Hamiltonian with integrated local gauge spaces, may be
an interesting path forward.

Beyond this originally considered multiplet basis of electric irreps, a
cornucopia of bases and strategies for quantum simulation with local non-
Abelian gauge symmetry continues to be developed—an abundance crucial for the
hardware-software codesign necessary to efficiently interface with rapidly
developing quantum technologies and crucial for assessing the effects of
systematics when calculations extend beyond the reach of classical
confirmation. One alternate formulation of non-Abelian LGT amenable for
quantum simulation (though not originally designed for this application) is
quantum link models (QLMs) Horn (1981); Orland and Rohrlich (1990);
Chandrasekharan and Wiese (1997b); Brower  _et al._ (1999, 2004); Wiese (2006,
2020), with a number of concrete proposals for implementation on quantum
devices e.g., Refs. Mezzacapo  _et al._ (2015a); Bernien  _et al._ (2017);
Shen  _et al._ (2020); Surace  _et al._ (2020a). The idea of QLMs is that the
intrinsic building blocks of the lattice are few-spin systems with appropriate
symmetry and commutation relations. While the link Hilbert spaces are never
extended as a function of the coupling, making them attractive for near term
quantum simulation Stannigel  _et al._ (2014); Mezzacapo  _et al._ (2015b);
Singh and Chandrasekharan (2019); Singh (2019); Surace  _et al._ (2020b);
Bhattacharya  _et al._ (2021); Shen  _et al._ (2020); Van Damme  _et al._
(2020), effective degrees of freedom of a continuous field at each spatial
site in a corresponding Kogut-Susskind lattice may be interpreted as
distributed across a coupling-dependent volume of the lattice. Though their
approach to the continuum and the associated systematic errors remain to be
quantified within simulations, RG-flow of QLMs is expected to recover the
predictions of QCD if they reside in the same universality class and with
appropriate simulation tunings to the critical point. Progress has been made
more broadly using spin systems as sandboxes for exploring algorithms and
techniques to facilitate quantum simulations of field theories. While ground
state preparation is of considerable interest for spin systems, they also
admit scattering of wavepackets or kinks, and allow for detailed studies of
the evolution of entanglement and energy density e.g., Tan  _et al._ (2021);
Verdel  _et al._ (2020); Milsted  _et al._ (2020); Gustafson  _et al._ (2021).
Spin systems also provide a laboratory for building “particle detectors”
within simulations. Recent work using trapped ion and superconducting qubits
systems has shown that measuring time delays in wavepacket scattering between
interacting and non-interacting theories can be used to extract elastic phase
shifts even with NISQ devices, a step toward calculating S-matrix elements
from real-time dynamics Gustafson  _et al._ (2021). Interestingly, the
entanglement entropy is found to increase through multiple elastic collisions;
adding a cubic interaction enabled the production of ballistic particles
during kink collisions, providing parallels with fragmentation in hadronic
collisions Milsted  _et al._ (2020). Studies of confinement between magnetic
domain walls have been carried out in spin models, where it has been shown
that confinement suppresses information propagation Tan  _et al._ (2021), with
obvious impacts beyond the SM.

Another approach for reducing the near-term qubit requirements for
representing gauge fields in 1-dim, and the approach used in the first quantum
simulation of the Schwinger model Martinez  _et al._ (2016), is to integrate
out the gauge field, trading non-local interactions for only fermionic degrees
of freedom. This simplification expresses the lattice only through its
fermionic degrees of freedom, alleviating the necessity of digitizing the
continuous manifold of the local gauge field. While the purely-fermionic
formulation is only available in 1-dim (where the gauge field is non-
dynamical) Zohar (2021), it has been demonstrated that a formulation in terms
of purely bosonic degrees of freedom is both possible and likely to be
experimentally fruitful Zohar and Cirac (2018, 2019); Aidelsburger  _et al._
(2021). There are also harmonic-oscillator-inspired formulations of gauge
theories that may be amenable to quantum simulation including orbifold
lattices Buser  _et al._ (2020) and Schwinger bosons Schwinger (1965); Mathur
_et al._ (2010). The latter of these formulations has evolved into the loop-
string-hadron (LSH) formulation Raychowdhury and Stryker (2018, 2020)
utilizing fundamentally gauge invariant degrees of freedom to analytically
trade non-Abelian Gauss law constraints for Abelian ones, a feature that is
likely to be advantageous in experimental connections e.g., cold atom quantum
simulators have demonstrated reliable expression of local Abelian symmetries
González-Cuadra  _et al._ (2017); Mil  _et al._ (2020); Ott  _et al._ (2020);
Dasgupta and Raychowdhury (2020). Also underway are explorations utilizing the
light front formalism Kreshchuk  _et al._ (2020); Echevarria  _et al._ (2020);
Kreshchuk  _et al._ (2021a, b), or using measurement-based techniques and
quantum sensors to extract correlation functions and the generating functional
Zache  _et al._ (2020); Bermudez  _et al._ (2017); Martín-Vázquez  _et al._
(2021). Furthermore, there are hybrid proposals that naturally map the gauge
symmetry onto a fundamental conservation of angular momentum in the
architecture of ultracold atoms Zohar  _et al._ (2013a, b), or that map the
bosonic modes of the gauge field to local phonon modes in the architecture of
trapped ions Davoudi  _et al._ (2021b). Beyond the basic realization that
quantum devices will always be embedded in large classical ecosystems for
device operation, error correction, and optimal use of quantum resources, such
works illuminate the potential of hybrid and heterogeneous architectures as
potentially advantageous for simulations of LGTs.

In addition to the quantum resources required for performance of time
evolution on idealized quantum hardware, the choice of basis or field
representation can dramatically affect the ratio of gauge-invariant to gauge-
variant Hilbert space, modifying the sensitivity that a calculation will have
to inevitable errors and noise experienced by the quantum device. For example,
gauge field integration available in (1+1)-dim removes all gauge-variant space
from the computational Hilbert space of the hardware Martinez  _et al._
(2016), while the local group integrations of the multiplet basis or
transformations to gauge-invariant degrees of freedom with Abelian constraints
in the LSH formulation both dramatically reduce, but do not completely remove,
the gauge violating Hilbert space. Beyond simple dimensionality, mappings can
vary in their available techniques for error mitigation. In particular,
paralleling analyses of quantum error correction codes, the array of gauge
field quantum representations will exhibit varying “code distances”, or error
separations between gauge-invariant states through the gauge-variant volume,
determining the ability to reliably correct gauge-violating errors if they are
detected Stryker (2019). Of particular interest in this direction has been the
dynamical generation or protection of gauge invariance from coherent sources
of error Goldman and Dalibard (2014); Halimeh and Hauke (2020); Kasper  _et
al._ (2020); Lamm  _et al._ (2020); Halimeh  _et al._ (2020a); Van Damme  _et
al._ (2020); Halimeh  _et al._ (2020b); Van Damme  _et al._ (2021) through the
introduction of low-body potential terms, creating effective Hamiltonians with
the desired gauge invariance emerging robustly. Optimizing theoretical and
software strategies for the simulation of gauge theories in the high-
dimensional landscape of state preparation, time evolution, and measurement
resources, robustness to quantum noise, and isolation of the gauge invariant
space will be an active process in perpetual communication with progress in
experiment and device design.

###  V.2 Quantum Fields for Quantum Information

As a purist motivation for focusing on the quantum simulation of fields, it is
expected that the quantum mechanical complexity at play in atomic scale and
larger systems will be commensurate with that in subatomic and smaller
systems. Thus, precision control of the former may enable efficient
calculation of emergent properties of the latter, in particular with resources
that no longer scale exponentially with volume. A concrete example of this
commensurate scaling can be seen in the identification of vacuum-to-vacuum
matrix elements in a self-interacting scalar field with classical external
sources to be a BQP-complete problem Jordan  _et al._ (2018). In this
direction, the relevance of fields for quantum simulation is not limited to
relativistic fields describing the dynamics of the building blocks of the SM,
but also applies to non-relativistic systems, such as EFT descriptions of low-
energy physics. From a less-purist perspective, the variety of qubit arrays
that are being developed in laboratories around the world e.g., 2-dim arrays
of implanted atoms in instrumented silicon substrates, ultracold atoms in
optical lattices, ion lattices, superconducting circuit ladders and arrays of
Josephson junctions, are the closest physical systems to latticized quantum
fields that have been created to date. As such, the BQP-completeness of the
scalar field Jordan  _et al._ (2018) potentially indicates a natural language
for quantum computation—any problem (physics or beyond) that can be
efficiently calculated on a quantum device can be efficiently mapped to a
scattering problem in scalar field theory, which itself is efficient to
implement on a quantum device Jordan  _et al._ (2012, 2014). One step further
lies the connection with quantum error detection/correction, where the quantum
information used to compute is embedded in topological excitations of a gauge
field or a bulk holographic dual Almheiri  _et al._ (2015); Pastawski  _et
al._ (2015), both providing a redundancy to enable robustness to local
environmental perturbations. So, whether one is interested in fundamental
simulations of nature or large-scale quantum computations more broadly, the
QFT language provides a natural framework.

Typically considered to be in the realm of condensed matter, quantum spin
liquids represent an important connection between research areas, bringing
together QFTs, spin models and the distribution and protection of quantum
information (for a review, see Ref. Savary and Balents (2016)). While there
are a number of candidate systems to provide logical qubits and quantum
memories, as reviewed in Ref. Campbell  _et al._ (2017), Kitaev’s Toric Code
Kitaev (2003) (and subsequent surface codes exploring alternate boundary
conditions Kitaev (1997b); Bravyi and Kitaev (1998); Dennis  _et al._ (2002);
Fowler  _et al._ (2012)) represents perhaps the first and clearest connection
between these areas of research. In its simplest form, a 2-dim system of
spin-${\frac{1}{2}}$ spins defining the links of a square lattice is subject
to a Hamiltonian with vertex operators (stars), plaquette operators, and
periodic boundary conditions. The ground state of the system is topologically
ordered and consequently highly entangled, supporting two logical qubits
defined by the action of Wilson-loop operators in the two directions. These
topological states are robust against errors (local perturbations) below some
threshold, which can be corrected by measuring $n-4$ independent stabilizers
(commuting star and plaquette operators) and using decoders to determine an
effective set of operations to restore the system. Reducing the number of
independent stabilizers allows an increased dimension of the code space at the
expense of error robustness. The quasi-particle picture is a convenient way to
understand the behavior of these systems and errors. Localized errors create
quasi-particles in the system and error correction corresponds to annihilating
quasi particles, returning the systems back into the code space (ideally
without inadvertently producing a logical error). Excitations above these
ground states are (non-local) pairs of electric charges and pairs of magnetic
vortices. The anyonic nature of these excitations and their braiding points to
the high-degree of underlying entanglement of the ground state(s), and as
such, local operations are unable to resolve which state of the (logical)
ground-state space the system is in. An auxillary spin can be included at each
lattice vertex to transform this system into a $Z_{2}$ LGT, of similar form as
the Kogut-Susskind Hamiltonian for SU(N) LGTs (for a review, see Ref. Kitaev
and Laumann (2009)), and generalizing in detail to SU(N) gauge theories Mathur
and Sreeraj (2016). The Toric Code is also established in 3-dim and 4-dim
using hypercubic (and other) Euclidean space lattice geometries. A major
challenge to the execution of circuits on logical qubits is the required
action of gates across the physical qubits defining the logical qubits. This
development is mature for the Toric Code, see e.g., Refs. Dennis  _et al._
(2002); Jochym-O’Connor and Yoder (2021), and also for other encodings, such
as color codes Bombin and Martin-Delgado (2006).

Interestingly from an SM physics perspective, higher-dimension SU(N) LGTs, due
to the phenomena of confinement manifest in non-Abelian gauge theories, have
utility in defining logical qubits due to their innate mechanisms to localize
qubit errors within “hadrons” Bombín (2015); Brown  _et al._ (2016a) (a