††thanks: leading contributor to analytic developments in this manuscript.

# Loop-string-hadron approach to SU(3) lattice Yang-Mills theory:
Gauge invariant Hilbert space of a trivalent vertex

Saurabh V. Kadam<EMAIL_ADDRESS>InQubator for Quantum Simulation (IQuS),
Department of Physics, University of Washington, Seattle, WA 98195, USA
Aahiri Naskar<EMAIL_ADDRESS>Department of Physics, BITS-
Pilani, K K Birla Goa Campus, Zuarinagar, Goa 403726, India.  Indrakshi
Raychowdhury<EMAIL_ADDRESS>Department of Physics, BITS-
Pilani, K K Birla Goa Campus, Zuarinagar, Goa 403726, India.  Center for
Research in Quantum Information and Technology, Birla Institute of Technology
and Science Pilani, K K Birla Goa Campus, 403726, India  Jesse R. Stryker
<EMAIL_ADDRESS>Physics Division, Lawrence Berkeley National Laboratory,
Berkeley, CA 94720, USA.  Maryland Center for Fundamental Physics and
Department of Physics, University of Maryland, College Park, MD 20742, USA.

###### Abstract

The construction of gauge invariant states of SU(3) lattice gauge theories has
garnered new interest in recent years, but implementing them is complicated by
the need for SU(3) Clebsch-Gordon coefficients. In the loop-string-hadron
(LSH) approach to lattice gauge theories, the elementary excitations are
strictly gauge invariant, and constructing the basis requires no knowledge of
Clebsch-Gordon coefficients. Originally developed for SU(2), the LSH
formulation was recently generalized to SU(3), but limited to one spatial
dimension. In this work, we generalize the LSH approach to constructing the
basis of SU(3) gauge invariant states at a trivalent vertex – the essential
building block to multidimensional space. A direct generalization from the
SU(2) vertex yields a legitimate basis; however, in certain sectors of the
Hilbert space, the naive LSH basis vectors so defined suffer from being
nonorthogonal. The issues with orthogonality are directly related to the
“missing label” or “outer multiplicity” problem associated with SU(3) tensor
products, and may also be phrased in terms of Littlewood-Richardson
coefficients or the need for a “seventh Casimir” operator. The states that are
unaffected by the problem are orthonormalized in closed form. For the sectors
that are afflicted, we discuss the nonorthogonal bases and their
orthogonalization. A few candidates for seventh Casimir operators are readily
constructed from the suite of LSH gauge-singlet operators. The diagonalization
of a seventh Casimir represents one prescriptive solution towards obtaining a
complete orthonormal basis, but a closed-form general solution remains to be
found.

††preprint: IQuS@UW-21-086

##  I Introduction

Gauge theories have been the subject of intense focus in fundamental physics
for many decades now, yet certain avenues of exploration have remained stuck
in nascent stages despite the advent of computers, supercomputers, and most
recently exascale computing. Quantum chromodynamics (QCD), describing the
strong force, is a gauge theory of direct relevance to the physics of our
universe – and also the backdrop for a variety of interesting physics
scenarios that are computationally intractable Gross  _et al._ (2023). The
development of quantum technology is initiating a paradigm shift in studying
gauge theories Bauer  _et al._ (2023a); Di Meglio  _et al._ (2023); Bauer  _et
al._ (2023b); Bañuls and Cichy (2020); Meurice  _et al._ (2022) where novel
quantum simulation/computation algorithms are being developed and are being
implemented Banerjee  _et al._ (2012, 2013); Huffman  _et al._ (2022); Klco
_et al._ (2018, 2020); Ciavarella  _et al._ (2021); Stetina  _et al._ (2022);
Ciavarella and Chernyshev (2022); Farrell  _et al._ (2023a); Turro  _et al._
(2024); Ciavarella and Bauer (2024); Farrell  _et al._ (2023b, 2024a, 2024b);
Haase  _et al._ (2021); Dasgupta and Raychowdhury (2022); Bennewitz  _et al._
(2024); Davoudi  _et al._ (2024); Belyansky  _et al._ (2024); Davoudi  _et
al._ (2023a, 2020); Mueller  _et al._ (2023); Davoudi  _et al._ (2023b);
Nguyen  _et al._ (2022); Atas  _et al._ (2023, 2021); Martinez  _et al._
(2016); Kasper  _et al._ (2017); Kane  _et al._ (2022); Mil  _et al._ (2020);
Muschik  _et al._ (2017); Meth  _et al._ (2023); Zhang  _et al._ (2023a);
Paulson  _et al._ (2021); Guo  _et al._ (2024); Crippa  _et al._ (2024);
Angelides  _et al._ (2023); Chakraborty  _et al._ (2022); Paulson  _et al._
(2021); Halimeh  _et al._ (2023); Angelides  _et al._ (2023); de Jong  _et
al._ (2022); Lamm  _et al._ (2019); Shaw  _et al._ (2020); Kan and Nam (2021);
Cohen  _et al._ (2021); Murairi  _et al._ (2022); Charles  _et al._ (2024);
Mildenberger  _et al._ (2022); Halimeh  _et al._ (2022); Zhang  _et al._
(2023b); Kane  _et al._ (2024); Yang  _et al._ (2020); Zohar  _et al._ (2013a,
b); Hauke  _et al._ (2013); Wiese (2013); Marcos  _et al._ (2013); Zohar  _et
al._ (2016); Yang  _et al._ (2016); Bender  _et al._ (2018); Luo  _et al._
(2020); Notarnicola  _et al._ (2020); Surace  _et al._ (2020); Surace and
Lerose (2021); Kasper  _et al._ (2020); Aidelsburger  _et al._ (2021); Surace
_et al._ (2024); Zache  _et al._ (2023a); Popov  _et al._ (2024); González-
Cuadra  _et al._ (2022); Illa  _et al._ (2024); Zache  _et al._ (2018); Armon
_et al._ (2021); Byrnes and Yamamoto (2006); González-Cuadra  _et al._ (2017);
Tagliacozzo  _et al._ (2013a, b); Zhou  _et al._ (2022); Yamamoto and Doi
(2022); Zohar and Reznik (2011); Zohar  _et al._ (2017); Alexandru  _et al._
(2022); Lamm  _et al._ (2024, 2020a) to push the boundary of knowledge Feynman
(1982); Lloyd (1996); Preskill (2021). Understanding gauge theories by their
quantum simulation/computation requires systematic development of suitable
frameworks Carena  _et al._ (2022); Bañuls  _et al._ (2017); Raychowdhury and
Stryker (2020a); Kadam  _et al._ (2023); Pardo  _et al._ (2023);
Chandrasekharan and Wiese (1997); Bauer and Grabowska (2023); Zohar and
Burrello (2015); Zohar and Cirac (2018, 2019); Alexandru  _et al._ (2024,
2019); Ji  _et al._ (2023, 2020); Kavaki and Lewis (2024); D’Andrea  _et al._
(2024); Romiti and Urbach (2024); Zache  _et al._ (2023b); Wiese (2021);
Hartung  _et al._ (2022) that are qubit-friendly and cost-effective Müller and
Yao (2023); Ciavarella and Bauer (2024); Haase  _et al._ (2021); Kane  _et
al._ (2022); Surace  _et al._ (2024); Zache  _et al._ (2023a); González-Cuadra
_et al._ (2022); Illa  _et al._ (2024); Zohar  _et al._ (2017); Lamm  _et al._
(2024). The theoretical frameworks are the starting point for algorithm
developments for various tasks such as state preparation, time evolution,
symmetry protection and other necessary operations, finally leading to
calculation of the observables via performing quantum measurements Davoudi
_et al._ (2024); Farrell  _et al._ (2024c); Kane  _et al._ (2024); Stryker
(2021); Lamm  _et al._ (2019); Surace and Lerose (2021); Nguyen  _et al._
(2022); Gustafson and Lamm (2023); Lamm  _et al._ (2020b); Raychowdhury and
Stryker (2020b); Mathew and Raychowdhury (2022, 2024); Muschik  _et al._
(2017); Atas  _et al._ (2021, 2023); Farrell  _et al._ (2024a); Grabowska  _et
al._ (2022).

Quantum computation for a physical system is naturally expressed in a
Hamiltonian framework, that requires a choice of basis. For gauge theories,
specifically when the gauge group is continuous and non-Abelian, choosing a
suitable basis stands is a nontrivial task. The Hamiltonian description of
gauge theories is equipped with a local constraint structure which is to be
preserved in the dynamics as well Kogut and Susskind (1975). A preferred
choice of basis that optimizes spaces resources to study quantum dynamics is a
gauge invariant one Davoudi  _et al._ (2021). However, the gauge invariance
for a pure gauge theory is only obtained at the cost of nonlocality. Moreover,
the gauge-invariant nonlocal basis spans the space of all Wilson loops in an
overcomplete manner, making the Hamiltonian calculation difficult. The
recently developed loop-string-hadron (LSH) approach in SU(2) gauge theory
addresses these issues by providing a local and orthonormal gauge invariant
basis, valid in any dimension Raychowdhury and Stryker (2020a). Generalizing
the framework to SU(3) is a crucial next step if it is to aid with quantum
simulating/computing quantum chromodynamics in the foreseeable future, but
doing so involves a series of nontrivial intermediate steps that are currently
being pursued Kadam  _et al._ (2023). The current work provides an
intermediate step by prescribing a local and orthonormal basis applicable to
pure SU(3) gauge theory in higher dimensions.

The foundation of the LSH framework for SU(2) gauge theory lies in the
prepotential formulation of the same, a reformulation of the original Kogut-
Susskind formulation of Hamiltonian lattice gauge theories Kogut and Susskind
(1975) that is also referred to as “Schwinger bosons.” The canonically
conjugate variables in the Hamiltonian formulation are the on-site electric
fields, the generators of the algebra, and ‘link operators’ which are elements
of the gauge group residing on each link of the lattice. While the eigenbasis
of the Hamiltonian is essentially never known in closed form, one may work
with either the well-understood ‘electric basis’ or the ‘magnetic/group
element basis’ to study the dynamics of the theory. The former serves as the
most conventional basis for lattice gauge theories, and is referred to as the
strong coupling eigenbasis. The Hamiltonian framework of gauge theory also
possesses an additional structure dictated by the Gauss law constraints. The
Hamiltonian commutes with these constraints, implying the dynamics to remain
gauge invariant.

The search for a suitable gauge invariant basis for calculating Hamiltonian
dynamics with optimal resources using a classical or quantum computer has been
pursued for many years by gauge theorists with varying interest. Wilson loops,
particle-pair flux strings, and hadrons belong to the gauge invariant Hilbert
space for a gauge theory. The prepotential formulation based on Schwinger
bosons was developed to address the issues with nonlocality and proliferation
of the loop basis Mathur (2005, 2007); Mathur  _et al._ (2010); Mathur and Sen
(2001); Raychowdhury (2013). Properly adapted, the SU(2) prepotential
formulation can be used to construct a local, orthonormal, gauge invariant
basis that solves nonlocality and the proliferation of loop degrees of
freedom, and when combined with staggered matter yields the complete LSH
framework for any spatial dimension. Generalization of the same to SU(3)
involves multiple layers of complication. A recent work Kadam  _et al._ (2023)
provided a complete SU(3) LSH framework for a one-dimensional spatial lattice
with staggered fermions. The present work is the first building block to the
SU(3) LSH basis construction of higher-dimensional theories.

For a $d$-dimensional Cartesian lattice, a site is connected with $2d$ number
of links, each carrying gauge degrees of freedom or an irreducible
representation (irrep). For SU(2) gauge theory, each link carries an angular
momentum irrep that transforms like a rigid rotor under the two SU(2)
generators at the two neighboring sites connected by that particular link.
Forming an on-site gauge singlet requires solving all the Gauss law
constraints (three components for SU(2)), which amounts to adding up $2d$
number of angular momentum irreps to form a net-zero angular momentum irrep,
following the rules of angular momentum addition. This task can be decomposed
into adding two irreps at a time as the fundamental building block. This is
geometrically realized as the “point splitting” scheme of LSH framework, where
a site of a square (cubic) lattice is split into two (four) trivalent
vertices, i.e., vertices attached to three edges. A trivalent vertex provides
the next complicated scenario from a one-dimensional spatial lattice, where
each vertex is only bivalent. Trivalent vertices naturally arise if the
continuum theory is discretized on a hexagonal or “honeycomb” lattice from the
outset. In a recent study Kavaki and Lewis (2024), trivalent vertices for 3D
lattices in SU(2) gauge theory were studied, taking them as a starting point
to ultimately arrive at triamond lattices as a novel alternative
discretization for the continuum theory. In this work, we look into the
simplest case of constructing gauge invariant states for SU(3) gauge theory of
a trivalent vertex, that is the simplest scenario beyond 1+1 dimensions and a
building block to full-fledged multidimensional lattices. Our analysis
demonstrates how already at a trivalent vertex it is a nontrivial task to
establish a basis that is suitable for computation (i.e., is complete and
orthonormalized), and we offer a prescriptive solution to that problem that
has no need for the infamous Clebsch-Gordon coefficients of SU(3).

The organization of the paper is as follows: The foundation of the LSH
framework, SU(3) irreducible Schwinger bosons, is briefly reviewed in section
II. This construction is necessary because it solves the
$\mathrm{Sp}(2,\mathbb{R})$ multiplicity problem that arises from naively
constructing SU(3) representations out of ordinary Schwinger bosons. The next
task is combining SU(3) irreps to form gauge-invariant singlets. The
combination of SU(3) irreps is characterized by Littlewood-Richardson
coefficients, the values of which imply further multiplicities when forming
singlets out of three SU(3) irreps. In section III, the local singlet basis is
defined, starting from the naive basis construction, examining some of its
properties, and ultimately leading to some proposals for “seventh Casimir”
operators as a means to defining a minimal and orthogonal basis.111A previous
work Anishetty and Sreeraj (2019) aimed to obtain local and orthonormal basis
for SU(3) by generalizing SU(2) along the same lines. However, effort of
reproducing the results has revealed several inconsistencies both in
orthogonality and calculated normalization factors, as elaborated in the body
of this manuscript.  The “seventh Casimir” candidate stands crucial because
the more familiar Casimir quantum numbers, which are interchangeable with
three irrep labels $(p_{I},q_{I})$, for $I=1,2,3$ meeting at a trivalent
vertex, fail to serve as a complete set of commuting observables and are
insufficient to characterize an orthogonal basis. We discuss the significance
of this analysis and state the roadmap for proceeding in the future in VI.

##  II SU(3) Prepotentials: Irreducible Schwinger Bosons

As mentioned before, LSH framework is derived from the prepotential framework,
which also goes by the name of Schwinger bosons. SU(3) prepotentials are
different than the naive Schwinger boson representation of SU(3) algebra,
which are adequate for SU(2). In this section we briefly introduce SU(3)
irreducible Schwinger bosons which serves as prepotentials for the theory and
is used later in this work to construct gauge singlet local basis.

Before proceeding to SU(3), it is instructive to have a quick look at the
original SU(2) Schwinger boson construction for higher dimensional pure gauge
theory, which directly maps to the SU(2) LSH basis-construction in arbitrary
dimension. For a site $x$ of a $d$-dimensional spatial lattice, the electric
fields $E^{a}(x,I)$ are associated with the links connected to that particular
site along directions $I=1,2,...,2d$. The electric field satisfy SU(2)
algebra:

$\displaystyle\left[E^{a}(x,I),E^{b}(x^{\prime},I^{\prime})\right]=i\epsilon_{abc}E^{c}(x,I)\delta_{I,I^{\prime}}\delta_{x,x^{\prime}}$
(1)

where, $a,b,c$ cn take values $1,2,3$.

The same algebra is satisfied by the construction:

$\displaystyle
E^{a}(x,I)=a^{\dagger}_{\alpha}(x,I)\left(\frac{\sigma^{a}}{2}\right)^{\alpha}_{\beta}a^{\beta}(x,I)$
(2)

where, $a^{\dagger}_{\alpha}(x,I)$’s are the independent Schwinger boson
doublets $(\alpha,\beta=1,2)$ associated with each link $(x,I)$ and satisfy:

$\displaystyle\left[a^{\alpha}(x,I),a^{\dagger}_{\beta}(x^{\prime},I^{\prime})\right]=\delta_{\beta,\alpha}\delta_{x,x^{\prime}}\delta_{I,I^{\prime}}~{}~{}~{}~{}\&~{}~{}~{}~{}\left[a^{\dagger}_{\alpha}(x,I),a^{\dagger}_{\beta}(x^{\prime},I^{\prime})\right]=0$
(3)

The strong coupling basis states or the irreps of SU(2), characterized by
$|j,m\rangle_{(x,I)}$ are constructed in terms of the SU(2) Schwinger bosons
as:

$\displaystyle|j,m\rangle_{(x,I)}\equiv\frac{1}{\sqrt{(j+m)!(j-m!})}\left(a^{\dagger}_{1}(x,I)\right)^{j+m}\left(a^{\dagger}_{2}(x,I)\right)^{j-m}|0,0\rangle,$
(4)

At a trivalent vertex $x$, one can construct the following gauge singlet
creation operators out of Schwinger bosons or prepotentials for SU(2) gauge
theory:

$\displaystyle\epsilon^{\alpha\beta}a^{\dagger}_{\alpha}(x,1)a^{\dagger}_{\beta}(x,2)~{}~{},~{}~{}\epsilon^{\alpha\beta}a^{\dagger}_{\alpha}(x,2)a^{\dagger}_{\beta}(x,3)~{}~{}\&~{}~{}\epsilon^{\alpha\beta}a^{\dagger}_{\alpha}(x,3)a^{\dagger}_{\beta}(x,1)$
(5)

The on-site loop Hilbert space is spanned by the basis vectors:

$\displaystyle|l_{12},l_{23},l_{31}\rangle_{x}=\mathcal{N}(\epsilon^{\alpha\beta}a^{\dagger}_{\alpha}(x,1)a^{\dagger}_{\beta}(x,2))^{l_{12}}(\epsilon^{\alpha\beta}a^{\dagger}_{\alpha}(x,2)a^{\dagger}_{\beta}(x,3))^{l_{23}}(\epsilon^{\alpha\beta}a^{\dagger}_{\alpha}(x,3)a^{\dagger}_{\beta}(x,1))^{l_{31}}|0\rangle_{x}$
(6)

Here, $|0\rangle$ is the Schwinger boson vacuum which is annihilated by all
the Schwinger boson annihilation operators present at that site. Note that,
there exist gauge sing

The loop degrees of freedom combined with the suitable point splitting scheme
for a particular dimension yield exactly the orthonormal and physical degrees
of freedom of a SU(2) gauge theory. The net electric flux can also describe
the gauge singlet state defined above values $j_{I}$, on each link $I=1,2,3$
connected to the trivalent vertex. The relation between the net flux on link
and the loop quantum numbers are given by:

$\displaystyle
l_{12}=j_{1}+j_{2}-j_{3}~{}~{},~{}~{}l_{23}=j_{2}+j_{3}-j_{1}~{}~{},~{}~{}l_{31}=j_{3}+j_{1}-j_{2}$
(7) or, $\displaystyle
2j_{1}=l_{12}+l_{31}~{}~{}~{}~{},~{}~{}~{}~{}2j_{2}=l_{12}+l_{23}~{}~{}~{}~{},~{}~{}~{}~{}2j_{3}=l_{23}+l_{31}~{}~{},~{}~{}$

Thus the local gauge invariant states are fully characterized by only the
SU(2) Casimirs, and the magnetic quantum numbers of irrep do not appear as
expected. This illustrates one-to-one mapping between loop quantum numbers
$|l_{12},l_{23},l_{31}\rangle$ and the gauge invariant orthonormal basis
constructed from 3 on-link irreps
$|j_{1},m_{1}\rangle\times|j_{2},m_{2}\rangle\times|j_{3},m_{3}\rangle$
leading to gauge singlet orthonormal basis states
$|j_{1},j_{2},j_{12}=j_{3},j_{123}=0,m_{123}=0\rangle\equiv|j_{1},j_{2},j_{3}\rangle$.
This is a signature of no Mandelstam constraint being present as loops are
orthonormal for a trivalent vertex.

It is insightful to return to square lattice and reinvestigate the situation.
This isomorphism between electric flux $\\{j_{I}\\}$ values and loop quantum
numbers $\\{l_{IJ}\\}$ is not there beyond a trivalent vertex. This is a
signature of Mandelstam constraint being present which is well studied in
literature Mathur (2007). As an example, for a site on a $2$-d lattice, one
has $4$ electric flux values $j_{1},~{}j_{2},j_{3},~{}j_{4}$ and six loop
quantum numbers $l_{12},~{}l_{13},~{}l_{14},~{}l_{23},~{}l_{24},~{}l_{34}$.
They are related as:

$\displaystyle
j_{1}=l_{12}+l_{13}+l_{14}~{}~{},~{}~{}j_{2}=l_{12}+l_{23}+l_{24}~{}~{},~{}~{}j_{3}=l_{13}+l_{23}+l_{34}$

The above set of algebraic relations allows at least the following
flexibilities in the loop quantum numbers so that the net fluxes on the link
remain the same.

$\displaystyle l_{12},~{}l_{13}+p,~{}l_{14}-p,~{}l_{23}-p,~{}l_{24}+p,l_{34}$
(8)

This is related to the existence of a Mandelstam constraint depicting
overcompleteness of the loop basis in this case, which was absent for a
trivalent vertex for SU(2).

(a) Prepotentials are Harmonic oscillator doublets or Schwinger bosons
assigned to each link originating from a particular site, all of which
transform as fundamental irrep $2$ of SU(2) group associated with the
respective link label.

(b) Irreducible prepotentials are irreducible Schwinger bosons or a pair of
modified harmonic oscillator triplets assigned to each link originating from a
particular site, which transform as fundamental irreps $3$ and $3^{*}$ of that
on-site SU(3).

Figure 1: Prepotential content at a trivalent vertex for pure (a) SU(2) and
(b) SU(3). Pictorial representation of the on-site singlets are given in Fig.
2. An on-site gauge invariant object can be represented by a fully
antisymmetrized column of two (for SU(2)) or three (for SU(3)) Young Tableaux
boxes. Mathematically the singlet construction refers to
(a) for SU(2): (i)
$\epsilon^{\alpha\beta}a^{\dagger}_{\alpha}(I)a^{\dagger}_{\beta}(J)$ as given
in (5) for $I,J=1,2,3$;
(b) for SU(3): (i)
$\epsilon^{\alpha\beta\gamma}A^{\dagger}_{\alpha}[I]\epsilon_{\beta\gamma\sigma}B^{\dagger\sigma}[J]=\delta^{\alpha}_{\sigma}A^{\dagger}_{\alpha}[I]B^{\dagger\sigma}[J]$,
(ii)
$\epsilon^{\alpha\beta\gamma}A^{\dagger}_{\alpha}[I]A^{\dagger}_{\beta}[J]A^{\dagger}_{\gamma}[K]$,
(iii)
$\epsilon_{\alpha\beta\gamma}B^{\dagger\alpha}[I]B^{\dagger\beta}[J]B^{\dagger\gamma}[K]$
as given in (III) for $I,J=1,2,3$.

The current work focuses on constructing SU(3) singlet states at a trivalent
vertex. The simple yet elegant construction for SU(2) mentioned above may be
generalized in a straightforward way for SU(3) gauge theories, but there that
comes with subtleties to be detailed in the later part of this manuscript. In
this section, we briefly review the generalization of Schwinger boson
construction for SU(3): the SU(3) prepotentials constructed out of irreducible
Schwinger bosons.

The prepotential formulation of the SU(3) gauge group, which is a rank-two
group with a three-dimensional fundamental representation, requires two
independent Schwinger boson triplets at each ends of each link. Considering a
site $x$, and links along directions $I=1,2,3$, the Schwinger bosons are
denoted by $a^{\dagger}_{\alpha}(x,I)$ and $b^{\dagger\alpha}(x,I)$. Here,
$\alpha$ and $\beta$ are the color indices that take integer values from 1 to
3. The SU(3) Schwinger bosons obey the ordinary bosonic commutation relations:

$\displaystyle\big{[}a^{\alpha}(x,I),a^{\alpha^{\prime}}(x^{\prime},I^{\prime})\big{]}$
$\displaystyle=\big{[}b_{\alpha}(x,I),b_{\alpha^{\prime}}(x^{\prime},I^{\prime})\big{]}=0,$
(9)
$\displaystyle\big{[}a^{\alpha}(x,I),b_{\alpha^{\prime}}(x^{\prime},I^{\prime})\big{]}$
$\displaystyle=\big{[}a^{\alpha}(x,I),b^{\dagger\alpha^{\prime}}(x^{\prime},I^{\prime})\big{]}=0,$
(10)
$\displaystyle\big{[}a^{\alpha}(x,I),a^{\dagger}_{\alpha^{\prime}}(x^{\prime},I^{\prime})\big{]}$
$\displaystyle=\big{[}b_{\alpha^{\prime}}(x,I),b^{\dagger\alpha}(x^{\prime},I^{\prime})\big{]}=\delta_{\alpha^{\prime}}^{\alpha}\delta_{xx^{\prime}}\delta_{II^{\prime}},$
(11)

where $I$ and $I^{\prime}$ can each take values $1,2,3$ for a trivalent
vertex.

Following the Schwinger boson construction for SU(3) generators, the
chromoelectric fields at the site $x$, for the link could be defined in terms
of the prepotentials as

$\displaystyle E^{\rm a}(x,I)$
$\displaystyle=a^{\dagger}_{\alpha}(x,I)\,\big{(}T^{\rm
a}\big{)}^{\alpha}{}_{\beta}\,a^{\beta}(x,I)-b^{\dagger\alpha}(x,I)\,\big{(}T^{*\rm
a}\big{)}_{\alpha}{}^{\beta}\,b_{\beta}(x,I),$ (12)

where $T^{\rm a}$ are one-half times the SU(3) Gell-Mann matrices. As shown in
Anishetty  _et al._ (2009, 2010), this set of ordinary Schwinger bosons is not
suitable for the construction of the gauge theory Hilbert space in the above
form. Following (4), a similar monomial of the Schwinger bosons
$a^{\dagger}_{\alpha}$ and $b^{\dagger\beta}$ will likewise create
representations of SU(3), however they are generally not irreducible Mukunda
and Pandit (1965); Chaturvedi and Mukunda (2002). The simplest case of the
$(1,1)$ irrep of SU(3) is naively constructed as
$a^{\dagger}_{\alpha}b^{\dagger\beta}|0,0\rangle$, however its proper
construction is instead

$|1,1\rangle_{\alpha}^{\beta}=a^{\dagger}_{\alpha}b^{\dagger\beta}|0,0\rangle-\frac{\delta_{\alpha}^{\beta}}{3}(a^{\dagger}\cdot
b^{\dagger})|0,0\rangle,$ (13)

such that the irrep is traceless, which is a fundamental property of any
irrep. Above, $\alpha\,,\beta=1,2,3$ are the color indices, $a^{\dagger}\cdot
b^{\dagger}\equiv\sum_{\gamma=1}^{3}a^{\dagger}_{\gamma}b^{\dagger\gamma}$,
and the $(0,0)$ irrep state, $|0,0\rangle$, satisfies
$a^{\alpha}|0,0\rangle=b_{\alpha}|0,0\rangle=0$ for all values of $\alpha$.
For a general irrep $(p,q)$, one has to extract out all the traces from the
monomial state $a^{\dagger}_{\alpha_{1}}\cdots
a^{\dagger}_{\alpha_{p}}b^{\dagger\beta_{1}}\cdots
b^{\dagger\beta_{q}}|0,0\rangle$ to satisfy the tracelessness condition (see,
for example, equation (35) of Ref. Mathur and Sen (2001)). Such a traceless
construction is cumbersome to use, and on top of that there is also an
$\mathrm{Sp}(2,\mathbb{R})$ multiplicity problem because $(a^{\dagger}\cdot
b^{\dagger})^{\rho}\displaystyle|P,Q\rangle_{\vec{\alpha}}^{\vec{\beta}}$ for
any positive integer $\rho$ transforms in the same way under SU(3) as the
$(p,q)$ irrep Chaturvedi and Mukunda (2002).

A solution to the $\mathrm{Sp}(2,\mathbb{R})$ multiplicity problem is obtained
in terms of the irreducible Schwinger boson (ISB) construction Anishetty  _et
al._ (2009), where the state in (13) can actually be constructed as a monomial
of ISBs:

$|1,1\rangle_{\alpha}^{\beta}=A^{\dagger}_{\alpha}B^{\dagger\beta}|0,0\rangle$
(14)

Above, $A^{\dagger}_{\alpha}$ and $B^{\dagger\beta}$ are the ISB creation
operators, constructed in terms of the naive SU(3) Schwinger bosons as

$\displaystyle A^{\dagger}_{\alpha}$ $\displaystyle\equiv$ $\displaystyle
a^{\dagger}_{\alpha}-\frac{1}{P+Q+1}(a^{\dagger}\cdot b^{\dagger})b_{\alpha},$
(15) $\displaystyle B^{\dagger\alpha}$ $\displaystyle\equiv$ $\displaystyle
b^{\dagger\alpha}-\frac{1}{P+Q+1}(a^{\dagger}\cdot b^{\dagger})a^{\alpha}.$
(16)

In (15,16), we have introduced the number operators

$P\equiv a^{\dagger}\cdot
a=\sum_{\alpha=1}^{3}a^{\dagger}_{\alpha}\,a^{\alpha}\quad\text{and}\quad
Q\equiv b^{\dagger}\cdot b=\sum_{\beta=1}^{3}b^{\dagger\beta}\,b_{\beta}$ (17)

Construction of irreps as monomials of ISBs as shown in (14) is valid for any
arbitrary irreps and is given by

$|p,q\rangle_{\vec{\alpha}}^{\vec{\beta}}=\mathcal{N}A^{\dagger}_{\alpha_{1}}\ldots
A^{\dagger}_{\alpha_{p}}B^{\dagger\beta_{1}}\ldots
B^{\dagger\beta_{q}}|0\rangle$ (18)

Here $\mathcal{N}$ corresponds to the normalization factor, the indices
$\alpha_{i}$ with $i=1,\cdots,p$ and $\beta_{j}$ with $j=1,\cdots,q$ can take
integer values between 1 to 3, and $\alpha_{i}$ and $\beta_{j}$ determine the
isospin and hypercharge of the irrep state Mukunda and Pandit (1965);
Chaturvedi and Mukunda (2002). The monomial states constructed in Eq. (18) are
free from any multiplicity problem as the states satisfy

$A^{\dagger}\cdot
B^{\dagger}\,|p,q\rangle_{\vec{\alpha}}^{\vec{\beta}}=0\quad\text{and}\quad
A\cdot B\,|p,q\rangle_{\vec{\alpha}}^{\vec{\beta}}=0,$ (19)

which ensures that they are restricted to the $\rho=0$ subspace or
equivalently the kernel of the operator $a\cdot b$; see Ref. Anishetty  _et
al._ (2010) for details. This implies that the operators $A^{\dagger}\cdot
B^{\dagger}$ and $A\cdot B$ are effectively the null operators within the
Hilbert space spanned by states in Eq. (18). This leads to the following
modified commutation relations for ISBs:

$\displaystyle[A^{\alpha},A^{\dagger}_{\beta}]\simeq\left(\delta^{\alpha}_{\beta}-\frac{1}{P+Q+2}B^{\dagger\alpha}B_{\beta}\right),$
(20)
$\displaystyle[B_{\alpha},B^{\dagger\beta}]\simeq\left(\delta_{\alpha}^{\beta}-\frac{1}{P+Q+2}A^{\dagger}_{\alpha}A^{\beta}\right),$
(21)
$\displaystyle[A^{\alpha},B^{\dagger\beta}]\simeq-\frac{1}{P+Q+2}B^{\dagger\alpha}A^{\beta},$
(22)
$\displaystyle[B_{\alpha},A^{\dagger}_{\beta}]\simeq-\frac{1}{P+Q+2}A^{\dagger}_{\alpha}B_{\beta},$
(23)

along with

$\displaystyle[A^{\dagger}_{\alpha},A^{\dagger}_{\beta}]=[A^{\alpha},A^{\beta}]=[B_{\alpha},B_{\beta}]=[B^{\dagger\alpha},B^{\dagger\beta}]=[A^{\alpha},B_{\beta}]=[A^{\dagger}_{\alpha},B^{\dagger\beta}]=0,$
(24)

where $\simeq$ indicates that the above set of commutation relations are valid
within the vector subspace spanned by SU(3) irreps defined in Eq. (18). Note
that, in the same subspace, the number operators for $A$-type and $B-$type
ISBs satisfy

$A^{\dagger}\cdot A\simeq P\quad\text{and}\quad B^{\dagger}\cdot B\simeq Q\,.$
(25)

The eigenvalues of $A^{\dagger}\cdot A$ and $B^{\dagger}\cdot B$ acting on an
SU(3) irrep are given by $p$ and $q$ respectively, which serve as the quantum
numbers to characterize the irrep as well.

##  III SU(3)-invariant basis at the vertex

(a) SU(2), being a rank 1 group, contains only a single fundamental irrep.
This is translated to the fact that SU(2) loops/ Wilson lines are
directionless. At a trivalent vertex, flux can flow along only three
directions as shown, consistent with on-site SU(2) singlets constructed using
prepotentials as in (5).

(b) SU(3), being a rank 2 group, contains two fundamental irreps $(1,0)$ and
$(0,1)$. With the convention that SU(3) loops/ Wilson lines are always
directed from a $(0,1)$ to a $(1,0)$, at a trivalent vertex, flux can flow
along six possible directions, as well can emerge/absorb from all three
directions, consistent with on-site SU(3) singlets constructed using
prepotentials as in (III).

Figure 2: On-site singlets constructed out of prepotential operators for SU(2)
and SU(3) pure gauge trivalent vertex. Young-Tableaux representation of the
same is discussed in Fig. 1.

Equipped with the irreducible Schwinger bosons, the construction of on-site
SU(2) gauge singlet pure-creation operators at a trivalent vertex $x$ as given
in (5) can be generalized to SU(3) as below:

$\displaystyle
L^{\dagger}_{12}=A^{\dagger}_{\alpha}(1)B^{\dagger\alpha}(2)~{}~{}$
$\displaystyle~{}~{}L^{\dagger}_{21}=A^{\dagger}_{\alpha}(2)B^{\dagger\alpha}(1)$
$\displaystyle
L^{\dagger}_{23}=A^{\dagger}_{\alpha}(2)B^{\dagger\alpha}(3)~{}~{}$
$\displaystyle~{}~{}L^{\dagger}_{32}=A^{\dagger}_{\alpha}(3)B^{\dagger\alpha}(2)$
$\displaystyle
L^{\dagger}_{31}=A^{\dagger}_{\alpha}(3)B^{\dagger\alpha}(1)~{}~{}$
$\displaystyle~{}~{}L^{\dagger}_{13}=A^{\dagger}_{\alpha}(1)B^{\dagger\alpha}(3)$
$\displaystyle
T^{\dagger}_{A}=\epsilon^{\alpha\beta\gamma}A^{\dagger}_{\alpha}(1)A^{\dagger}_{\beta}(2)A^{\dagger}_{\gamma}(3)~{}~{}$
$\displaystyle~{}~{}T^{\dagger}_{B}=\epsilon_{\alpha\beta\gamma}B^{\dagger\alpha}(1)B^{\dagger\beta}(2)B^{\dagger\gamma}(3)$

When applied to the Hilbert space created by ISBs, the above set of gauge
singlet variables satisfy

$\displaystyle T^{\dagger}_{A}T^{\dagger}_{B}\simeq
L^{\dagger}_{12}L^{\dagger}_{23}L^{\dagger}_{31}+L^{\dagger}_{21}L^{\dagger}_{32}L^{\dagger}_{13}.$
(27)

The above identity can be checked with help from the algebraic identity

$\displaystyle\epsilon_{\alpha_{1}\beta_{1}\gamma_{1}}\epsilon^{\alpha_{2}\beta_{2}\gamma_{2}}=\left|\begin{array}[]{ccc}\delta_{\alpha_{1}}^{\alpha_{2}}&\delta_{\alpha_{1}}^{\beta_{2}}&\delta_{\alpha_{1}}^{\gamma_{2}}\\\
\delta_{\beta_{1}}^{\alpha_{2}}&\delta_{\beta_{1}}^{\beta_{2}}&\delta_{\beta_{1}}^{\gamma_{2}}\\\
\delta_{\gamma_{1}}^{\alpha_{2}}&\delta_{\gamma_{1}}^{\beta_{2}}&\delta_{\gamma_{1}}^{\gamma_{2}}\\\
\end{array}\right|$ (31)

and the properties of irreducible Schwinger bosons as given in (19).

Figure 3: Pictorial representation of the identity (27), which implies that an
equal amount of $T_{A}$ & $T_{B}$ excitation present at a site is equivalent
to having a linear combination of flux lines flowing around the site in a
cyclic and anti-cyclic manner.

In addition to the purely creation-type operators listed in (III) and their
purely annihilation adjoints, there exist numerous other mixed-type gauge
singlet operators. A few of these arise in normalization evalutions of
Appendix A, while the others will be the subject of future work. Next, we
discuss how to construct basis states out of these gauge singlet operators
acting on the total bosonic vacuum.

We define the naive LSH states as

$\displaystyle\left.\left|\ell_{12}\,\ell_{23}\,\ell_{31};\ell_{21}\,\ell_{32}\,\ell_{13};t\right\rangle\right\rangle$
$\displaystyle\equiv
L_{12}^{\dagger\,\ell_{12}}L_{23}^{\dagger\,\ell_{23}}L_{31}^{\dagger\,\ell_{31}}L_{21}^{\dagger\,\ell_{21}}L_{32}^{\dagger\,\ell_{32}}L_{13}^{\dagger\,\ell_{13}}\times\begin{cases}T_{A}^{\dagger\,t}\ket{0},&t\geq
0\\\ T_{B}^{\dagger\,-t}\ket{0},&t<0\end{cases}$ (32) $\displaystyle\ell_{IJ}$
$\displaystyle\in\\{0,\ 1,\ 2,\ 3,\,\ldots\\},$ (33) $\displaystyle t$
$\displaystyle\in\\{0,\ \pm 1,\ \pm 2,\ \pm 3,\ \ldots\\}$ (34) 222In this
work, any operator raised to the zero power is understood to be shorthand for
the identity operator: $(T_{A}^{\dagger})^{0}=(T_{B}^{\dagger})^{0}\equiv 1$.

. Above, the notation $\left.\left|\cdots\right\rangle\right\rangle$ is used
to indicate a ket that is not necessarily normalized to unity. In this
notation, the total vacuum ket $\ket{0}$ may also be written as

$\displaystyle\left.\left|0\right\rangle\right\rangle$
$\displaystyle\equiv\ket{0}=\left.\left|0\,0\,0;0\,0\,0;0\right\rangle\right\rangle.$
(35)

The vacuum is normalized to unity by definition: $\braket{0}{0}=1$.

It is no surprise that the kets of (32) are generally not normalized, but what
is less obvious is the fact that they are not all orthogonal – a fact that was
overlooked in Ref. Anishetty and Sreeraj (2019). Thus, the LSH “quantum
numbers” are not always “good” quantum numbers. It is for this reason that we
refer to the states in (32) as the “naive” LSH states. The simplest example
where nonorthogonality can be found is in the sector of
$p_{1}=q_{1}=p_{2}=q_{2}=p_{3}=q_{3}=1$, as depicted in Fig. 4. The issue of
nonorthogonality is of practical importance for at least two reasons: i)
Orthogonality is a great aid to normalization evaluations, and wrongly
assuming orthogonality will lead to incorrect normalizations Anishetty and
Sreeraj (2019). Such normalization factors arise when evaluating matrix
elements of operators. ii) One would like an orthogonal basis for quantum
computation, such that the basis states are distinguishable and can be mapped
onto distinct measurements of qubits/qudits.

Figure 4: The bra, $\bra{0\,0\,0;1\,1\,1;0}$, and the ket,
$\ket{1\,1\,1;0\,0\,0;0}$ are two normalized naive LSH states characterized by
$p_{1}=q_{1}=p_{2}=q_{2}=p_{3}=q_{3}=1$. Although they have distinct LSH
labels, explicit calculation confirms that these states are neither parallel
nor orthogonal [the angle between them is $\cos^{-1}(-2/7)$] and presents the
simplest counterexample to total orthogonality of the $\\{\ell_{ij}\\}$ and
$t$ quantum numbers that was claimed in Anishetty and Sreeraj (2019).

Despite issues of nonorthogonality, we have found that the above set
nonetheless constitutes a valid basis for the Hilbert space of gauge-singlet
states of the vertex. That is, (32) defines a complete basis for the Hilbert
space that is linearly independent. This empirical observation is supported by
our computer-assisted investigations, and is discussed more below. Note that
the naive basis does not admit any states containing both $T_{A}^{\dagger}$
and $T_{B}^{\dagger}$ excitations explicitly; this is because the identity
(27) implies that whenever both excitations are present, the state can be
decomposed into a linear combination of states with suitably incremented
$\ell_{IJ}$ quantum numbers. The omission of LSH states with simultaneous
$T_{A}^{\dagger}$ and $T_{B}^{\dagger}$ creation operators therefore serves to
fulfill the requirement of linear independence, and below we confirm that it
leads to the correct number of states.

Nonorthogonality can only arise between states sharing the same
$(p_{1},q_{1},p_{2},q_{2},p_{3},q_{3})$ quantum numbers (as is the case in
Fig. 4). We shall frequently refer to this collection of six quantum numbers,
so we here introduce the shorthand notation

$\displaystyle\overset{\longrightarrow}{p\,q}$
$\displaystyle\equiv(p_{1},q_{1},p_{2},q_{2},p_{3},q_{3}).$ (36)

The $\overset{\longrightarrow}{p\,q}$ quantum numbers for a given naive basis
ket are obtained from the naive LSH quantum numbers as

$\displaystyle p_{1}$ $\displaystyle=\ell_{12}+\ell_{13}+|t|\,\theta(t),$
(37a) $\displaystyle q_{1}$
$\displaystyle=\ell_{31}+\ell_{21}+|t|\,\theta(-t),$ (37b) $\displaystyle
p_{2}$ $\displaystyle=\ell_{23}+\ell_{21}+|t|\,\theta(t),$ (37c)
$\displaystyle q_{2}$ $\displaystyle=\ell_{12}+\ell_{32}+|t|\,\theta(-t),$
(37d) $\displaystyle p_{3}$
$\displaystyle=\ell_{31}+\ell_{32}+|t|\,\theta(t),$ (37e) $\displaystyle
q_{3}$ $\displaystyle=\ell_{23}+\ell_{13}+|t|\,\theta(-t),$ (37f)

where $\theta$ denotes the Heaviside function. Inspection of (37) reveals a
symmetry of $\overset{\longrightarrow}{p\,q}$ under the following
transformation:

$\displaystyle\bigl{(}\ell_{12},\,\ell_{23},\,\ell_{31};\ell_{21},\,\ell_{32},\,\ell_{13};t\bigr{)}\to$
$\displaystyle\bigl{(}\ell_{12},\,\ell_{23},\,\ell_{31};\ell_{21},\,\ell_{32},\,\ell_{13};t\bigr{)}-n(1,1,1;-1,-1,-1;0)$
$\displaystyle=\bigl{(}\ell_{12}-n,\,\ell_{23}-n,\,\ell_{31}-n;\ell_{21}+n,\,\ell_{32}+n,\,\ell_{13}+n;t\bigr{)},$
(38)

where $n$ is any integer satisfying

$\displaystyle-\min(\ell_{21},\,\ell_{32},\,\ell_{13})\leq$ $\displaystyle
n\leq\min(\ell_{12},\,\ell_{23},\,\ell_{31}),$ (39)

ensuring that all loop numbers remain nonnegative. This indicates that the
quantum numbers $\overset{\longrightarrow}{p\,q}$ are insufficient to
completely characterize the states. Thus, nonorthogonality may arise if two
naive basis kets are related by (III). Moreover, it will be argued below that
all naive basis kets belonging to a common $\overset{\longrightarrow}{p\,q}$
sector must by related by (III).

Some choices of $\overset{\longrightarrow}{p\,q}$ are, however, free from
problems with orthogonality, because the subspace they characterize is one-
dimensional. This is the situation for an LSH state that satisifies
$\min(\ell_{12},\,\ell_{23},\,\ell_{31})=\min(\ell_{21},\,\ell_{32},\,\ell_{13})=0$.
We shall refer to these one-dimensional subspaces as being _nondegenerate_ ,
in reference to the $\overset{\longrightarrow}{p\,q}$ quantum numbers.
Multidimensional subspaces, will be referred to as being _degenerate_ , again
in reference to the $\overset{\longrightarrow}{p\,q}$ quantum numbers). In the
degenerate sectors, another quantum number is necessary to distinguish
orthogonal states.

The need for seven quantum numbers to characterize any general state of the
vertex can be understood by the following argument: The characterization of
the state of an individual link end requires five pieces of data,
conventionally chosen as $p$, $q$, the total isospin, the third component of
isospin, and the hypercharge. The 15 degrees of freedom belonging to three
link ends are then tied together and constrained by eight components of
Gauss’s law, leading to $15-8=7$ remnant gauge invariant degrees of freedom.
It is well-known that $p$ and $q$ can be inverted to give eigenvalues of the
quadratic and cubic Casimir, operators, hence the specification of
$\overset{\longrightarrow}{p\,q}$ is equivalent to knowing the eigenvalues of
six distinct Casimir operators. We shall use the term “seventh Casimir” to
refer to some other indepedent operator that that can be simultaneously
diagonalized with the first six, with its eigenvalues being sufficient to
identify orthogonal states. Formally, the seventh Casimir should be a
Hermitian operator that commutes with the other six Casimir operators and has
a nondegenerate spectrum when projected into any given
$\overset{\longrightarrow}{p\,q}$ sector.

##  IV Nondegenerate basis states

Whenever $\dim(\overset{\longrightarrow}{p\,q})=1$, the only LSH state
belonging to the one-dimensional subspace is automatically orthogonal to all
other naive basis states, and it can be normalized in closed form. The
nondegenerate basis states may be categorized into three classes. To enumerate
them, note that the criterion
$\min(\ell_{12},\,\ell_{23},\,\ell_{31})=\min(\ell_{21},\,\ell_{32},\,\ell_{13})=0$
implies that one of $\\{\ell_{12},\,\ell_{23},\,\ell_{31}\\}$ and one of
$\\{\ell_{21},\,\ell_{32},\,\ell_{13}\\}$ must be zero. The vertex labels
$(I,J,K)$ can always be associated with legs 1, 2, and 3 such that $\ell_{IK}$
denotes one of the vanishing $\ell$-type quantum numbers. We then obtain three
classes according to the choice of which member of
$\\{\ell_{JI},\ell_{KJ},\ell_{IK}\\}$ is also zero.

  * •

Class I: States of the form
$\left.\left|\ell_{IJ},\ell_{JK};\ell_{JI},\ell_{KJ};t\right\rangle\right\rangle$,
i.e.,

$\displaystyle\left.\left|\ell_{IJ},\ell_{JK},\ell_{JI},\ell_{KJ};t\right\rangle\right\rangle$
$\displaystyle=(L_{IJ}^{\dagger})^{\ell_{IJ}}(L_{JK}^{\dagger})^{\ell_{JK}}(L_{JI}^{\dagger})^{\ell_{JI}}(L_{KJ}^{\dagger})^{\ell_{KJ}}\times\begin{cases}T_{A}^{\dagger\,t}\ket{0},&t\geq
0\\\ T_{B}^{\dagger\,-t}\ket{0},&t<0.\end{cases}$ (40)

  * •

Class IIa: States of the form
$\left.\left|\ell_{IJ},\ell_{JK},\ell_{JI},\ell_{IK};t\right\rangle\right\rangle$,
i.e.,

$\displaystyle\left.\left|\ell_{IJ},\ell_{JK},\ell_{JI},\ell_{IK};t\right\rangle\right\rangle$
$\displaystyle=(L_{IJ}^{\dagger})^{\ell_{IJ}}(L_{JK}^{\dagger})^{\ell_{JK}}(L_{JI}^{\dagger})^{\ell_{JI}}(L_{IK}^{\dagger})^{\ell_{IK}}\times\begin{cases}T_{A}^{\dagger\,t}\ket{0},&t\geq
0\\\ T_{B}^{\dagger\,-t}\ket{0},&t<0.\end{cases}$ (41)

  * •

Class IIb: States of the form
$\left.\left|\ell_{IJ},\ell_{JK},\ell_{KJ},\ell_{IK};t\right\rangle\right\rangle$,
i.e.,

$\displaystyle\left.\left|\ell_{IJ},\ell_{JK},\ell_{KJ},\ell_{IK};t\right\rangle\right\rangle$
$\displaystyle=(L_{IJ}^{\dagger})^{\ell_{IJ}}(L_{JK}^{\dagger})^{\ell_{JK}}(L_{KJ}^{\dagger})^{\ell_{KJ}}(L_{IK}^{\dagger})^{\ell_{IK}}\times\begin{cases}T_{A}^{\dagger\,t}\ket{0},&t\geq
0\\\ T_{B}^{\dagger\,-t}\ket{0},&t<0.\end{cases}$ (42)

Above, we have written the naive basis by suppressing the two $\ell$-type
quantum numbers that must be zero. Classes IIa and IIb are related to each
other by a symmetry: $A\leftrightarrow B$ for the Schwinger bosons and
$i\leftrightarrow k$ for the vertex legs. A pictorial representation of some
representative states in these nondegenerate classes is given in Fig. 5.

Figure 5: One-dimensional nondegenerate subspaces are states where at least
two $\ell_{IJ}$ quantum numbers are zero. Suppressing the $t$ excitation, the
rest of the distribution of nonzero can be pictorially represented as in the
above examples.

Note that nondegenerate basis kets with more than the minimal number of
vanishing $\ell$-type quantum numbers belong to multiple classes; e.g., states
of the form $\left.\left|\ell_{12};t\right\rangle\right\rangle$ belong to all
three classes, while states of the form
$\left.\left|\ell_{23},\ell_{31},\ell_{32};t\right\rangle\right\rangle$ belong
to Classes I and IIa.

The naive basis kets that are nondegenerate can be normalized in closed form
by evaluating the norm-squared of the states, using recursion, on a class-by-
class basis. The process to do this is provided in some detail in Appendix A.
For Class I, the squared norms are evaluated to be

$\displaystyle\left\langle\braket{\ell_{IJ},\ell_{JK},\ell_{JI},\ell_{KJ},t}{\ell_{IJ},\ell_{JK},\ell_{JI},\ell_{KJ},t}\right\rangle$
$\displaystyle=\tfrac{1}{2}(\ell_{IJ}+\ell_{JI}+\ell_{JK}+\ell_{KJ}+|t|+2)\ell_{IJ}!\,\ell_{JK}!\,\ell_{JI}!\,\ell_{KJ}!\,|t|!\,(\ell_{IJ}+\ell_{KJ}+|t|+1)!\,(\ell_{JK}+\ell_{JI}+|t|+1)!\,.$
(43)

For Classes IIa and IIb, the squared norms are evaluated to be

$\displaystyle\left\langle\braket{\ell_{IJ},\ell_{JK},\ell_{JI},\ell_{IK},t}{\ell_{IJ},\ell_{JK},\ell_{JI},\ell_{IK},t}\right\rangle$
$\displaystyle=\frac{1}{2}\tfrac{(\ell_{IJ}+\ell_{JK}+\ell_{JI}+\ell_{IK}+|t|+2)\,\ell_{IJ}!\,\ell_{JK}!\,\ell_{JI}!\,\ell_{IK}!\,|t|!\,(\ell_{IJ}+\ell_{IK}+|t|+1)!\,(\ell_{JK}+\ell_{JI}+|t|+1)!\,\binom{\ell_{IJ}+\ell_{JK}+\ell_{JI}+\ell_{IK}+|t|+1}{\ell_{IK}}}{\binom{\ell_{IJ}+\ell_{JI}+\ell_{IK}+|t|+1}{\ell_{IK}}}\,,$
(44)
$\displaystyle\left\langle\braket{\ell_{IJ},\ell_{JK},\ell_{KJ},\ell_{IK},t}{\ell_{IJ},\ell_{JK},\ell_{KJ},\ell_{IK},t}\right\rangle$
$\displaystyle=\frac{1}{2}\tfrac{(\ell_{IJ}+\ell_{JK}+\ell_{KJ}+\ell_{IK}+|t|+2)\,\ell_{IJ}!\,\ell_{JK}!\,\ell_{KJ}!\,\ell_{IK}!\,|t|!\,(\ell_{IJ}+\ell_{KJ}+|t|+1)!\,(\ell_{JK}+\ell_{IK}+|t|+1)!\,\binom{\ell_{IJ}+\ell_{JK}+\ell_{KJ}+\ell_{IK}+|t|+1}{\ell_{IK}}}{\binom{\ell_{JK}+\ell_{KJ}+\ell_{IK}+|t|+1}{\ell_{IK}}}\,.$
(45)

Equipped with these norms, we can express orthonormal states as

$\displaystyle\ket{\ell_{IJ},\ell_{JK},\ell_{JI},\ell_{KJ},t}=\tfrac{(L_{IJ}^{\dagger})^{\ell_{IJ}}(L_{JK}^{\dagger})^{\ell_{JK}}(L_{JI}^{\dagger})^{\ell_{JI}}(L_{KJ}^{\dagger})^{\ell_{KJ}}\ket{0}}{\sqrt{\tfrac{1}{2}(\ell_{IJ}+\ell_{JI}+\ell_{JK}+\ell_{KJ}+|t|+2)\ell_{IJ}!\,\ell_{JK}!\,\ell_{JI}!\,\ell_{KJ}!\,|t|!\,(\ell_{IJ}+\ell_{KJ}+|t|+1)!\,(\ell_{JK}+\ell_{JI}+|t|+1)!}},$
(46) $\displaystyle\ket{\ell_{IJ},\ell_{JK},\ell_{JI},\ell_{IK};t}=$
$\displaystyle\quad\tfrac{(L_{IJ}^{\dagger})^{\ell_{IJ}}(L_{JK}^{\dagger})^{\ell_{JK}}(L_{JI}^{\dagger})^{\ell_{JI}}(L_{IK}^{\dagger})^{\ell_{IK}}\ket{0}}{\sqrt{\frac{1}{2}\tfrac{(\ell_{IJ}+\ell_{JK}+\ell_{JI}+\ell_{IK}+|t|+2)\,\ell_{IJ}!\,\ell_{JK}!\,\ell_{JI}!\,\ell_{IK}!\,|t|!\,(\ell_{IJ}+\ell_{IK}+|t|+1)!\,(\ell_{JK}+\ell_{JI}+|t|+1)!\,\binom{\ell_{IJ}+\ell_{JK}+\ell_{JI}+\ell_{IK}+|t|+1}{\ell_{IK}}}{\binom{\ell_{IJ}+\ell_{JI}+\ell_{IK}+|t|+1}{\ell_{IK}}}}},$
(47) $\displaystyle\ket{\ell_{IJ},\ell_{JK},\ell_{KJ},\ell_{IK},t}=$
$\displaystyle\quad\tfrac{(L_{IJ}^{\dagger})^{\ell_{IJ}}(L_{JK}^{\dagger})^{\ell_{JK}}(L_{KJ}^{\dagger})^{\ell_{KJ}}(L_{IK}^{\dagger})^{\ell_{IK}}\ket{0}}{\sqrt{\frac{1}{2}\tfrac{(\ell_{IJ}+\ell_{JK}+\ell_{KJ}+\ell_{IK}+|t|+2)\,\ell_{IJ}!\,\ell_{JK}!\,\ell_{KJ}!\,\ell_{IK}!\,|t|!\,(\ell_{JK}+\ell_{IK}+|t|+1)!\,(\ell_{IJ}+\ell_{KJ}+|t|+1)!\,\binom{\ell_{IJ}+\ell_{JK}+\ell_{KJ}+\ell_{IK}+|t|+1}{\ell_{IK}}}{\binom{\ell_{JK}+\ell_{KJ}+\ell_{IK}+|t|+1}{\ell_{IK}}}}}.$
(48)

Above, the unit-normalized basis kets have been written with the notation
$\ket{\cdots}$ to distinguish them from the naive basis kets
$\left.\left|\cdots\right\rangle\right\rangle$. The above formulas have been
confirmed to agree with software-assisted implementations for a wide variety
of choices for the quantum numbers.

##  V Degenerate subspaces

###  V.1 SU(3) multiplicities, or Littlewood-Richardson coefficients

We have noted that, in the case of SU(2), the spins (the irreps) $j_{i}$ for
$i=1,2,3$ are enough to completely characterize the gauge-invariant state of a
trivalent vertex. This is because if spins $j_{1}$, $j_{2}$, and $j_{3}$ can
be added to form spin 0, then there is a unique way to do this by way of the
$3j$ symbols. An equivalent statement is that if the tensor product of
$j_{1}\otimes j_{2}$ contains $j_{3}$, then there is one and only one copy of
$j_{3}$.

For general groups, the tensor product of two irreps $\lambda$ and $\mu$ may
contain irreps for which there are multiple copies:

$\displaystyle\lambda\otimes\mu$
$\displaystyle=\bigoplus_{\nu}d_{\lambda,\mu}^{\nu}\,\nu$ (49)

where $d_{\lambda,\mu}^{\nu}$ is the multiplicity of irrep $\nu$, a
nonnegative integer also known as a Littlewood-Richardson coefficient (LRC)
Littlewood and Richardson (1934); Knuth (1970); Gopalkrishna Gadiyar and
Sharatchandra (1992); Crampé  _et al._ (2023). This is most familiar in the
case of SU(3) in the example of $(1,1)\otimes(1,1)$, which contains two copies
of the (1,1) octet:

$d_{(1,1),(1,1)}^{(1,1)}=2.$

For SU(2), the nonzero LRCs are always equal to one. But for SU(3), the
nonzero LRCs can be any positive integer. Three irreps $(p_{i},q_{i})$, for
$i=1,2,3$, meeting at the vertex then characterize a subspace with dimension
equal to the LRC:

$\displaystyle\dim(\overset{\longrightarrow}{p\,q})$
$\displaystyle=d_{(p_{1},q_{1}),(p_{2},q_{2})}^{(q_{3},p_{3})}=d_{(p_{1},q_{1}),(p_{3},q_{3})}^{(q_{2},p_{2})}=d_{(p_{2},q_{2}),(p_{3},q_{3})}^{(q_{1},p_{1})}.$
(50)

Since $\dim(\overset{\longrightarrow}{p\,q})>1$ is possible, the
$(p_{i},q_{i})$ do not in general provide a full characterization of a state,
which is to say the irreps alone do not constitute a complete set of commuting
observables.

In order to have a basis for the subspace characterized by the
$(p_{i},q_{i})$, one must identify a number of linearly independent states
equal to the associated LRC. Our reference point for evaluating LRCs is the
Littlewood-Richardson Calculator lrcalc library available in SageMath The Sage
Developers (2022), with suitable processing of the output to make it
applicable to SU(3). Numerical checks across all
$\overset{\longrightarrow}{p\,q}$ sectors we have explored indicate that LRCs
can be correctly predicted by assuming that all naive LSH basis kets belonging
to a given sector $\overset{\longrightarrow}{p\,q}$ form a basis for that
sector, and then counting up the number of basis kets. We will now present how
that counting is done.

Suppose that we have a valid choice of $\overset{\longrightarrow}{p\,q}$,
i.e., a choice of $\overset{\longrightarrow}{p\,q}$ such that
$\dim(\overset{\longrightarrow}{p\,q})\geq 1$, and we wish to find all LSH
basis kets that belong to that sector. We effectively want to invert (37) to
find solutions to the $\ell_{IJ}$ and $t$ in terms of
$\overset{\longrightarrow}{p\,q}$. It is easy to solve for $t$ uniquely
because (37) can all be combined to obtain

$\displaystyle t=\frac{1}{3}(p_{1}+p_{2}+p_{3}-q_{1}-q_{2}-q_{3}).$ (51)

What is not uniquely determined is the choice of $\ell_{IJ}$’s.

Knowing $t$, and how it contributes to the $p_{i}$ and $q_{i}$ values, we may
focus on the contributions to $p_{i}$ and $q_{i}$ coming from the
$\ell_{IJ}$’s:

$\displaystyle p_{1}^{\prime}$
$\displaystyle=\ell_{12}+\ell_{13}=p_{1}-|t|\,\theta(t),$ (52a) $\displaystyle
q_{1}^{\prime}$ $\displaystyle=\ell_{31}+\ell_{21}=q_{1}-|t|\,\theta(-t),$
(52b) $\displaystyle p_{2}^{\prime}$
$\displaystyle=\ell_{23}+\ell_{21}=p_{2}-|t|\,\theta(t),$ (52c) $\displaystyle
q_{2}^{\prime}$ $\displaystyle=\ell_{12}+\ell_{32}=q_{2}-|t|\,\theta(-t),$
(52d) $\displaystyle p_{3}^{\prime}$
$\displaystyle=\ell_{31}+\ell_{32}=p_{3}-|t|\,\theta(t),$ (52e) $\displaystyle
q_{3}^{\prime}$ $\displaystyle=\ell_{23}+\ell_{13}=q_{3}-|t|\,\theta(-t).$
(52f)

It is easy to see that there is a linear dependence among these equations
because $\sum_{i}p_{i}^{\prime}=\sum_{i}q_{i}^{\prime}$. We can therefore
remove one equation, say, that for $q_{3}^{\prime}$:

$\displaystyle p_{1}^{\prime}$ $\displaystyle=\ell_{12}+\ell_{13},$ (53a)
$\displaystyle q_{1}^{\prime}$ $\displaystyle=\ell_{31}+\ell_{21},$ (53b)
$\displaystyle p_{2}^{\prime}$ $\displaystyle=\ell_{23}+\ell_{21},$ (53c)
$\displaystyle q_{2}^{\prime}$ $\displaystyle=\ell_{12}+\ell_{32},$ (53d)
$\displaystyle p_{3}^{\prime}$ $\displaystyle=\ell_{31}+\ell_{32}.$ (53e)

This system of five equations has six unknown $\ell_{IJ}$ variables, so it is
reasonable to expect a family of solutions. Since all the $\ell_{IJ}$ are on
equal footings, let us then choose to treat $\ell_{31}$ as a parameter. We can
then invert the five equations to obtain

$\displaystyle\ell_{12}$
$\displaystyle=\ell_{31}-p_{3}^{\prime}+q_{2}^{\prime},$ (54a)
$\displaystyle\ell_{23}$
$\displaystyle=\ell_{31}+p_{2}^{\prime}-q_{1}^{\prime},$ (54b)
$\displaystyle\ell_{21}$ $\displaystyle=-\ell_{31}+q_{1}^{\prime},$ (54c)
$\displaystyle\ell_{32}$ $\displaystyle=-\ell_{31}+p_{3}^{\prime},$ (54d)
$\displaystyle\ell_{13}$
$\displaystyle=-\ell_{31}+p_{1}^{\prime}+p_{3}^{\prime}-q_{2}^{\prime}.$ (54e)

We are now free to choose $\ell_{31}$, so long as all of the other
$\ell_{IJ}$’s evaluate to nonnegative integers. Using the above equations, the
requirements that $\ell_{12}$, $\ell_{23}$ and $\ell_{31}$ be nonnegative give

$\displaystyle\ell_{31}-p_{3}^{\prime}+q_{2}^{\prime}$ $\displaystyle\geq 0,$
(55) $\displaystyle\ell_{31}+p_{2}^{\prime}-q_{1}^{\prime}$ $\displaystyle\geq
0,$ (56) $\displaystyle\ell_{31}$ $\displaystyle\geq 0,$ (57)
$\displaystyle\Rightarrow\ell_{31}$
$\displaystyle\geq\max(p_{3}^{\prime}-q_{2}^{\prime},q_{1}^{\prime}-p_{2}^{\prime},0).$
(58)

Meanwhile, the requirements that $\ell_{21}$, $\ell_{32}$ and $\ell_{13}$ be
nonnegative give

$\displaystyle-\ell_{31}+q_{1}^{\prime}$ $\displaystyle\geq 0,$ (59)
$\displaystyle-\ell_{31}+p_{3}^{\prime}$ $\displaystyle\geq 0,$ (60)
$\displaystyle-\ell_{31}+p_{1}^{\prime}+p_{3}^{\prime}-q_{2}^{\prime}$
$\displaystyle\geq 0,$ (61) $\displaystyle\Rightarrow\ell_{31}$
$\displaystyle\leq\min(q_{1}^{\prime},p_{3}^{\prime},p_{1}^{\prime}+p_{3}^{\prime}-q_{2}^{\prime}).$
(62)

In the end, we are not interested in the actual value of $\ell_{31}$, but
rather the number of valid solutions. The above bounds indicate that the
number of solutions is

$\displaystyle\dim(p_{1},q_{1},p_{2},q_{2},p_{3},q_{3})$
$\displaystyle=1+\min(q_{1}^{\prime},p_{3}^{\prime},p_{1}^{\prime}+p_{3}^{\prime}-q_{2}^{\prime})-\max(p_{3}^{\prime}-q_{2}^{\prime},q_{1}^{\prime}-p_{2}^{\prime},0).$
(63)

To obtain this result, we assumed that the choices of $(p_{i},q_{i})$ were
valid for forming a singlet configuration. This derivation in fact gives a
test for detecting valid choices of $(p_{i},q_{i})$: if the calculated $t$ is
an integer, and the limits for $\ell_{31}$ are sensible [i.e.,
$\min(q_{1}^{\prime},p_{3}^{\prime},p_{1}^{\prime}+p_{3}^{\prime}-q_{2}^{\prime})\geq\max(p_{3}^{\prime}-q_{2}^{\prime},q_{1}^{\prime}-p_{2}^{\prime},0)\geq
0$], then there exists an LSH basis state with the given
$\overset{\longrightarrow}{p\,q}$ quantum numbers, and therefore they are
valid.

As mentioned earlier, we have found the LSH-state-counting method of
evaluating $\dim(\overset{\longrightarrow}{p\,q})$ to agree with the output
derived from lrcalc in every sector we have checked. These data can be
aggregated into a summary of the total Hilbert space dimensionality of states
up to and including some cutoff value
$p_{\textrm{max}}=q_{\textrm{max}}=\Lambda$. Table 1 presents these
dimensionalities, obtained by both methods of counting, up to $\Lambda=10$.

$\Lambda=p_{\textrm{max}}=q_{\textrm{max}}$ | Dimensionality
---|---
1 | 20
2 | 185
3 | 1,023
4 | 4,072
5 | 12,937
6 | 34,940
7 | 83,454
8 | 181,107
9 | 363,910
10 | 686,531
Table 1: The total number of independent states for the trivalent vertex,
subject to a cutoff on irreps:
$\max(p_{1},q_{1},p_{2},q_{2},p_{3},q_{3})\leq\Lambda$.

One other observation is in order. Earlier, we remarked that all naive LSH
states belonging to a given $\overset{\longrightarrow}{p\,q}$ sector must be
related by (III). This is apparent from the solution to $\ell_{IJ}$’s given in
(54), where we see that increasing the value of $\ell_{31}$ by one induces the
same change to $\ell_{12}$ and $\ell_{23}$, while simultaneously decreasing
$\ell_{21}$, $\ell_{32}$, and $\ell_{13}$ by one.

###  V.2 Overlap matrices

Within those subspaces for which
$\dim(p_{1},q_{1},p_{2},q_{2},p_{3},q_{3})>1$, we find that the naive LSH
basis states generally fail to be orthogonal, presenting an obstacle to a
complete and orthonormal basis suitable for computation. To orthogonalize the
degenerate subspaces, the most obvious brute-force solution is to apply the
Gram-Schmidt procedure on the set of naive basis states. If one knows the
overlaps between all naive basis states belonging to a given sector, they will
have sufficient information to construct an orthonormal basis. In principle,
Gram-Schmidt orthogonalization of each $\overset{\longrightarrow}{p\,q}$
sector constitutes a solution to the complete construction of an orthonormal
basis. However, such an approach offers little insight into the nature of the
problem and has no connection to a seventh Casimir.

An ideal choice for the seventh Casimir would be a Hermitian operator that i)
commutes with all $P_{i}$ and $Q_{i}$, ii) always has a nondegenerate spectrum
once $\overset{\longrightarrow}{p\,q}$ is fixed [that way no further labels
are needed], and iii) has eigenstates that can be constructed systematically.
Using the LSH operators, several choices can easily be made that manifestly
satisfy (i). To satisfy (ii), we have deferred to computer algebra software to
confirm the nondegenerate spectrum, and the choices we have examined seem to
generally satisfy this criterion. We have not been successful in finding a
solution to (iii); this problem remains open and we return to it in Sec. VI.

Below we tabulate the overlaps between naive LSH basis kets in a variety of
sectors of fixed $\overset{\longrightarrow}{p\,q}$. This serves a few
purposes: follow-up works can be done with exact results to compare with, the
values themselves may be used as a reference, and patterns might later be
identified that help toward an analytic resolution to the orthogonalization
problem. Below we display results for a selection of 2- and 3-dimensional
sectors, and some additional data for 4-dimensional sectors may be found in
Appendix B.

$\overset{\longrightarrow}{p\,q}=(1,1,1,1,1,1)$:

$\displaystyle\left(\begin{array}[]{c}\left\langle\left\langle
1\,1\,1;0\,0\,0;0\right|\right.\\\ \left\langle\left\langle
0\,0\,0;1\,1\,1;0\right|\right.\end{array}\right)\left(\begin{array}[]{ccc}\left.\left|1\,1\,1;0\,0\,0;0\right\rangle\right\rangle,&\left.\left|0\,0\,0;1\,1\,1;0\right\rangle\right\rangle\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{cc}\frac{56}{3}&\frac{-16}{3}\\\
\frac{-16}{3}&\frac{56}{3}\end{array}\right)$ (69)

$\overset{\longrightarrow}{p\,q}=(2,1,2,1,2,1)$:

$\displaystyle\left(\begin{array}[]{c}\left\langle\left\langle
1\,1\,1;0\,0\,0;1\right|\right.\\\ \left\langle\left\langle
0\,0\,0;1\,1\,1;1\right|\right.\end{array}\right)\left(\begin{array}[]{ccc}\left.\left|1\,1\,1;0\,0\,0;1\right\rangle\right\rangle,&\left.\left|0\,0\,0;1\,1\,1;1\right\rangle\right\rangle\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{cc}315&-45\\\ -45&315\\\
\end{array}\right)$ (75)

$\overset{\longrightarrow}{p\,q}=(2,1,1,2,1,1)$:

$\displaystyle\left(\begin{array}[]{c}\left\langle\left\langle
2\,1\,1;0\,0\,0;0\right|\right.\\\ \left\langle\left\langle
1\,0\,0;1\,1\,1;0\right|\right.\end{array}\right)\left(\begin{array}[]{ccc}\left.\left|2\,1\,1;0\,0\,0;0\right\rangle\right\rangle,&\left.\left|1\,0\,0;1\,1\,1;0\right\rangle\right\rangle\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{cc}130&-35\\\ -35&\frac{185}{2}\\\
\end{array}\right)$ (81)

$\overset{\longrightarrow}{p\,q}=(2,1,2,2,1,2)$:

$\displaystyle\left(\begin{array}[]{c}\left\langle\left\langle
2\,2\,1;0\,0\,0;0\right|\right.\\\ \left\langle\left\langle
1\,1\,0;1\,1\,1;0\right|\right.\end{array}\right)\left(\begin{array}[]{ccc}\left.\left|2\,2\,1;0\,0\,0;0\right\rangle\right\rangle,&\left.\left|1\,1\,0;1\,1\,1;0\right\rangle\right\rangle\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{cc}864&-216\\\ -216&432\\\
\end{array}\right)$ (87)

$\overset{\longrightarrow}{p\,q}=(2,2,2,2,2,2)$:

$\displaystyle\left(\begin{array}[]{c}\left\langle\left\langle
2\,2\,2;0\,0\,0;0\right|\right.\\\ \left\langle\left\langle
1\,1\,1;1\,1\,1;0\right|\right.\\\ \left\langle\left\langle
0\,0\,0;2\,2\,2;0\right|\right.\end{array}\right)\left(\begin{array}[]{ccc}\left.\left|2\,2\,2;0\,0\,0;0\right\rangle\right\rangle,&\left.\left|1\,1\,1;1\,1\,1;0\right\rangle\right\rangle,&\left.\left|0\,0\,0;2\,2\,2;0\right\rangle\right\rangle\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{ccc}\frac{136512}{25}&-\frac{31104}{25}&\frac{15552}{25}\\\
-\frac{31104}{25}&\frac{47088}{25}&-\frac{31104}{25}\\\
\frac{15552}{25}&-\frac{31104}{25}&\frac{136512}{25}\end{array}\right)$ (95)

$\overset{\longrightarrow}{p\,q}=(3,2,2,3,2,2)$:

$\displaystyle\left(\begin{array}[]{c}\left\langle\left\langle
3\,2\,2;0\,0\,0;0\right|\right.\\\ \left\langle\left\langle
2\,1\,1;1\,1\,1;0\right|\right.\\\ \left\langle\left\langle
1\,0\,0;2\,2\,2;0\right|\right.\end{array}\right)\left(\begin{array}[]{ccc}\left.\left|3\,2\,2;0\,0\,0;0\right\rangle\right\rangle,&\left.\left|2\,1\,1;1\,1\,1;0\right\rangle\right\rangle,&\left.\left|1\,0\,0;2\,2\,2;0\right\rangle\right\rangle\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{ccc}\frac{350784}{5}&-\frac{74592}{5}&\frac{36288}{5}\\\
-\frac{74592}{5}&\frac{101136}{5}&-\frac{65184}{5}\\\
\frac{36288}{5}&-\frac{65184}{5}&\frac{200256}{5}\end{array}\right)$ (103)

###  V.3 Seventh Casimir candidates and orthogonalization

We have noted that a good seventh quantum number or seventh Casimir would be a
Hermitian operator that conserves all the $p_{i}$ and $q_{i}$, and has a
nondegenerate spectrum within any fixed $\overset{\longrightarrow}{p\,q}$
sector. To conserve all the $(p_{i},q_{i})$, note that LSH gauge-singlet
operators all induce definite changes on $\overset{\longrightarrow}{p\,q}$;
for example, $L_{12}^{\dagger}$ applied to a ket increases $p_{1}$ and $q_{2}$
by 1 and does not change any of $\\{p_{2},q_{1},p_{3},q_{3}\\}$, while $T_{A}$
decreases all $p_{i}$ by one and leaves all $q_{i}$ alone. One can then
multiply operators together to form a candidate seventh Casimir that makes no
net change on any of the $p_{i}$ or $q_{i}$.

Additionally, it may be useful to construct the seventh Casimir such that it
annihilates states that are necessarily nondegenerate. Noting that degeneracy
requires all $p_{i}\geq 1$ and $q_{i}\geq 1$, this property can be enforced by
putting to the right-hand side of the operator some product that lowers all
$p_{i}$ and $q_{i}$ by one; three obvious choices are $L_{12}L_{23}L_{31}$,
$L_{21}L_{32}L_{13}$, and $T_{A}T_{B}$. If we then multiply with the adjoint
of such a product, we will have something that conserves the $P_{i}$ and
$Q_{i}$, and is manifestly Hermitian:
$(L_{12}L_{23}L_{31})^{\dagger}L_{12}L_{23}L_{31}$,
$(L_{21}L_{32}L_{13})^{\dagger}L_{21}L_{32}L_{13}$, and
$(T_{A}T_{B})^{\dagger}T_{A}T_{B}$ are all candidates for the seventh Casimir.
In the following, we will focus on the choice

$\displaystyle C_{T}$ $\displaystyle\equiv(T_{A}T_{B})^{\dagger}T_{A}T_{B}.$
(104)

This is not the only candidate we have studied, but in the low-dimensional
subspaces we have calculated it seems to have matrix elements and eigenstates
that are relatively simple when compared to other candidates. Furthermore, the
use of $T_{A}T_{B}$ seems like a natural choice given its appearance in the
special constraint (27).

To diagonalize (i.e., find the eigenbasis of) $C_{T}$ in a given
$\overset{\longrightarrow}{p\,q}$ sector, it is sufficient to evaluate the
overlap matrix in that sector and the matrix elements of $C_{T}$ between the
naive basis states. We let $O$ and $\tilde{C}$ denote the
$\dim(\overset{\longrightarrow}{p\,q})\times\dim(\overset{\longrightarrow}{p\,q})$
matrices whose elements are defined by

$\displaystyle O_{ij}$
$\displaystyle\equiv\left\langle\braket{i}{j}\right\rangle,$ (105)
$\displaystyle\tilde{C}_{ij}$ $\displaystyle\equiv\left\langle\left\langle
i\right|\right.C_{T}\left.\left|j\right\rangle\right\rangle,$ (106)

where $\left.\left|j\right\rangle\right\rangle$ is the $j^{\mathrm{th}}$ naive
basis ket in the sector $\overset{\longrightarrow}{p\,q}$ as previously
ordered. The matrices $O$ and $\tilde{C}$ can be used to obtain the matrix
representation of $C_{T}$ with respect to the naive basis, that is, the
$\dim(\overset{\longrightarrow}{p\,q})\times\dim(\overset{\longrightarrow}{p\,q})$
matrix $[C_{T}]_{\mathrm{naive}}$ whose elements
$([C_{T}]_{\mathrm{naive}})_{ij}$ are defined by

$\displaystyle C_{T}\left.\left|j\right\rangle\right\rangle$
$\displaystyle\equiv\sum_{i}([C_{T}]_{\mathrm{naive}})_{ij}\left.\left|i\right\rangle\right\rangle.$
(107)

The relationship among these three matrices is
$\tilde{C}=O[C_{T}]_{\mathrm{naive}}$, or

$\displaystyle[C_{T}]_{\mathrm{naive}}$ $\displaystyle=O^{-1}\tilde{C}.$ (108)

One then finds the eigenvalues and eigenvectors of $[C_{T}]_{\mathrm{naive}}$,
where of course the eigenvectors will have coordinates with respect to the
naive basis kets.

For normalization, let $\vec{v}_{i}$ denote the eigenvectors of
$[C_{T}]_{\mathrm{naive}}$. If $S$ is a matrix with components

$\displaystyle S_{ij}$ $\displaystyle=(\vec{v}_{j})_{i},$ (109)

so that the unnormalized eigenstates are

$\displaystyle\left.\left|\phi_{i}\right\rangle\right\rangle$
$\displaystyle=\sum_{j}S_{ji}\left.\left|j\right\rangle\right\rangle,$ (110)

then the eigenstates’ squared norms are evaluated as

$\displaystyle\left\langle\braket{\phi_{i}}{\phi_{i}}\right\rangle$
$\displaystyle=(S^{\dagger}OS)_{ii}.\qquad\textrm{(no sum)}$ (111)

Thus, the normalized eigenstate is

$\displaystyle\ket{\phi_{i}}$
$\displaystyle=\frac{1}{\sqrt{(S^{\dagger}OS)_{ii}}}\sum_{j}S_{ji}\left.\left|j\right\rangle\right\rangle.$
(112)

Below we present the diagonalization of $C_{T}$ for a variety of low-
dimensional sectors. The results are presented as follows: In the sector
$(p_{1},q_{1},p_{2},q_{2},p_{3},q_{3})$, the eigenvalues of $C_{T}$ are
denoted by $\mathrm{Spec}_{p_{1}q_{1}p_{2}q_{2}p_{3}q_{3}}(C_{T})$ and
presented in increasing order. The eigenstates of $C_{T}$,
$\left.\left|\phi_{n}\right\rangle\right\rangle_{p_{1}q_{1}p_{2}q_{2}p_{3}q_{3}}$,
are ordered in the same way, with $n=1$
($n=\dim(p_{1},q_{1},p_{2},q_{2},p_{3},q_{3})$) identifying the eigenvector
with the lowest (highest) eigenvalue.

Note that in all sectors we have explored, we have confirmed the eigenvalues
of $C_{T}$ to be nondegenerate, and that the lowest eigenvalue is zero. (We
have also confirmed some of the nondegenerate states to be in the nullspace of
$C_{T}$, including nontrivial examples where
$\min(p_{1},q_{1},p_{2},q_{2},p_{3},q_{3})\geq 1$.) We believe these
properties of $C_{T}$—nondegeneracy, and a lowest eigenvalue equal to zero—are
applicable to any given $\overset{\longrightarrow}{p\,q}$ sector, but we do
not have a proof.

$\displaystyle\mathrm{Spec}_{111111}\bigl{(}C_{T}\bigr{)}$
$\displaystyle=\left\\{0,\frac{80}{3}\right\\},$ (113)
$\displaystyle\left(\begin{array}[]{c}\left.\left|\phi_{1}\right\rangle\right\rangle_{111111}\\\
\left.\left|\phi_{2}\right\rangle\right\rangle_{111111}\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{cc}1&-1\\\ 1&1\\\
\end{array}\right)\left(\begin{array}[]{c}\left.\left|1\,1\,1;0\,0\,0;0\right\rangle\right\rangle\\\
\left.\left|0\,0\,0;1\,1\,1;0\right\rangle\right\rangle\\\
\end{array}\right).$ (120)

$\displaystyle\mathrm{Spec}_{212121}\bigl{(}C_{T}\bigr{)}$
$\displaystyle=\left\\{0,90\right\\},$ (121)
$\displaystyle\left(\begin{array}[]{c}\left.\left|\phi_{1}\right\rangle\right\rangle_{212121}\\\
\left.\left|\phi_{2}\right\rangle\right\rangle_{212121}\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{cc}1&-1\\\ 1&1\\\
\end{array}\right)\left(\begin{array}[]{c}\left.\left|1\,1\,1;0\,0\,0;1\right\rangle\right\rangle\\\
\left.\left|0\,0\,0;1\,1\,1;1\right\rangle\right\rangle\\\
\end{array}\right).$ (128)

$\displaystyle\mathrm{Spec}_{211211}\bigl{(}C_{T}\bigr{)}$
$\displaystyle=\left\\{0,\frac{305}{6}\right\\},$ (129)
$\displaystyle\left(\begin{array}[]{c}\left.\left|\phi_{1}\right\rangle\right\rangle_{211211}\\\
\left.\left|\phi_{2}\right\rangle\right\rangle_{211211}\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{cc}-\frac{23}{38}&1\\\ 1&1\\\
\end{array}\right)\left(\begin{array}[]{c}\left.\left|2\,1\,1;0\,0\,0;0\right\rangle\right\rangle\\\
\left.\left|1\,0\,0;1\,1\,1;0\right\rangle\right\rangle\end{array}\right).$
(136)

$\displaystyle\mathrm{Spec}_{212212}\bigl{(}C_{T}\bigr{)}$
$\displaystyle=\left\\{0,108\right\\},$ (137)
$\displaystyle\left(\begin{array}[]{c}\ket{\phi_{1}}_{212212}\\\
\ket{\phi_{2}}_{212212}\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{cc}\sqrt{\frac{1}{672}}&0\\\
0&\sqrt{\frac{1}{864}}\\\
\end{array}\right)\left(\begin{array}[]{cc}-\frac{1}{3}&1\\\ 1&1\\\
\end{array}\right)\left(\begin{array}[]{c}\left.\left|2\,2\,1;0\,0\,0;0\right\rangle\right\rangle\\\
\left.\left|1\,1\,0;1\,1\,1;0\right\rangle\right\rangle\end{array}\right).$
(146)

$\displaystyle\mathrm{Spec}_{222222}\bigl{(}C_{T}\bigr{)}$
$\displaystyle=\left\\{0,\frac{1008}{5},\frac{45684}{125}\right\\},$ (147)
$\displaystyle\left(\begin{array}[]{c}\left.\left|\phi_{1}\right\rangle\right\rangle_{222222}\\\
\left.\left|\phi_{2}\right\rangle\right\rangle_{222222}\\\
\left.\left|\phi_{3}\right\rangle\right\rangle_{222222}\\\ \end{array}\right)$
$\displaystyle=\left(\begin{array}[]{ccc}-\tfrac{37}{208}&1&-\tfrac{37}{208}\\\
1&0&-1\\\ \tfrac{1}{2}&1&\tfrac{1}{2}\\\
\end{array}\right)\left(\begin{array}[]{c}\left.\left|2\,2\,2;0\,0\,0;0\right\rangle\right\rangle\\\
\left.\left|1\,1\,1;1\,1\,1;0\right\rangle\right\rangle\\\
\left.\left|0\,0\,0;2\,2\,2;0\right\rangle\right\rangle\\\
\end{array}\right).$ (157)

$\displaystyle\mathrm{Spec}_{322322}\bigl{(}C_{T}\bigr{)}$
$\displaystyle=\left\\{0,\tfrac{7}{900}\left(59609-\sqrt{383666161}\right),\tfrac{7}{900}\left(59609+\sqrt{383666161}\right)\right\\},$
(158)
$\displaystyle\left(\begin{array}[]{c}\left.\left|\phi_{1}\right\rangle\right\rangle_{322322}\\\
\left.\left|\phi_{2}\right\rangle\right\rangle_{322322}\\\
\left.\left|\phi_{3}\right\rangle\right\rangle_{322322}\\\ \end{array}\right)$
$\displaystyle=\left(\begin{array}[]{ccc}-\frac{64}{501}&1&-\frac{101}{334}\\\
\frac{6869-\sqrt{383666161}}{24500}&\frac{31369-\sqrt{383666161}}{24500}&1\\\
\frac{\sqrt{383666161}-6865}{24504}&1&\frac{31369-\sqrt{383666161}}{24504}\\\
\end{array}\right)\left(\begin{array}[]{c}\left.\left|3\,2\,2;0\,0\,0;0\right\rangle\right\rangle\\\
\left.\left|2\,1\,1;1\,1\,1;0\right\rangle\right\rangle\\\
\left.\left|1\,0\,0;2\,2\,2;0\right\rangle\right\rangle\\\
\end{array}\right).$ (168)

##  VI Discussion

In this paper, we have presented a solution to the construction of SU(3)
gauge-singlet operators and an orthonormal basis at a trivalent vertex. We
have done this without any need for SU(3) Clebsch-Gordon coefficients, and the
underlying Hilbert space in which calculations are ultimately performed is
nothing but the Hilbert space of a collection of harmonic oscillators that are
all on the same footing. The use of irreducible Schwinger bosons at the
operator level was necessary to avoid $\mathrm{Sp}(2,\mathbb{R})$ redundancies
that would arise from creating representations out of ordinary bosonic
creation and annihilation operators.

Certain features carried over from the SU(2) loop-string-hadron formulation:
the construction of elementary excitations by contracting fundamental and/or
antifundamental bosonic creation operators together via SU(3) invariant
tensors again turns out the provide an elegant way of writing down a complete
basis. In SU(3), the LSH states can be categorized into a few different
classes depending on the which $\ell_{IJ}$ are vanishing. For the subspace of
states belonging to what we have called Classes I, IIa, and IIb, we have
calculated a closed-form analytic solution to the orthonormal basis states.
These states bear a clear resemblance to the orthonormal loop states of SU(2)
Yang-Mills Raychowdhury (2019); Raychowdhury and Stryker (2020a).

States living outside of Classes I and II, however, differ in an important way
from the states of the SU(2) theory. For these states, the irreps of the
vertex legs provide an incomplete characterization of the state, and one must
appeal to a seventh Casimir – a degree of freedom that we interpret as being
‘internal’ to the vertex. We are not the first group to explore this ‘missing
label’ or ‘multiplicity problem’, that is an old topic of interest in the
group theory of SU(3). As mentioned, one may always deal with the multiplicity
problem by applying Gram-Schmidt orthogonalization on the naive LSH basis
states, but we propose to alternatively find and diagonalize a Hermitian
operator whose spectrum is nondegenerate within any given
$\overset{\longrightarrow}{p\,q}$ sector. With the tools available to us from
the loop-string-hadron approach, we could easily identify a few candidate
Hermitian operators to serve as the seventh Casimir. The candidate that we
have focused on, $C_{T}$, seems well-motivated theoretically, and appears to
have some nice properties. In all examples we have checked, for a given
$\overset{\longrightarrow}{p\,q}$, we have found $C_{T}$ to be nondegenerate
and to have zero as its lowest eigenvalue. In some of the low-dimensional
sectors, we have found a number of eigenstates of $C_{T}$ whose coefficients
in the naive basis have simple ratios. The departures from the simple ratios
of coefficients in the higher-dimensional sectors could be a hint that the
ideal choice of seventh Casimir is related to our $C_{T}$ but somehow
different. Unfortunately, evaluating the matrix elements of $C_{T}$ gets
computationally expensive quickly as a function of the subspace dimension,
such that finding the eigenstates becomes a much slower process than simple
Gram-Schmidt orthogonalization.

We note that ultimately what is sought is a systematically constructible
orthonormal basis; having this would allow one to evaluate the matrix elements
of any gauge-singlet contracted operator in full generality, at which point
the underlying basis for computation could be entirely transitioned into seven
LSH quantum numbers and one could retire the system of eighteen harmonic
oscillators. We are optimistic that a closed-form solution does exist,
although we have not been able to find it yet. We think finding the
eigenstates of a well-chosen Hermitian operator is a promising path toward the
solution. We speculate that what may be missing is some kind of ladder
operator that would enable one to reach all eigenstates in a
$\overset{\longrightarrow}{p\,q}$ sector once one of them has been found. The
fact that there is always a zero in the spectrum of our $C_{T}$ seems to
support this idea; this would correspond to the state at the “bottom of the
ladder.” It is perhaps not an accident that the lowest eigenvector of $C_{T}$
in the example sectors we have shown always had rational coefficients in the
naive basis (up to overall normalization), while we have confirmed higher
states to often have irrational coefficients. Another attractive possibility
is that there may be a correspondence of states between different
$\overset{\longrightarrow}{p\,q}$ sectors; the knowledge of an orthogonal
basis in some sector might be used to construct the orthogonal basis of
another sector. Yet another possibility is that there may be ladder operators
that connect orthogonal states in different $\overset{\longrightarrow}{p\,q}$
sectors.

In future work, we will further examine the properties of LSH gauge-singlet
operators—particularly those that appear in the plaquette operators of higher-
dimensional theories–in combination with orthonormal bases that are suitable
for computation. Additionally, we seek to understand the details of how “point
splitting” should work for SU(3) vertices of valency four or six, which are
relevant to square and cubic lattices.

##  VII Acknowledgments

The authors would like to thank Anthony Ciavarella, Zohreh Davoudi, David B.
Kaplan, John Lombard and Himadri Mukherjee for insightful conversations at
various points throughout this work. Work by JRS was supported by the U.S.
Department of Energy (DOE), Office of Science under contract DE-
AC02-05CH11231, partially through Quantum Information Science Enabled
Discovery (QuantISED) for High Energy Physics (KA2401032). JRS and SK both
received support from the U.S. Department of Energy’s Office of Science Early
Career Award under award DE-SC0020271, for theoretical developments for
simulating lattice gauge theories on quantum computers. SK acknowledges
support by the U.S. DOE, Office of Science, Office of Nuclear Physics,
InQubator for Quantum Simulation (IQuS) (award no. DE-SC0020970), and by the
DOE QuantISED program through the theory consortium “Intersections of QIS and
Theoretical Particle Physics” at Fermilab (Fermilab subcontract no. 666484).
SK further acknowledges the support from the Department of Physics and the
College of Arts and Sciences at the University of Washington. Research of IR
is supported by the OPERA award (FR/SCM/11-Dec-2020/PHY) from BITS-Pilani, the
Start-up Research Grant (SRG/2022/000972) and Core-Research Grant
(CRG/2022/007312) from ANRF, India and the cross-discipline research fund
(C1/23/185) from BITS-Pilani. AN is supported by the Start-up Research Grant
(SRG/2022/000972) from ANRF, India received by IR.

## References

  * Gross  _et al._ (2023) Franz Gross _et al._ , “50 Years of Quantum Chromodynamics,” Eur. Phys. J. C 83, 1125 (2023), arXiv:2212.11107 [hep-ph] .
  * Bauer  _et al._ (2023a) Christian W. Bauer _et al._ , “Quantum Simulation for High-Energy Physics,” PRX Quantum 4, 027001 (2023a), arXiv:2204.03381 [quant-ph] .
  * Di Meglio  _et al._ (2023) Alberto Di Meglio _et al._ , “Quantum Computing for High-Energy Physics: State of the Art and Challenges. Summary of the QC4HEP Working Group,” (2023), arXiv:2307.03236 [quant-ph] .
  * Bauer  _et al._ (2023b) Christian W. Bauer, Zohreh Davoudi, Natalie Klco, and Martin J. Savage, “Quantum simulation of fundamental particles and forces,” Nature Rev. Phys. 5, 420–432 (2023b), arXiv:2404.06298 [hep-ph] .
  * Bañuls and Cichy (2020) Mari Carmen Bañuls and Krzysztof Cichy, “Review on Novel Methods for Lattice Gauge Theories,” Rept. Prog. Phys. 83, 024401 (2020), arXiv:1910.00257 [hep-lat] .
  * Meurice  _et al._ (2022) Yannick Meurice, Ryo Sakai, and Judah Unmuth-Yockey, “Tensor lattice field theory for renormalization and quantum computing,” Rev. Mod. Phys. 94, 025005 (2022), arXiv:2010.06539 [hep-lat] .
  * Banerjee  _et al._ (2012) D. Banerjee, M. Dalmonte, M. Muller, E. Rico, P. Stebler, U. J. Wiese, and P. Zoller, “Atomic Quantum Simulation of Dynamical Gauge Fields coupled to Fermionic Matter: From String Breaking to Evolution after a Quench,” Phys. Rev. Lett. 109, 175302 (2012), arXiv:1205.6366 [cond-mat.quant-gas] .
  * Banerjee  _et al._ (2013) D. Banerjee, M. Bögli, M. Dalmonte, E. Rico, P. Stebler, U. J. Wiese, and P. Zoller, “Atomic Quantum Simulation of U(N) and SU(N) Non-Abelian Lattice Gauge Theories,” Phys. Rev. Lett. 110, 125303 (2013), arXiv:1211.2242 [cond-mat.quant-gas] .
  * Huffman  _et al._ (2022) Emilie Huffman, Miguel García Vera, and Debasish Banerjee, “Toward the real-time evolution of gauge-invariant $\mathbb{Z}_{2}$ and $U(1)$ quantum link models on noisy intermediate-scale quantum hardware with error mitigation,” Phys. Rev. D 106, 094502 (2022), arXiv:2109.15065 [quant-ph] .
  * Klco  _et al._ (2018) N. Klco, E. F. Dumitrescu, A. J. McCaskey, T. D. Morris, R. C. Pooser, M. Sanz, E. Solano, P. Lougovski, and M. J. Savage, “Quantum-classical computation of Schwinger model dynamics using quantum computers,” Phys. Rev. A 98, 032331 (2018), arXiv:1803.03326 [quant-ph] .
  * Klco  _et al._ (2020) Natalie Klco, Jesse R. Stryker, and Martin J. Savage, “SU(2) non-Abelian gauge field theory in one dimension on digital quantum computers,” Phys. Rev. D 101, 074512 (2020), arXiv:1908.06935 [quant-ph] .
  * Ciavarella  _et al._ (2021) Anthony Ciavarella, Natalie Klco, and Martin J. Savage, “Trailhead for quantum simulation of SU(3) Yang-Mills lattice gauge theory in the local multiplet basis,” Phys. Rev. D 103, 094501 (2021), arXiv:2101.10227 [quant-ph] .
  * Stetina  _et al._ (2022) Torin F. Stetina, Anthony Ciavarella, Xiaosong Li, and Nathan Wiebe, “Simulating Effective QED on Quantum Computers,” Quantum 6, 622 (2022), arXiv:2101.00111 [quant-ph] .
  * Ciavarella and Chernyshev (2022) Anthony N. Ciavarella and Ivan A. Chernyshev, “Preparation of the SU(3) lattice Yang-Mills vacuum with variational quantum methods,” Phys. Rev. D 105, 074504 (2022), arXiv:2112.09083 [quant-ph] .
  * Farrell  _et al._ (2023a) Roland C. Farrell, Ivan A. Chernyshev, Sarah J. M. Powell, Nikita A. Zemlevskiy, Marc Illa, and Martin J. Savage, “Preparations for quantum simulations of quantum chromodynamics in 1+1 dimensions. I. Axial gauge,” Phys. Rev. D 107, 054512 (2023a), arXiv:2207.01731 [quant-ph] .
  * Turro  _et al._ (2024) Francesco Turro, Anthony Ciavarella, and Xiaojun Yao, “Classical and quantum computing of shear viscosity for (2+1)D SU(2) gauge theory,” Phys. Rev. D 109, 114511 (2024), arXiv:2402.04221 [hep-lat] .
  * Ciavarella and Bauer (2024) Anthony N. Ciavarella and Christian W. Bauer, “Quantum Simulation of SU(3) Lattice Yang Mills Theory at Leading Order in Large N,” (2024), arXiv:2402.10265 [hep-ph] .
  * Farrell  _et al._ (2023b) Roland C. Farrell, Ivan A. Chernyshev, Sarah J. M. Powell, Nikita A. Zemlevskiy, Marc Illa, and Martin J. Savage, “Preparations for quantum simulations of quantum chromodynamics in 1+1 dimensions. II. Single-baryon $\beta$-decay in real time,” Phys. Rev. D 107, 054513 (2023b), arXiv:2209.10781 [quant-ph] .
  * Farrell  _et al._ (2024a) Roland C. Farrell, Marc Illa, Anthony N. Ciavarella, and Martin J. Savage, “Quantum simulations of hadron dynamics in the Schwinger model using 112 qubits,” Phys. Rev. D 109, 114510 (2024a), arXiv:2401.08044 [quant-ph] .
  * Farrell  _et al._ (2024b) Roland C. Farrell, Marc Illa, and Martin J. Savage, “Steps Toward Quantum Simulations of Hadronization and Energy-Loss in Dense Matter,” (2024b), arXiv:2405.06620 [quant-ph] .
  * Haase  _et al._ (2021) Jan F. Haase, Luca Dellantonio, Alessio Celi, Danny Paulson, Angus Kan, Karl Jansen, and Christine A. Muschik, “A resource efficient approach for quantum and classical simulations of gauge theories in particle physics,” Quantum 5, 393 (2021), arXiv:2006.14160 [quant-ph] .
  * Dasgupta and Raychowdhury (2022) Raka Dasgupta and Indrakshi Raychowdhury, “Cold-atom quantum simulator for string and hadron dynamics in non-Abelian lattice gauge theory,” Phys. Rev. A 105, 023322 (2022), arXiv:2009.13969 [hep-lat] .
  * Bennewitz  _et al._ (2024) Elizabeth R. Bennewitz _et al._ , “Simulating Meson Scattering on Spin Quantum Simulators,” (2024), arXiv:2403.07061 [quant-ph] .
  * Davoudi  _et al._ (2024) Zohreh Davoudi, Chung-Chun Hsieh, and Saurabh V. Kadam, “Scattering wave packets of hadrons in gauge theories: Preparation on a quantum computer,” (2024), arXiv:2402.00840 [quant-ph] .
  * Belyansky  _et al._ (2024) Ron Belyansky, Seth Whitsitt, Niklas Mueller, Ali Fahimniya, Elizabeth R. Bennewitz, Zohreh Davoudi, and Alexey V. Gorshkov, “High-Energy Collision of Quarks and Mesons in the Schwinger Model: From Tensor Networks to Circuit QED,” Phys. Rev. Lett. 132, 091903 (2024), arXiv:2307.02522 [quant-ph] .
  * Davoudi  _et al._ (2023a) Zohreh Davoudi, Alexander F. Shaw, and Jesse R. Stryker, “General quantum algorithms for Hamiltonian simulation with applications to a non-Abelian lattice gauge theory,” Quantum 7, 1213 (2023a), arXiv:2212.14030 [hep-lat] .
  * Davoudi  _et al._ (2020) Zohreh Davoudi, Mohammad Hafezi, Christopher Monroe, Guido Pagano, Alireza Seif, and Andrew Shaw, “Towards analog quantum simulations of lattice gauge theories with trapped ions,” Phys. Rev. Res. 2, 023015 (2020), arXiv:1908.03210 [quant-ph] .
  * Mueller  _et al._ (2023) Niklas Mueller, Joseph A. Carolan, Andrew Connelly, Zohreh Davoudi, Eugene F. Dumitrescu, and Kübra Yeter-Aydeniz, “Quantum Computation of Dynamical Quantum Phase Transitions and Entanglement Tomography in a Lattice Gauge Theory,” PRX Quantum 4, 030323 (2023), arXiv:2210.03089 [quant-ph] .
  * Davoudi  _et al._ (2023b) Zohreh Davoudi, Niklas Mueller, and Connor Powers, “Towards Quantum Computing Phase Diagrams of Gauge Theories with Thermal Pure Quantum States,” Phys. Rev. Lett. 131, 081901 (2023b), arXiv:2208.13112 [hep-lat] .
  * Nguyen  _et al._ (2022) Nhung H. Nguyen, Minh C. Tran, Yingyue Zhu, Alaina M. Green, C. Huerta Alderete, Zohreh Davoudi, and Norbert M. Linke, “Digital Quantum Simulation of the Schwinger Model and Symmetry Protection with Trapped Ions,” PRX Quantum 3, 020324 (2022), arXiv:2112.14262 [quant-ph] .
  * Atas  _et al._ (2023) Yasar Y. Atas, Jan F. Haase, Jinglei Zhang, Victor Wei, Sieglinde M. L. Pfaendler, Randy Lewis, and Christine A. Muschik, “Simulating one-dimensional quantum chromodynamics on a quantum computer: Real-time evolutions of tetra- and pentaquarks,” Phys. Rev. Res. 5, 033184 (2023), arXiv:2207.03473 [quant-ph] .
  * Atas  _et al._ (2021) Yasar Y. Atas, Jinglei Zhang, Randy Lewis, Amin Jahanpour, Jan F. Haase, and Christine A. Muschik, “SU(2) hadrons on a quantum computer via a variational approach,” Nature Commun. 12, 6499 (2021), arXiv:2102.08920 [quant-ph] .
  * Martinez  _et al._ (2016) E. A. Martinez _et al._ , “Real-time dynamics of lattice gauge theories with a few-qubit quantum computer,” Nature 534, 516–519 (2016), arXiv:1605.04570 [quant-ph] .
  * Kasper  _et al._ (2017) V. Kasper, F. Hebenstreit, F. Jendrzejewski, M. K. Oberthaler, and J. Berges, “Implementing quantum electrodynamics with ultracold atomic systems,” New J. Phys. 19, 023030 (2017), arXiv:1608.03480 [cond-mat.quant-gas] .
  * Kane  _et al._ (2022) Christopher Kane, Dorota M. Grabowska, Benjamin Nachman, and Christian W. Bauer, “Efficient quantum implementation of 2+1 U(1) lattice gauge theories with Gauss law constraints,” (2022), arXiv:2211.10497 [quant-ph] .
  * Mil  _et al._ (2020) Alexander Mil, Torsten V. Zache, Apoorva Hegde, Andy Xia, Rohit P. Bhatt, Markus K. Oberthaler, Philipp Hauke, Jürgen Berges, and Fred Jendrzejewski, “A scalable realization of local U(1) gauge invariance in cold atomic mixtures,” Science 367, 1128–1130 (2020), arXiv:1909.07641 [cond-mat.quant-gas] .
  * Muschik  _et al._ (2017) Christine Muschik, Markus Heyl, Esteban Martinez, Thomas Monz, Philipp Schindler, Berit Vogell, Marcello Dalmonte, Philipp Hauke, Rainer Blatt, and Peter Zoller, “U(1) Wilson lattice gauge theories in digital quantum simulators,” New J. Phys. 19, 103020 (2017), arXiv:1612.08653 [quant-ph] .
  * Meth  _et al._ (2023) Michael Meth _et al._ , “Simulating 2D lattice gauge theories on a qudit quantum computer,” (2023), arXiv:2310.12110 [quant-ph] .
  * Zhang  _et al._ (2023a) Jinglei Zhang, Ryan Ferguson, Stefan Kühn, Jan F. Haase, C. M. Wilson, Karl Jansen, and Christine A. Muschik, “Simulating gauge theories with variational quantum eigensolvers in superconducting microwave cavities,” Quantum 7, 1148 (2023a), arXiv:2108.08248 [quant-ph] .
  * Paulson  _et al._ (2021) Danny Paulson _et al._ , “Simulating 2D Effects in Lattice Gauge Theories on a Quantum Computer,” PRX Quantum 2, 030334 (2021), arXiv:2008.09252 [quant-ph] .
  * Guo  _et al._ (2024) Yibin Guo, Takis Angelides, Karl Jansen, and Stefan Kühn, “Concurrent VQE for Simulating Excited States of the Schwinger Model,” (2024), arXiv:2407.15629 [quant-ph] .
  * Crippa  _et al._ (2024) Arianna Crippa, Simone Romiti, Lena Funcke, Karl Jansen, Stefan Kühn, Paolo Stornati, and Carsten Urbach, “Towards determining the (2+1)-dimensional Quantum Electrodynamics running coupling with Monte Carlo and quantum computing methods,” (2024), arXiv:2404.17545 [hep-lat] .
  * Angelides  _et al._ (2023) Takis Angelides, Pranay Naredi, Arianna Crippa, Karl Jansen, Stefan Kühn, Ivano Tavernelli, and Derek S. Wang, “First-Order Phase Transition of the Schwinger Model with a Quantum Computer,” (2023), arXiv:2312.12831 [hep-lat] .
  * Chakraborty  _et al._ (2022) Bipasha Chakraborty, Masazumi Honda, Taku Izubuchi, Yuta Kikuchi, and Akio Tomiya, “Classically emulated digital quantum simulation of the Schwinger model with a topological term via adiabatic state preparation,” Phys. Rev. D 105, 094503 (2022), arXiv:2001.00485 [hep-lat] .
  * Halimeh  _et al._ (2023) Jad C. Halimeh, Monika Aidelsburger, Fabian Grusdt, Philipp Hauke, and Bing Yang, “Cold-atom quantum simulators of gauge theories,” (2023), arXiv:2310.12201 [cond-mat.quant-gas] .
  * de Jong  _et al._ (2022) Wibe A. de Jong, Kyle Lee, James Mulligan, Mateusz Płoskoń, Felix Ringer, and Xiaojun Yao, “Quantum simulation of nonequilibrium dynamics and thermalization in the Schwinger model,” Phys. Rev. D 106, 054508 (2022), arXiv:2106.08394 [quant-ph] .
  * Lamm  _et al._ (2019) Henry Lamm, Scott Lawrence, and Yukari Yamauchi (NuQS), “General Methods for Digital Quantum Simulation of Gauge Theories,” Phys. Rev. D 100, 034518 (2019), arXiv:1903.08807 [hep-lat] .
  * Shaw  _et al._ (2020) Alexander F. Shaw, Pavel Lougovski, Jesse R. Stryker, and Nathan Wiebe, “Quantum Algorithms for Simulating the Lattice Schwinger Model,” Quantum 4, 306 (2020), arXiv:2002.11146 [quant-ph] .
  * Kan and Nam (2021) Angus Kan and Yunseong Nam, “Lattice Quantum Chromodynamics and Electrodynamics on a Universal Quantum Computer,” (2021), arXiv:2107.12769 [quant-ph] .
  * Cohen  _et al._ (2021) Thomas D. Cohen, Henry Lamm, Scott Lawrence, and Yukari Yamauchi (NuQS), “Quantum algorithms for transport coefficients in gauge theories,” Phys. Rev. D 104, 094514 (2021), arXiv:2104.02024 [hep-lat] .
  * Murairi  _et al._ (2022) Edison M. Murairi, Michael J. Cervia, Hersh Kumar, Paulo F. Bedaque, and Andrei Alexandru, “How many quantum gates do gauge theories require?” Phys. Rev. D 106, 094504 (2022), arXiv:2208.11789 [hep-lat] .
  * Charles  _et al._ (2024) Clement Charles, Erik J. Gustafson, Elizabeth Hardt, Florian Herren, Norman Hogan, Henry Lamm, Sara Starecheski, Ruth S. Van de Water, and Michael L. Wagman, “Simulating Z2 lattice gauge theory on a quantum computer,” Phys. Rev. E 109, 015307 (2024), arXiv:2305.02361 [hep-lat] .
  * Mildenberger  _et al._ (2022) Julius Mildenberger, Wojciech Mruczkiewicz, Jad C. Halimeh, Zhang Jiang, and Philipp Hauke, “Probing confinement in a $\mathbb{Z}_{2}$ lattice gauge theory on a quantum computer,” (2022), arXiv:2203.08905 [quant-ph] .
  * Halimeh  _et al._ (2022) Jad C. Halimeh, Ian P. McCulloch, Bing Yang, and Philipp Hauke, “Tuning the Topological $\theta$-Angle in Cold-Atom Quantum Simulators of Gauge Theories,” PRX Quantum 3, 040316 (2022), arXiv:2204.06570 [cond-mat.quant-gas] .
  * Zhang  _et al._ (2023b) Wei-Yong Zhang _et al._ , “Observation of microscopic confinement dynamics by a tunable topological $\theta$-angle,” (2023b), arXiv:2306.11794 [cond-mat.quant-gas] .
  * Kane  _et al._ (2024) Christopher F. Kane, Niladri Gomes, and Michael Kreshchuk, “Nearly optimal state preparation for quantum simulations of lattice gauge theories,” Phys. Rev. A 110, 012455 (2024), arXiv:2310.13757 [quant-ph] .
  * Yang  _et al._ (2020) Bing Yang, Hui Sun, Robert Ott, Han-Yi Wang, Torsten V. Zache, Jad C. Halimeh, Zhen-Sheng Yuan, Philipp Hauke, and Jian-Wei Pan, “Observation of gauge invariance in a 71-site Bose–Hubbard quantum simulator,” Nature 587, 392–396 (2020), arXiv:2003.08945 [cond-mat.quant-gas] .
  * Zohar  _et al._ (2013a) Erez Zohar, J. Ignacio Cirac, and Benni Reznik, “Simulating (2+1)-Dimensional Lattice QED with Dynamical Matter Using Ultracold Atoms,” Phys. Rev. Lett. 110, 055302 (2013a), arXiv:1208.4299 [quant-ph] .
  * Zohar  _et al._ (2013b) Erez Zohar, J. Ignacio Cirac, and Benni Reznik, “Cold-Atom Quantum Simulator for SU(2) Yang-Mills Lattice Gauge Theory,” Phys. Rev. Lett. 110, 125304 (2013b), arXiv:1211.2241 [quant-ph] .
  * Hauke  _et al._ (2013) Philipp Hauke, David Marcos, Marcello Dalmonte, and Peter Zoller, “Quantum simulation of a lattice Schwinger model in a chain of trapped ions,” Phys. Rev. X 3, 041018 (2013), arXiv:1306.2162 [cond-mat.quant-gas] .
  * Wiese (2013) Uwe-Jens Wiese, “Ultracold Quantum Gases and Lattice Systems: Quantum Simulation of Lattice Gauge Theories,” Annalen Phys. 525, 777–796 (2013), arXiv:1305.1602 [quant-ph] .
  * Marcos  _et al._ (2013) D. Marcos, P. Rabl, E. Rico, and P. Zoller, “Superconducting Circuits for Quantum Simulation of Dynamical Gauge Fields,” Phys. Rev. Lett. 111, 110504 (2013), arXiv:1306.1674 [cond-mat.mes-hall] .
  * Zohar  _et al._ (2016) Erez Zohar, J. Ignacio Cirac, and Benni Reznik, “Quantum Simulations of Lattice Gauge Theories using Ultracold Atoms in Optical Lattices,” Rept. Prog. Phys. 79, 014401 (2016), arXiv:1503.02312 [quant-ph] .
  * Yang  _et al._ (2016) Dayou Yang, Gouri Shankar Giri, Michael Johanning, Christof Wunderlich, Peter Zoller, and Philipp Hauke, “Analog quantum simulation of (1+1)-dimensional lattice QED with trapped ions,” Phys. Rev. A 94, 052321 (2016), arXiv:1604.03124 [quant-ph] .
  * Bender  _et al._ (2018) Julian Bender, Erez Zohar, Alessandro Farace, and J. Ignacio Cirac, “Digital quantum simulation of lattice gauge theories in three spatial dimensions,” New J. Phys. 20, 093001 (2018), arXiv:1804.02082 [quant-ph] .
  * Luo  _et al._ (2020) Di Luo, Jiayu Shen, Michael Highman, Bryan K. Clark, Brian DeMarco, Aida X. El-Khadra, and Bryce Gadway, “Framework for simulating gauge theories with dipolar spin systems,” Phys. Rev. A 102, 032617 (2020), arXiv:1912.11488 [quant-ph] .
  * Notarnicola  _et al._ (2020) Simone Notarnicola, Mario Collura, and Simone Montangero, “Real-time-dynamics quantum simulation of (1+1)-dimensional lattice QED with Rydberg atoms,” Phys. Rev. Res. 2, 013288 (2020), arXiv:1907.12579 [cond-mat.quant-gas] .
  * Surace  _et al._ (2020) Federica M. Surace, Paolo P. Mazza, Giuliano Giudici, Alessio Lerose, Andrea Gambassi, and Marcello Dalmonte, “Lattice gauge theories and string dynamics in Rydberg atom quantum simulators,” Phys. Rev. X 10, 021041 (2020), arXiv:1902.09551 [cond-mat.quant-gas] .
  * Surace and Lerose (2021) Federica Maria Surace and Alessio Lerose, “Scattering of mesons in quantum simulators,” New J. Phys. 23, 062001 (2021), arXiv:2011.10583 [cond-mat.quant-gas] .
  * Kasper  _et al._ (2020) Valentin Kasper, Gediminas Juzeliunas, Maciej Lewenstein, Fred Jendrzejewski, and Erez Zohar, “From the Jaynes–Cummings model to non-abelian gauge theories: a guided tour for the quantum engineer,” New J. Phys. 22, 103027 (2020), arXiv:2006.01258 [quant-ph] .
  * Aidelsburger  _et al._ (2021) Monika Aidelsburger _et al._ , “Cold atoms meet lattice gauge theory,” Phil. Trans. Roy. Soc. Lond. A 380, 20210064 (2021), arXiv:2106.03063 [cond-mat.quant-gas] .
  * Surace  _et al._ (2024) Federica Maria Surace, Pierre Fromholz, Francesco Scazza, and Marcello Dalmonte, “Scalable, ab initio protocol for quantum simulating SU($N$)$\times$U(1) Lattice Gauge Theories,” Quantum 8, 1359 (2024), arXiv:2310.08643 [cond-mat.quant-gas] .
  * Zache  _et al._ (2023a) Torsten V. Zache, Daniel González-Cuadra, and Peter Zoller, “Fermion-qudit quantum processors for simulating lattice gauge theories with matter,” Quantum 7, 1140 (2023a), arXiv:2303.08683 [quant-ph] .
  * Popov  _et al._ (2024) Pavel P. Popov, Michael Meth, Maciej Lewenstein, Philipp Hauke, Martin Ringbauer, Erez Zohar, and Valentin Kasper, “Variational quantum simulation of U(1) lattice gauge theories with qudit systems,” Phys. Rev. Res. 6, 013202 (2024), arXiv:2307.15173 [quant-ph] .
  * González-Cuadra  _et al._ (2022) Daniel González-Cuadra, Torsten V. Zache, Jose Carrasco, Barbara Kraus, and Peter Zoller, “Hardware Efficient Quantum Simulation of Non-Abelian Gauge Theories with Qudits on Rydberg Platforms,” Phys. Rev. Lett. 129, 160501 (2022), arXiv:2203.15541 [quant-ph] .
  * Illa  _et al._ (2024) Marc Illa, Caroline E. P. Robin, and Martin J. Savage, “Qu8its for quantum simulations of lattice quantum chromodynamics,” Phys. Rev. D 110, 014507 (2024), arXiv:2403.14537 [quant-ph] .
  * Zache  _et al._ (2018) T. V. Zache, F. Hebenstreit, F. Jendrzejewski, M. K. Oberthaler, J. Berges, and P. Hauke, “Quantum simulation of lattice gauge theories using Wilson fermions,” Quantum Sci. Technol. 3, 034010 (2018), arXiv:1802.06704 [cond-mat.quant-gas] .
  * Armon  _et al._ (2021) Tsafrir Armon, Shachar Ashkenazi, Gerardo García-Moreno, Alejandro González-Tudela, and Erez Zohar, “Photon-Mediated Stroboscopic Quantum Simulation of a Z2 Lattice Gauge Theory,” Phys. Rev. Lett. 127, 250501 (2021), arXiv:2107.13024 [quant-ph] .
  * Byrnes and Yamamoto (2006) Tim Byrnes and Yoshihisa Yamamoto, “Simulating lattice gauge theories on a quantum computer,” Phys. Rev. A 73, 022328 (2006), arXiv:quant-ph/0510027 .
  * González-Cuadra  _et al._ (2017) Daniel González-Cuadra, Erez Zohar, and J. Ignacio Cirac, “Quantum Simulation of the Abelian-Higgs Lattice Gauge Theory with Ultracold Atoms,” New J. Phys. 19, 063038 (2017), arXiv:1702.05492 [quant-ph] .
  * Tagliacozzo  _et al._ (2013a) L. Tagliacozzo, A. Celi, P. Orland, and M. Lewenstein, “Simulations of non-Abelian gauge theories with optical lattices,” Nature Commun. 4, 2615 (2013a), arXiv:1211.2704 [cond-mat.quant-gas] .
  * Tagliacozzo  _et al._ (2013b) L. Tagliacozzo, A. Celi, A. Zamora, and M. Lewenstein, “Optical Abelian Lattice Gauge Theories,” Annals Phys. 330, 160–191 (2013b), arXiv:1205.0496 [cond-mat.quant-gas] .
  * Zhou  _et al._ (2022) Zhao-Yu Zhou, Guo-Xian Su, Jad C. Halimeh, Robert Ott, Hui Sun, Philipp Hauke, Bing Yang, Zhen-Sheng Yuan, Jürgen Berges, and Jian-Wei Pan, “Thermalization dynamics of a gauge theory on a quantum simulator,” Science 377, abl6277 (2022), arXiv:2107.13563 [cond-mat.quant-gas] .
  * Yamamoto and Doi (2022) Arata Yamamoto and Takumi Doi, “Toward nuclear physics from lattice QCD on quantum computers,” (2022), arXiv:2211.14550 [hep-lat] .
  * Zohar and Reznik (2011) Erez Zohar and Benni Reznik, “Confinement and lattice QED electric flux-tubes simulated with ultracold atoms,” Phys. Rev. Lett. 107, 275301 (2011), arXiv:1108.1562 [quant-ph] .
  * Zohar  _et al._ (2017) Erez Zohar, Alessandro Farace, Benni Reznik, and J. Ignacio Cirac, “Digital lattice gauge theories,” Phys. Rev. A 95, 023604 (2017), arXiv:1607.08121 [quant-ph] .
  * Alexandru  _et al._ (2022) Andrei Alexandru, Paulo F. Bedaque, Ruairí Brett, and Henry Lamm, “Spectrum of digitized QCD: Glueballs in a S(1080) gauge theory,” Phys. Rev. D 105, 114508 (2022), arXiv:2112.08482 [hep-lat] .
  * Lamm  _et al._ (2024) Henry Lamm, Ying-Ying Li, Jing Shu, Yi-Lin Wang, and Bin Xu, “Block Encodings of Discrete Subgroups on Quantum Computer,” (2024), arXiv:2405.12890 [hep-lat] .
  * Lamm  _et al._ (2020a) Henry Lamm, Scott Lawrence, and Yukari Yamauchi (NuQS), “Parton physics on a quantum computer,” Phys. Rev. Res. 2, 013272 (2020a), arXiv:1908.10439 [hep-lat] .
  * Feynman (1982) Richard P. Feynman, “Simulating physics with computers,” Int. J. Theor. Phys. 21, 467–488 (1982).
  * Lloyd (1996) Seth Lloyd, “Universal quantum simulators,” Science 273, 1073–1078 (1996), https://www.science.org/doi/pdf/10.1126/science.273.5278.1073 .
  * Preskill (2021) John Preskill, “Quantum computing 40 years later,” (2021), arXiv:2106.10522 [quant-ph] .
  * Carena  _et al._ (2022) Marcela Carena, Henry Lamm, Ying-Ying Li, and Wanqiang Liu, “Improved Hamiltonians for Quantum Simulations of Gauge Theories,” Phys. Rev. Lett. 129, 051601 (2022), arXiv:2203.02823 [hep-lat] .
  * Bañuls  _et al._ (2017) Mari Carmen Bañuls, Krzysztof Cichy, J. Ignacio Cirac, Karl Jansen, and Stefan Kühn, “Efficient basis formulation for 1+1 dimensional SU(2) lattice gauge theory: Spectral calculations with matrix product states,” Phys. Rev. X 7, 041046 (2017), arXiv:1707.06434 [hep-lat] .
  * Raychowdhury and Stryker (2020a) Indrakshi Raychowdhury and Jesse R. Stryker, “Loop, string, and hadron dynamics in SU(2) Hamiltonian lattice gauge theories,” Phys. Rev. D 101, 114502 (2020a), arXiv:1912.06133 [hep-lat] .
  * Kadam  _et al._ (2023) Saurabh V. Kadam, Indrakshi Raychowdhury, and Jesse R. Stryker, “Loop-string-hadron formulation of an SU(3) gauge theory with dynamical quarks,” Phys. Rev. D 107, 094513 (2023), arXiv:2212.04490 [hep-lat] .
  * Pardo  _et al._ (2023) Guy Pardo, Tomer Greenberg, Aryeh Fortinsky, Nadav Katz, and Erez Zohar, “Resource-efficient quantum simulation of lattice gauge theories in arbitrary dimensions: Solving for Gauss’s law and fermion elimination,” Phys. Rev. Res. 5, 023077 (2023), arXiv:2206.00685 [quant-ph] .
  * Chandrasekharan and Wiese (1997) S. Chandrasekharan and U. J. Wiese, “Quantum link models: A Discrete approach to gauge theories,” Nucl. Phys. B 492, 455–474 (1997), arXiv:hep-lat/9609042 .
  * Bauer and Grabowska (2023) Christian W. Bauer and Dorota M. Grabowska, “Efficient representation for simulating U(1) gauge theories on digital quantum computers at all values of the coupling,” Phys. Rev. D 107, L031503 (2023), arXiv:2111.08015 [hep-ph] .
  * Zohar and Burrello (2015) Erez Zohar and Michele Burrello, “Formulation of lattice gauge theories for quantum simulations,” Phys. Rev. D 91, 054506 (2015), arXiv:1409.3085 [quant-ph] .
  * Zohar and Cirac (2018) Erez Zohar and J. Ignacio Cirac, “Eliminating fermionic matter fields in lattice gauge theories,” Phys. Rev. B 98, 075119 (2018), arXiv:1805.05347 [quant-ph] .
  * Zohar and Cirac (2019) Erez Zohar and J. Ignacio Cirac, “Removing Staggered Fermionic Matter in $U(N)$ and $SU(N)$ Lattice Gauge Theories,” Phys. Rev. D 99, 114511 (2019), arXiv:1905.00652 [quant-ph] .
  * Alexandru  _et al._ (2024) Andrei Alexandru, Paulo F. Bedaque, Andrea Carosso, Michael J. Cervia, Edison M. Murairi, and Andy Sheng, “Fuzzy gauge theory for quantum computers,” Phys. Rev. D 109, 094502 (2024), arXiv:2308.05253 [hep-lat] .
  * Alexandru  _et al._ (2019) Andrei Alexandru, Paulo F. Bedaque, Siddhartha Harmalkar, Henry Lamm, Scott Lawrence, and Neill C. Warrington (NuQS), “Gluon Field Digitization for Quantum Computers,” Phys. Rev. D 100, 114501 (2019), arXiv:1906.11213 [hep-lat] .
  * Ji  _et al._ (2023) Yao Ji, Henry Lamm, and Shuchen Zhu (NuQS), “Gluon digitization via character expansion for quantum computers,” Phys. Rev. D 107, 114503 (2023), arXiv:2203.02330 [hep-lat] .
  * Ji  _et al._ (2020) Yao Ji, Henry Lamm, and Shuchen Zhu (NuQS), “Gluon Field Digitization via Group Space Decimation for Quantum Computers,” Phys. Rev. D 102, 114513 (2020), arXiv:2005.14221 [hep-lat] .
  * Kavaki and Lewis (2024) Ali H. Z. Kavaki and Randy Lewis, “From square plaquettes to triamond lattices for SU(2) gauge theory,” Commun. Phys. 7, 208 (2024), arXiv:2401.14570 [hep-lat] .
  * D’Andrea  _et al._ (2024) Irian D’Andrea, Christian W. Bauer, Dorota M. Grabowska, and Marat Freytsis, “New basis for Hamiltonian SU(2) simulations,” Phys. Rev. D 109, 074501 (2024), arXiv:2307.11829 [hep-ph] .
  * Romiti and Urbach (2024) Simone Romiti and Carsten Urbach, “Digitizing lattice gauge theories in the magnetic basis: reducing the breaking of the fundamental commutation relations,” Eur. Phys. J. C 84, 708 (2024), arXiv:2311.11928 [hep-lat] .
  * Zache  _et al._ (2023b) Torsten V. Zache, Daniel González-Cuadra, and Peter Zoller, “Quantum and Classical Spin-Network Algorithms for q-Deformed Kogut-Susskind Gauge Theories,” Phys. Rev. Lett. 131, 171902 (2023b), arXiv:2304.02527 [quant-ph] .
  * Wiese (2021) Uwe-Jens Wiese, “From quantum link models to D-theory: a resource efficient framework for the quantum simulation and computation of gauge theories,” Phil. Trans. A. Math. Phys. Eng. Sci. 380, 20210068 (2021), arXiv:2107.09335 [hep-lat] .
  * Hartung  _et al._ (2022) Tobias Hartung, Timo Jakobs, Karl Jansen, Johann Ostmeyer, and Carsten Urbach, “Digitising SU(2) gauge fields and the freezing transition,” Eur. Phys. J. C 82, 237 (2022), arXiv:2201.09625 [hep-lat] .
  * Müller and Yao (2023) Berndt Müller and Xiaojun Yao, “Simple Hamiltonian for quantum simulation of strongly coupled (2+1)D SU(2) lattice gauge theory on a honeycomb lattice,” Phys. Rev. D 108, 094505 (2023), arXiv:2307.00045 [quant-ph] .
  * Farrell  _et al._ (2024c) Roland C. Farrell, Marc Illa, Anthony N. Ciavarella, and Martin J. Savage, “Scalable Circuits for Preparing Ground States on Digital Quantum Computers: The Schwinger Model Vacuum on 100 Qubits,” PRX Quantum 5, 020315 (2024c), arXiv:2308.04481 [quant-ph] .
  * Stryker (2021) Jesse R. Stryker, “Shearing approach to gauge invariant Trotterization,” (2021), arXiv:2105.11548 [hep-lat] .
  * Gustafson and Lamm (2023) Erik J. Gustafson and Henry Lamm, “Robustness of Gauge Digitization to Quantum Noise,” (2023), arXiv:2301.10207 [hep-lat] .
  * Lamm  _et al._ (2020b) Henry Lamm, Scott Lawrence, and Yukari Yamauchi (NuQS), “Suppressing Coherent Gauge Drift in Quantum Simulations,” (2020b), arXiv:2005.12688 [quant-ph] .
  * Raychowdhury and Stryker (2020b) Indrakshi Raychowdhury and Jesse R. Stryker, “Solving Gauss’s Law on Digital Quantum Computers with Loop-String-Hadron Digitization,” Phys. Rev. Res. 2, 033039 (2020b), arXiv:1812.07554 [hep-lat] .
  * Mathew and Raychowdhury (2022) Emil Mathew and Indrakshi Raychowdhury, “Protecting local and global symmetries in simulating (1+1)D non-Abelian gauge theories,” Phys. Rev. D 106, 054510 (2022), arXiv:2206.07444 [hep-lat] .
  * Mathew and Raychowdhury (2024) Emil Mathew and Indrakshi Raychowdhury, “Protecting gauge symmetries in the the dynamics of SU(3) lattice gauge theories,”  (2024), 10.48550/arXiv.2404.12158, arXiv:2404.12158 [hep-lat] .
  * Grabowska  _et al._ (2022) Dorota M. Grabowska, Christopher Kane, Benjamin Nachman, and Christian W. Bauer, “Overcoming exponential scaling with system size in Trotter-Suzuki implementations of constrained Hamiltonians: 2+1 U(1) lattice gauge theories,” (2022), arXiv:2208.03333 [quant-ph] .
  * Kogut and Susskind (1975) John B. Kogut and Leonard Susskind, “Hamiltonian Formulation of Wilson’s Lattice Gauge Theories,” Phys. Rev. D 11, 395–408 (1975).
  * Davoudi  _et al._ (2021) Zohreh Davoudi, Indrakshi Raychowdhury, and Andrew Shaw, “Search for efficient formulations for Hamiltonian simulation of non-Abelian lattice gauge theories,” Phys. Rev. D 104, 074505 (2021), arXiv:2009.11802 [hep-lat] .
  * Mathur (2005) Manu Mathur, “Harmonic oscillator prepotentials in SU(2) lattice gauge theory,” J. Phys. A 38, 10015–10026 (2005), arXiv:hep-lat/0403029 .
  * Mathur (2007) Manu Mathur, “Loop Approach to Lattice Gauge Theories,” Nucl. Phys. B 779, 32–62 (2007), arXiv:hep-lat/0702007 .
  * Mathur  _et al._ (2010) Manu Mathur, Indrakshi Raychowdhury, and Ramesh Anishetty, “SU(N) Irreducible Schwinger Bosons,” J. Math. Phys. 51, 093504 (2010), arXiv:1003.5487 [math-ph] .
  * Mathur and Sen (2001) Manu Mathur and Diptiman Sen, “Coherent states for SU(3),” J. Math. Phys. 42, 4181–4196 (2001), arXiv:quant-ph/0012099 .
  * Raychowdhury (2013) Indrakshi Raychowdhury,  _Prepotential Formulation of Lattice Gauge Theories_ , Ph.D. thesis, Calcutta U. (2013).
  * Anishetty and Sreeraj (2019) Ramesh Anishetty and T. P. Sreeraj, “Addition of SU(3) generators and its Singlet Hilbert space,” J. Math. Phys. 60, 061701 (2019), arXiv:1903.07956 [math-ph] .
  * Anishetty  _et al._ (2009) Ramesh Anishetty, Manu Mathur, and Indrakshi Raychowdhury, “Irreducible SU(3) Schhwinger Bosons,” J. Math. Phys. 50, 053503 (2009), arXiv:0901.0644 [math-ph] .
  * Anishetty  _et al._ (2010) Ramesh Anishetty, Manu Mathur, and Indrakshi Raychowdhury, “Prepotential formulation of SU(3) lattice gauge theory,” J. Phys. A 43, 035403 (2010), arXiv:0909.2394 [hep-lat] .
  * Mukunda and Pandit (1965) N Mukunda and LK Pandit, “Tensor methods and a unified representation theory of su 3,” Journal of Mathematical Physics 6, 746–765 (1965).
  * Chaturvedi and Mukunda (2002) S. Chaturvedi and N. Mukunda, “The Schwinger SU(3) construction. 1. Multiplicity problem and relation to induced representations,” J. Math. Phys. 43, 5262–5277 (2002), arXiv:quant-ph/0204119 .
  * Littlewood and Richardson (1934) Dudley Ernest Littlewood and Archibald Read Richardson, “Group characters and algebra,” Philosophical Transactions of the Royal Society A 233, 99–141 (1934).
  * Knuth (1970) Donald E. Knuth, “Permutations, matrices, and generalized Young tableaux.” Pacific Journal of Mathematics 34, 709 – 727 (1970).
  * Gopalkrishna Gadiyar and Sharatchandra (1992) H. Gopalkrishna Gadiyar and H. S. Sharatchandra, “The Missing link: Operators for labeling multiplicity in the Clebsch-Gordan series,” J. Phys. A 25, L85–L88 (1992).
  * Crampé  _et al._ (2023) Nicolas Crampé, Loïc Poulain d’Andecy, and Luc Vinet, “The Missing Label of su3 and Its Symmetry,” Communications in Mathematical Physics 400, 179–213 (2023), arXiv:2110.03521 [math.RT] .
  * The Sage Developers (2022) The Sage Developers,  _SageMath, the Sage Mathematics Software System_ (2022), dOI 10.5281/zenodo.6259615.
  * Raychowdhury (2019) Indrakshi Raychowdhury, “Low energy spectrum of SU(2) lattice gauge theory: An alternate proposal via loop formulation,” Eur. Phys. J. C 79, 235 (2019), arXiv:1804.01304 [hep-lat] .

##  Appendix A Normalization of nondegenerate states

In this appendix, we provide partial proofs for the normalization of SU(3) LSH
basis kets at a trivalent vertex, for choices of
$\overset{\longrightarrow}{p\,q}=(p_{1},q_{1},p_{2},q_{2},p_{3},q_{3})$ that
are nondegenerate. (Analogous calculations were discussed in an appendix of
Ref. Raychowdhury and Stryker (2020a) for the SU(2) loop-string-hadron theory
with one flavor of fermions.) As discussed in the main text, such
nondegenerate states must have at least two vanishing $\ell_{IJ}$ quantum
numbers, and they were organized into classes I and II depending on which
quantum numbers out of $\\{\ell_{IJ}\\}$ and $t$ are vanishing. The full proof
for either class I or II, in each case involving five or more layers of proof-
by-induction, is lengthy and we do not reproduce the complete arguments here.
But to give a taste of how these calculations are done, below we present the
norm evaluation for naive basis states having up to two nonzero quantum
numbers, $t$ and a single $\ell_{IJ}$, which can serve as the starting point
for normalizing either class I or II.

In the norm evaluations, one encounters numerous gauge-singlet operators
beyond just the pure-creation gauge singlet operators defined (III) and their
pure-annihilation adjoints. For example, there are bilinear gauge singlet
operators constructed as a mixture of a creation and annihilation operators:

$\displaystyle N_{IJ}$ $\displaystyle\equiv
A^{\dagger}_{\alpha}(I)A^{\alpha}(J),$ (169) $\displaystyle M_{IJ}$
$\displaystyle\equiv B^{\dagger\alpha}(I)B_{\alpha}(J),$ (170)

where $I$ and $J$ label two different link ends attached to the vertex. For
the trivalent vertex, $I$ and $J$ each take values from 1 to 3 with $I\not=J$.
We will also make frequent reference to the irrep quantum numbers and a
function of them:

$\displaystyle P_{I}$
$\displaystyle=a(I)^{\dagger}_{\alpha}a(I)^{\alpha}\simeq
A(I)^{\dagger}_{\alpha}A(I)^{\alpha},$ (171) $\displaystyle Q_{I}$
$\displaystyle=b(I)^{\dagger\,\alpha}b(I)_{\alpha}\simeq
B(I)^{\dagger\,\alpha}B(I)_{\alpha},$ (172) $\displaystyle F_{I}$
$\displaystyle\equiv(P_{I}+Q_{I}+2)^{-1},$ (173)

where $\simeq$ means the identity is valid in the Hilbert space of irreducible
Schwinger bosons.

The first set of excitations we will normalize is the $t$-type ones:

$\displaystyle\left.\left|t\right\rangle\right\rangle$
$\displaystyle=(T_{A}^{\dagger})^{|t|\,\theta(t)}(T_{B}^{\dagger})^{|t|\,\theta(-t)}\ket{0}.$
(174)

It is convenient to first establish the following intermediate results.

######  Lemma A.1.

For two vertex legs $I$ and $J$, $I\neq J$, and any integer $t\geq 0$, we have

$\displaystyle N_{IJ}T_{A}^{\dagger\,t}\ket{0}=0.$ (175)

###### Proof.

We use induction on $t$.

When $t=0$, the $A(J)$ annihilation operators in $N_{IJ}$ are directly applied
to the vacuum state, giving $N_{IJ}\ket{0}=0$. Now taking $t\geq 1$, we assume
$N_{IJ}T_{A}^{\dagger\,t-1}\ket{0}=0$. Then we evaluate

$\displaystyle N_{IJ}T_{A}^{\dagger\,t}\ket{0}$
$\displaystyle=N_{IJ}T_{A}^{\dagger}T_{A}^{\dagger\,t-1}\ket{0}$ (176)
$\displaystyle=[N_{IJ},T_{A}^{\dagger}]T_{A}^{\dagger\,t-1}\ket{0},$ (177)

where we have used the inductive hypothesis for $t-1$. We need the commutator

$\displaystyle[N_{IJ},T_{A}^{\dagger}]$
$\displaystyle=[A(I)^{\dagger}_{\mu}A(J)^{\mu},\epsilon^{\alpha\beta\gamma}A(1)^{\dagger}_{\alpha}A(2)^{\dagger}_{\beta}A(3)^{\dagger}_{\gamma}]$
(178)
$\displaystyle=\epsilon^{\alpha\beta\gamma}A(I)^{\dagger}_{\mu}[A(J)^{\mu},A(1)^{\dagger}_{\alpha}A(2)^{\dagger}_{\beta}A(3)^{\dagger}_{\gamma}]$
(179)
$\displaystyle=\epsilon^{\alpha\beta\gamma}A(I)^{\dagger}_{\mu}\left([A(J)^{\mu},A(1)^{\dagger}_{\alpha}]A(2)^{\dagger}_{\beta}A(3)^{\dagger}_{\gamma}+A(1)^{\dagger}_{\alpha}[A(J)^{\mu},A(2)^{\dagger}_{\beta}]A(3)^{\dagger}_{\gamma}+\right.$
$\displaystyle\qquad\qquad\qquad\quad\left.+A(1)^{\dagger}_{\alpha}A(2)^{\dagger}_{\beta}[A(J)^{\mu},A(3)^{\dagger}_{\gamma}]\right)$
(180)
$\displaystyle\simeq\epsilon^{\alpha\beta\gamma}A(I)^{\dagger}_{\mu}\left(\delta_{J\,1}\bigl{(}\delta^{\mu}_{\alpha}-F_{1}B(1)^{\dagger\,\mu}B(1)_{\alpha}\bigr{)}A(2)^{\dagger}_{\beta}A(3)^{\dagger}_{\gamma}+\right.$
$\displaystyle\qquad\qquad\qquad\quad+\delta_{J\,2}A(1)^{\dagger}_{\alpha}\bigl{(}\delta^{\mu}_{\beta}-F_{2}B(2)^{\dagger\,\mu}B(2)_{\beta}\bigr{)}A(3)^{\dagger}_{\gamma}+$
$\displaystyle\qquad\qquad\qquad\quad\left.+\delta_{J\,3}A(1)^{\dagger}_{\alpha}A(2)^{\dagger}_{\beta}\bigl{(}\delta^{\mu}_{\gamma}-F_{3}B(3)^{\dagger\,\mu}B(3)_{\gamma}\bigr{)}\right).$
(181)

Looking at the first term of the last line, if $\delta_{J\,1}\neq 0$, then
$I=2$ or $I=3$; in either case, the term will contain two factors of
$A(I)^{\dagger}$ components, that is a symmetric tensor with two lower
indices. Contraction with the antisymmetric $\epsilon^{\alpha\beta\gamma}$
will then produce zero. The same logic can be applied to eliminate the second
and third terms. Thus,

$\displaystyle N_{IJ}T_{A}^{\dagger\,t}\ket{0}$
$\displaystyle=[N_{IJ},T_{A}^{\dagger}]T_{A}^{\dagger\,t-1}\ket{0}=(0)T_{A}^{\dagger\,t-1}\ket{0}=0.$
(182)

∎

The next result is proved by a similar, but not identical, argument.

######  Lemma A.2.

For two vertex legs $I$ and $J$, $I\neq J$, and any integer $p\geq 0$, we have

$\displaystyle N_{IJ}T_{B}^{\dagger\,p}\ket{0}=0.$ (183)

###### Proof.

We use induction on $p$.

When $p=0$, the $A(J)$ annihilation operators in $N_{IJ}$ are directly applied
to the vacuum state, giving $N_{IJ}\ket{0}=0$. Now taking $p\geq 1$, we assume
$N_{IJ}T_{B}^{\dagger\,p-1}\ket{0}=0$. Then we evaluate

$\displaystyle N_{IJ}T_{B}^{\dagger\,p}\ket{0}$
$\displaystyle=N_{IJ}T_{B}^{\dagger}T_{B}^{\dagger\,p-1}\ket{0}$ (184)
$\displaystyle=[N_{IJ},T_{B}^{\dagger}]T_{B}^{\dagger\,p-1}\ket{0},$ (185)

where we have used the inductive hypothesis for $p-1$. We need the commutator

$\displaystyle[N_{IJ},T_{B}^{\dagger}]$
$\displaystyle=[A(I)^{\dagger}_{\mu}A(J)^{\mu},\epsilon_{\alpha\beta\gamma}B(1)^{\dagger\,\alpha}B(2)^{\dagger\,\beta}B(3)^{\dagger\,\gamma}]$
(186)
$\displaystyle=\epsilon_{\alpha\beta\gamma}A(I)^{\dagger}_{\mu}[A(J)^{\mu},B(1)^{\dagger\,\alpha}B(2)^{\dagger\,\beta}B(3)^{\dagger\,\gamma}]$
(187)
$\displaystyle=\epsilon_{\alpha\beta\gamma}A(I)^{\dagger}_{\mu}\left([A(J)^{\mu},B(1)^{\dagger\,\alpha}]B(2)^{\dagger\,\beta}B(3)^{\dagger\,\gamma}+B(1)^{\dagger\,\alpha}[A(J)^{\mu},B(2)^{\dagger\,\beta}]B(3)^{\dagger\,\gamma}+\right.$
$\displaystyle\qquad\qquad\qquad\quad\left.+B(1)^{\dagger\,\alpha}B(2)^{\dagger\,\beta}[A(J)^{\mu},B(3)^{\dagger\,\gamma}]\right)$
(188)
$\displaystyle\simeq\epsilon^{\alpha\beta\gamma}A(I)^{\dagger}_{\mu}\left(\delta_{J\,1}\bigl{(}-F_{1}B(1)^{\dagger\,\mu}A(1)^{\alpha}\bigr{)}B(2)^{\dagger\,\beta}B(3)^{\dagger\,\gamma}+\delta_{J\,2}B(1)^{\dagger\,\alpha}\bigl{(}-F_{2}B(2)^{\dagger\,\mu}A(2)^{\beta}\bigr{)}B(3)^{\dagger\,\gamma}+\right.$
$\displaystyle\qquad\qquad\qquad\quad\left.+\delta_{J\,3}B(1)^{\dagger\,\alpha}B(2)^{\dagger\,\beta}\bigl{(}-F_{3}B(3)^{\dagger\,\mu}A(3)^{\gamma}\bigr{)}\right).$
(189)

Looking at the first term of the last line, we see a factor of $A(1)^{\alpha}$
with no components of $A(1)^{\dagger}$ anywhere to the right of it. Since
$[N_{IJ},T_{B}^{\dagger}]$ is being applied to the ket
$T_{B}^{\dagger\,p-1}\ket{0}$, a $P_{1}=0$ eigenstate, the term carrying
$A(1)^{\alpha}$ must evaluate to zero. The same logic can be applied to
eliminate the second and third terms. Thus,

$\displaystyle N_{IJ}T_{B}^{\dagger\,p}\ket{0}$
$\displaystyle=[N_{IJ},T_{B}^{\dagger}]T_{B}^{\dagger\,p-1}\ket{0}=(0)T_{B}^{\dagger\,p-1}\ket{0}=0.$
(190)

∎

The two Lemmas above are combined into the following Corollary.

######  Corollary A.3.

For two vertex legs $I$ and $J$, $I\neq J$, and any integer $t$, we have

$\displaystyle N_{IJ}\left.\left|t\right\rangle\right\rangle$
$\displaystyle=0.$ (191)

Equipped with A.3, we are ready to evaluate the normalization of
$\left.\left|t\right\rangle\right\rangle$ states.

######  Theorem A.4.

The norm-squared of states $\left.\left|t\right\rangle\right\rangle$ in (174)
is

$\displaystyle\left\langle\braket{t}{t}\right\rangle$
$\displaystyle=\tfrac{1}{2}(|t|+2)\,|t|!\,(|t|+1)!\,(|t|+1)!$
$\displaystyle=\tfrac{1}{2}(|t|+2)!\,(|t|+1)!\,|t|!\,.$ (192)

###### Proof.

Without loss of generality, assume $t\geq 0$; the proof for $t\leq 0$ follows
identical reasoning. We prove the formula by induction.

For the base case $t=0$, the squared norm evaluates as

$\displaystyle\left\langle\braket{0}{0}\right\rangle$
$\displaystyle=\braket{0}{0}=1,$ (193)

because the complete bosonic vacuum $\ket{0}$ is normalized to unity by
definition. This agrees with the right-hand side of (A.4) when evaluated for
$|t|=0$.

For the inductive step, take $t\geq 1$ and assume the formula holds for the
state $\left.\left|t-1\right\rangle\right\rangle$. The norm evaluation begins
with

$\displaystyle\left\langle\braket{t}{t}\right\rangle$
$\displaystyle=\left\langle\left\langle
t-1\right|\right.T_{A}T_{A}^{\dagger}\left.\left|t-1\right\rangle\right\rangle$
(194) $\displaystyle=\left\langle\left\langle
t-1\right|\right.T_{A}T_{A}^{\dagger\,t}\ket{0}$ (195)
$\displaystyle=\left\langle\left\langle
t-1\right|\right.\bigl{[}T_{A},T_{A}^{\dagger\,t}\bigr{]}\ket{0}$ (196)
$\displaystyle=\left\langle\left\langle
t-1\right|\right.\sum_{k=0}^{t-1}T_{A}^{\dagger\,t-1-k}\bigl{[}T_{A},T_{A}^{\dagger}\bigr{]}T_{A}^{\dagger\,k}\ket{0}.$
(197)

To proceed, one needs the commutator $\bigl{[}T_{A},T_{A}^{\dagger}\bigr{]}$;
but going into the evaluation of this commutator, one should take note of the
ket on the right to which this operator is applied:
$T_{A}^{\dagger\,k}\ket{0}$.

One may first expand out the definitions to obtain

$\displaystyle\left[T_{A},T_{A}^{\dagger}\right]$
$\displaystyle=\epsilon^{\lambda\mu\nu}\bigl{[}T_{A},\,A(1)^{\dagger}_{\lambda}A(2)^{\dagger}_{\mu}A(3)^{\dagger}_{\nu}\bigr{]}$
(198)
$\displaystyle=\epsilon^{\lambda\mu\nu}\left(A(1)^{\dagger}_{\lambda}A(2)^{\dagger}_{\mu}\bigl{[}T_{A},\,A(3)^{\dagger}_{\nu}\bigr{]}+A(1)^{\dagger}_{\lambda}\bigl{[}T_{A},\,A(2)^{\dagger}_{\mu}\bigr{]}A(3)^{\dagger}_{\nu}+\bigl{[}T_{A},\,A(1)^{\dagger}_{\lambda}\bigr{]}A(2)^{\dagger}_{\mu}A(3)^{\dagger}_{\nu}\right)$
(199)

Here we will use the information about the ket to the right when evaluating

$\displaystyle[T_{A},A(3)^{\dagger}_{\nu}]$
$\displaystyle=\epsilon_{\alpha\beta\gamma}[A(1)^{\alpha}A(2)^{\beta}A(3)^{\gamma},A(3)^{\dagger}_{\nu}]$
(200)
$\displaystyle=\epsilon_{\alpha\beta\gamma}A(1)^{\alpha}A(2)^{\beta}[A(3)^{\gamma},A(3)^{\dagger}_{\nu}]$
(201)
$\displaystyle\simeq\epsilon_{\alpha\beta\gamma}A(1)^{\alpha}A(2)^{\beta}(\delta^{\gamma}_{\nu}-F_{3}B(3)^{\dagger\gamma}B(3)_{\nu})$
(202)
$\displaystyle\to\epsilon_{\alpha\beta\gamma}A(1)^{\alpha}A(2)^{\beta}\delta^{\gamma}_{\nu}-0$
(203) $\displaystyle=\epsilon_{\alpha\beta\nu}A(1)^{\alpha}A(2)^{\beta},$
(204)

and similarly for the other two terms of (199). So now

$\displaystyle\left[T_{A},T_{A}^{\dagger}\right]$
$\displaystyle\to\epsilon^{\lambda\mu\nu}\bigl{(}A(1)^{\dagger}_{\lambda}A(2)^{\dagger}_{\mu}\epsilon_{\alpha\beta\nu}A(1)^{\alpha}A(2)^{\beta}+A(1)^{\dagger}_{\lambda}\epsilon_{\alpha\beta\mu}A(3)^{\alpha}A(1)^{\beta}A(3)^{\dagger}_{\nu}+$
$\displaystyle\qquad\qquad+\epsilon_{\alpha\beta\lambda}A(2)^{\alpha}A(3)^{\beta}A(2)^{\dagger}_{\mu}A(3)^{\dagger}_{\nu}\bigr{)}$
(205)

For the first term, it is easy to show that
$\epsilon^{\lambda\mu\nu}\epsilon_{\alpha\beta\nu}A(1)^{\alpha}A(2)^{\beta}=A(1)^{\lambda}A(2)^{\mu}-A(1)^{\mu}A(2)^{\lambda}$,
so that

$\displaystyle\epsilon^{\lambda\mu\nu}A(1)^{\dagger}_{\lambda}A(2)^{\dagger}_{\mu}\epsilon_{\alpha\beta\nu}A(1)^{\alpha}A(2)^{\beta}$
$\displaystyle=A(1)^{\dagger}\cdot A(1)A(2)^{\dagger}\cdot
A(2)-A(1)^{\dagger}_{\lambda}N_{21}A(2)^{\lambda}$ (206) $\displaystyle\simeq
P_{1}P_{2}-A(1)^{\dagger}_{\lambda}N_{21}A(2)^{\lambda}.$ (207)

Again using the information about the ket to the right, one can
straightforwardly show that $A(1)^{\dagger}_{\lambda}N_{21}A(2)^{\lambda}\to
N_{21}N_{12}-P_{2}$ in this expression. We have now “fully evaluated” the
first term of (199) as

$\displaystyle\epsilon^{\lambda\mu\nu}A(1)^{\dagger}_{\lambda}A(2)^{\dagger}_{\mu}\bigl{[}T_{A},\,A(3)^{\dagger}_{\nu}\bigr{]}$
$\displaystyle\to P_{1}P_{2}+P_{2}-N_{21}N_{12},$ (208)

by which we mean that it is written entirely in terms of LSH gauge singlet
operators.

For the second term of (199), similar steps and substitutions can be used to
obtain

$\displaystyle\epsilon^{\lambda\mu\nu}A(1)^{\dagger}_{\lambda}\bigl{[}T_{A},\,A(2)^{\dagger}_{\mu}\bigr{]}A(3)^{\dagger}_{\nu}$
$\displaystyle\simeq P_{1}A(3)\cdot A(3)^{\dagger}-N_{13}N_{31}.$ (209)

Here one should use another substitution that is valid here but not in
general: $A(3)\cdot A(3)^{\dagger}\to P_{3}+3$. So then the second term of
(199) is evaluated as

$\displaystyle\epsilon^{\lambda\mu\nu}A(1)^{\dagger}_{\lambda}\bigl{[}T_{A},\,A(2)^{\dagger}_{\mu}\bigr{]}A(3)^{\dagger}_{\nu}\to
P_{1}(P_{3}+3)-N_{13}N_{31}$ (210)

The third term of (199) can be evaluated using similar steps to those used
above, arriving at

$\displaystyle\epsilon^{\lambda\mu\nu}\bigl{[}T_{A},\,A(1)^{\dagger}_{\lambda}\bigr{]}A(2)^{\dagger}_{\mu}A(3)^{\dagger}_{\nu}\to(P_{2}+3)(P_{3}+3)-(P_{2}+3)-N_{32}N_{23}.$
(211)

Putting together the three intermediate results (208), (210), and (211), and
simplifying, we have shown that

$\displaystyle\bigl{[}T_{A},T_{A}^{\dagger}\bigr{]}$ $\displaystyle\to
P_{1}P_{2}+P_{2}P_{3}+P_{3}P_{1}+3\bigl{(}P_{1}+P_{2}+P_{3}\bigr{)}+6-N_{21}N_{12}-N_{32}N_{23}-N_{13}N_{31},$
(212)

for the purposes of (197). The final step to dealing with
$\bigl{[}T_{A},T_{A}^{\dagger}\bigr{]}$ is addressing the $N_{IJ}N_{JI}$
terms. It is at this point that we invoke A.3, which tells us that any
$N_{JI}$ applied to a state containing only $t$-type excitations will evaluate
to zero. Thus, the last three terms above are dropped.

Returning to the norm evaluation,

$\displaystyle\left\langle\braket{t}{t}\right\rangle$
$\displaystyle=\left\langle\left\langle
t-1\right|\right.\sum_{k=0}^{t-1}T_{A}^{\dagger\,t-1-k}\bigl{(}P_{1}P_{2}+P_{2}P_{3}+P_{3}P_{1}+3\bigl{(}P_{1}+P_{2}+P_{3}\bigr{)}+6\bigr{)}T_{A}^{\dagger\,k}\ket{0}$
(213) $\displaystyle=\left\langle\left\langle
t-1\right|\right.\sum_{k=0}^{t-1}T_{A}^{\dagger\,t-1-k}(k^{2}+k^{2}+k^{2}+3(k+k+k)+6)T_{A}^{\dagger\,k}\ket{0}$
(214)
$\displaystyle=\sum_{k=0}^{t-1}(3k^{2}+9k+6)\left\langle\braket{t-1}{t-1}\right\rangle$
(215) $\displaystyle=(t+2)(t+1)t\,\left\langle\braket{t-1}{t-1}\right\rangle.$
(216)

By hypothesis,
$\left\langle\braket{t-1}{t-1}\right\rangle=\tfrac{1}{2}(t+1)!\,t!\,(t-1)!$,
so that

$\displaystyle\left\langle\braket{t}{t}\right\rangle$
$\displaystyle=(t+2)(t+1)\,t\tfrac{1}{2}(t+1)!\,t!\,(t-1)!=\tfrac{1}{2}(t+2)!\,(t+1)!\,t!$
(217)

∎

The next type of excitation to be included and normalized is a single
$\ell$-type one:

$\displaystyle\left.\left|\ell_{IJ},t\right\rangle\right\rangle$
$\displaystyle=(L_{IJ}^{\dagger})^{\ell_{IJ}}\left.\left|t\right\rangle\right\rangle.$
(218)

Similar to the evaluation of $\left\langle\braket{t}{t}\right\rangle$, it is
convenient to first establish an intermediate result as follows.

######  Lemma A.5.

For two vertex legs $I$ and $J$, $I\neq J$, any integer $t$, and any integer
$\ell\geq 0$, we have

$\displaystyle N_{IJ}L_{IJ}^{\dagger\,\ell}T_{A}^{\dagger\,t}\ket{0}=0.$ (219)

###### Proof.

We use induction on $\ell$.

For the base case of $\ell=0$, the assertion is that
$N_{IJ}T_{A}^{\dagger\,t}\ket{0}=0$, which is nothing but the statement of
A.3.

Now we move on to $\ell\geq 1$. Evaluating gives

$\displaystyle N_{IJ}L_{IJ}^{\dagger\,\ell}T_{A}^{\dagger\,t}\ket{0}$
$\displaystyle=N_{IJ}L_{IJ}^{\dagger}L_{IJ}^{\dagger\,\ell-1}T_{A}^{\dagger\,t}\ket{0}$
(220)
$\displaystyle=[N_{IJ},L_{IJ}^{\dagger}]L_{IJ}^{\dagger\,\ell-1}T_{A}^{\dagger\,t}\ket{0},$
(221)

where we have used the inductive hypothesis for $\ell-1$. Using the commutator
identity $[N_{IJ},L_{IJ}^{\dagger}]\simeq-F_{J}L_{IJ}^{\dagger}N_{IJ}$, we
have

$\displaystyle N_{IJ}L_{IJ}^{\dagger\,\ell}T_{A}^{\dagger\,t}\ket{0}$
$\displaystyle=-F_{J}L_{IJ}^{\dagger}N_{IJ}L_{IJ}^{\dagger\,\ell-1}T_{A}^{\dagger\,t}\ket{0}$
(222) $\displaystyle=0,$ (223)

where we have again used the inductive hypothesis for $\ell-1$. This concludes
the proof. ∎

######  Theorem A.6.

The norm-squared of states $\left.\left|\ell_{IJ},t\right\rangle\right\rangle$
in (218) is

$\displaystyle\left\langle\braket{\ell_{IJ},t}{\ell_{IJ},t}\right\rangle$
$\displaystyle=\tfrac{1}{2}(\ell_{IJ}+|t|+2)\ell_{IJ}!\,|t|!\,(\ell_{IJ}+|t|+1)!\,(|t|+1)!$
$\displaystyle=\tfrac{1}{2}(\ell_{IJ}+|t|+2)!\,(|t|+1)!\,\ell_{IJ}!\,|t|!\,.$
(224)

###### Proof.

We prove the formula by induction on $\ell_{IJ}$.

For the base case $\ell_{IJ}=0$, the squared norm evaluates as

$\displaystyle\left\langle\braket{0,t}{0,t}\right\rangle$
$\displaystyle=\left\langle\braket{t}{t}\right\rangle$ (225)
$\displaystyle=\tfrac{1}{2}(|t|+2)!\,(|t|+1)!\,|t|!$ (226)

This agrees with the right-hand side of (A.6) when $\ell_{IJ}\to 0$.

We now proceed to the inductive step, taking $\ell_{IJ}\geq 1$ and assuming
that (A.6) holds for $\left.\left|\ell_{IJ}-1,t\right\rangle\right\rangle$.
The norm evaluation begins with

$\displaystyle\left\langle\braket{\ell_{IJ},t}{\ell_{IJ},t}\right\rangle$
$\displaystyle=\left\langle\left\langle\ell_{IJ}-1,t\right|\right.L_{IJ}L_{IJ}^{\dagger}\left.\left|\ell_{IJ}-1,t\right\rangle\right\rangle$
(227)
$\displaystyle=\left\langle\left\langle\ell_{IJ}-1,t\right|\right.L_{IJ}L_{IJ}^{\dagger\,\ell_{IJ}}\left.\left|0,t\right\rangle\right\rangle$
(228)
$\displaystyle=\left\langle\left\langle\ell_{IJ}-1,t\right|\right.\bigl{[}L_{IJ},L_{IJ}^{\dagger\,\ell_{IJ}}\bigr{]}\left.\left|0,t\right\rangle\right\rangle$
(229)
$\displaystyle=\left\langle\left\langle\ell_{IJ}-1,t\right|\right.\sum_{k=0}^{\ell_{IJ}-1}L_{IJ}^{\dagger\,\ell_{IJ}-1-k}\bigl{[}L_{IJ},L_{IJ}^{\dagger}\bigr{]}L_{IJ}^{\dagger\,k}\left.\left|0,t\right\rangle\right\rangle$
(230)

To proceed, one needs the commutator identity

$\displaystyle\bigl{[}L_{IJ},L_{IJ}^{\dagger}\bigr{]}$ $\displaystyle\simeq
3+P_{I}+Q_{J}-F_{I}F_{J}L_{JI}^{\dagger}L_{JI}-F_{I}M_{IJ}M_{JI}-F_{J}N_{JI}N_{IJ}.$
(231)

We will simplify the expression for $\bigl{[}L_{IJ},L_{IJ}^{\dagger}\bigr{]}$
by considering its action on the ket
$L_{IJ}^{\dagger\,k}\left.\left|0,t\right\rangle\right\rangle$:

  * •

$P_{I}+Q_{J}$ evaluates to $|t|+2k$.

  * •

$L_{JI}$ evaluates to zero because
$L_{IJ}^{\dagger\,k}\left.\left|0,t\right\rangle\right\rangle$ is a $P_{J}=0$
or $Q_{I}=0$ eigenstate (depending on the sign of $t$), in which case the
$A(J)$ or $B(I)$ annihilation components must evaluate to zero, respectively.
Therefore, we can drop $F_{I}F_{J}L_{JI}^{\dagger}L_{JI}$ from the commutator.

  * •

Considering the two terms $F_{I}M_{IJ}M_{JI}$ and $F_{J}N_{JI}N_{IJ}$, at
least one of these is necessarily zero, depending on the value of $t$; without