# Protecting gauge symmetries in the the dynamics of SU(3) lattice gauge
theories

Emil Mathew<EMAIL_ADDRESS>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,
Zuarinagar, Goa 403726, India

###### Abstract

Quantum simulation of the dynamics of a lattice gauge theory demands imposing
on-site constraints. Ideally, the dynamics remain confined within the physical
Hilbert space, where all the states satisfy those constraints. For non-Abelian
gauge theories, implementing local constraints is non-trivial, as is keeping
the dynamics confined in the physical Hilbert space, considering the erroneous
quantum devices. SU(3) gauge group, albeit crucial for describing the strong
interaction of nature, is notorious for studying via Hamiltonian simulation.
This work presents a couple of symmetry protection protocols for simulating
the exact dynamics of SU(3) gauge theory in $1+1$ dimension. The first
protocol doesn’t require imposing any local symmetry but relies on protecting
global symmetries, which are Abelian with a preferred choice of framework,
namely the loop-string-hadron framework. Generalization to a higher dimension
is possible, however, the protection scheme needs to be local for that case
but is still Abelian and thus advantageous. The symmetry protection schemes
presented here are important steps towards quantum simulating the full theory
of quantum chromodynamics.

###### Contents

  1. I Introduction
  2. II A suitable framework to study dynamics: LSH framework
    1. II.1 LSH basis
    2. II.2 LSH Hamiltonian
    3. II.3 Gauss law constraints in LSH framework
    4. II.4 Global symmetries of the LSH framework in $1+1$ dimension
  3. III Symmetry protection in erroneous dynamics
    1. III.1 Scheme I: LSH dynamics via global symmetry protection
    2. III.2 Scheme II: LSH dynamics via local symmetry protection scheme
  4. IV Conclusions and outlook
  5. A LSH SU(3) Dictionary
  6. B Results for different initial-states

##  I Introduction

The Hamiltonian formulation, first put forth by Kogut and Susskind Kogut and
Susskind (1975), offers a sign-problem-free approach to studying the real-time
dynamics of the gauge theories, which provides a promising framework for
computation in the upcoming quantum computation era.

The Hamiltonian obtained via temporal gauge fixing is defined on the spatial
lattice while the time remains continuous. However, the Hamiltonian dynamics
for gauge theory is constrained by a set of local constraints known as Gauss
law. The generators of the Gauss law constraints are mutually non-commuting
for non-Abelian gauge theories such as SU(2) or SU(3) gauge theories, the
later one being the underlying theory of the strong interaction of nature are
of crucial physical interest. The primary ingredients for SU(N) lattice gauge
theories are a Hamiltonian $\hat{H}$ and constraints $\hat{G}^{a}(r)$, which
satisfy:

$\displaystyle\left[H,G^{a}(r)\right]=0,~{}~{}\left[G^{a}(r),G^{b}(r^{\prime})\right]=i{f^{ab}}_{c}G^{c}(r)\delta_{r,r^{\prime}}$
(1)

$\forall r$ and $a=1,2,...,N^{2}-1$. The physical Hilbert space for a gauge
theory is generally chosen as the space of states, which are annihilated by
all the Gauss’ law operators $G^{a}(r)$ that generate gauge transformations
locally at each site $r$. In principle, one can choose other gauge symmetry
sectors (characterized by an integer quantum number of Gauss law generator) as
physical Hilbert space, yet the Hamiltonian dynamics remains confined to that
sector. The conventional choice of gauge invariant sector leads to Wilson
loops, strings, and hadrons as the physical degrees of freedom for the gauge
theory Gambini and Pullin (2000); Raychowdhury and Stryker (2020a); Mathur
(2007).

Performing Hamiltonian simulation in the gauge theory Hilbert space (without
imposing Gauss law constraint) involves huge computational costs being wasted
in gauge redundancy and is not at all suitable for quantum simulation of the
same as quantum resources are even more expensive and limited. Hamiltonian
simulation of gauge theories primarily demands (i) preparation of the initial
state in the gauge invariant Hilbert space/ physical Hilbert space (ii) and
guarantee protection of gauge invariance throughout the simulated dynamics.
Both of these tasks stand difficult in context to simulate the theory using
the conventional framework and on a quantum hardware, respectively.

The conventional framework for Hamiltonian lattice gauge theory is given by
the well-studied Kogut-Susskind Kogut and Susskind (1975) formulation.
Construction of a gauge invariant Hilbert space in the Kogut-Susskind
formulation is possible but expensive Davoudi  _et al._ (2021a). Another major
obstacle when working with the gauge invariant Hilbert space is that it
involves non-locality. Even in the $1+1$ dimension, one can effectively
eliminate all the gauge degrees of freedom for a gauge theory at the cost of
introducing non-local interaction among on-site fermionic degrees of freedom.
Over the last decade, there has been a renewed interest in Hamiltonian
simulations of lattice gauge theories Cirac and Zoller (2012); Dalmonte and
Montangero (2016); Banuls  _et al._ (2020); Aidelsburger  _et al._ (2022);
Zohar (2022); Bauer  _et al._ (2023). There have been a series of work on
addressing this issue in the context of low-dimensional theories, and towards
developing novel frameworks particularly 1+1 dimensions, Chandrasekharan and
Wiese (1997); Brower  _et al._ (1999); Zohar and Reznik (2011); Zohar and
Burrello (2015); Zohar  _et al._ (2015, 2017); Zohar and Cirac (2018, 2019);
Ciavarella  _et al._ (2021); Bañuls  _et al._ (2017); Kaplan and Stryker
(2020); Unmuth-Yockey (2019); Bender and Zohar (2020); Haase  _et al._ (2021);
Bauer and Grabowska (2023); Ashkenazi and Zohar (2022); Zache  _et al._
(2018); Buser  _et al._ (2021); Ji  _et al._ (2020); Pardo  _et al._ (2023);
Stryker (2021). On the same note, there have been a multitude of research done
into applying one or more of these formalism to describe digital as well as
analog simulation protocols in the current NISQ-era Zohar and Reznik (2011);
Banerjee  _et al._ (2012, 2013); Zohar  _et al._ (2013); Tagliacozzo  _et al._
(2013a); Zohar  _et al._ (2015); Tagliacozzo  _et al._ (2013b); Mazza  _et
al._ (2012); Kasper  _et al._ (2016); González-Cuadra  _et al._ (2017); Zohar
_et al._ (2017); Kasper  _et al._ (2017); Muschik  _et al._ (2017); Lamm  _et
al._ (2019); Shaw  _et al._ (2020); Armon  _et al._ (2021); Carena  _et al._
(2022); Davoudi  _et al._ (2020, 2021b); Klco  _et al._ (2018, 2020);
Raychowdhury and Stryker (2020b); Dasgupta and Raychowdhury (2022); Paulson
_et al._ (2021); Riechert  _et al._ (2022); Zhou  _et al._ (2022); Homeier
_et al._ (2023); Müller and Yao (2023); Zache  _et al._ (2023); Su  _et al._
(2024). The noisy nature of current quantum simulation technology demands
error mitigation/correction techniques to faithfully simulate gauge-invariant
dynamics. Over the years, there have been quite a few proposals suggesting
different ways to control this error Tran  _et al._ (2021); Halimeh and Hauke
(2020); Halimeh  _et al._ (2021, 2022); Schweizer  _et al._ (2019); Stannigel
_et al._ (2014); Kasper  _et al._ (2023); Stryker (2019). The different
proposals and error mitigation schemes led to several digital and analog
simulations of Abelian and non-Abelian lattice gauge theories over the last
few years Martinez  _et al._ (2016); Kokail  _et al._ (2019); Mil  _et al._
(2020); Semeghini  _et al._ (2021); Yang  _et al._ (2020); Atas  _et al._
(2021a, b); Mildenberger  _et al._ (2022); Riechert  _et al._ (2022); Alam
_et al._ (2022); Gustafson  _et al._ (2022); Ciavarella and Chernyshev (2022);
Illa and Savage (2022); Su  _et al._ (2023); Atas  _et al._ (2023a); Farrell
_et al._ (2023a); Rahman  _et al._ (2022); Zhang  _et al._ (2023); Farrell
_et al._ (2023b, c); Ciavarella (2023); Charles  _et al._ (2024); Mueller  _et
al._ (2023); Kavaki and Lewis (2024); Farrell  _et al._ (2024); Ciavarella and
Bauer (2024).

In higher dimensions, the Hamiltonian includes plaquette terms, which are
gauge invariant but involve the minimum number of sites required to trace a
closed path on the lattice. Quantum simulation of the non-local terms poses
additional challenges while maintaining non-Abelian gauge invariance locally
at each site. The current work addresses how these complications can be
avoided by using the loop- string-hadron (LSH) framework Raychowdhury (2019);
Kadam  _et al._ (2022) towards making concrete progress in quantum simulating
QCD.

The organization of the paper is as follows. In section II, we briefly review
the loop string hadron formalism for SU(3) gauge theory. In section III.1, we
focus on the global protection scheme valid for $1+1$ dimensional models.
Section III.2 discusses the alternate protection scheme, which involves local
protection yet is simpler than protecting all the non-Abelian symmetries of
the original theory and is valid in higher dimensions. Finally, we summarise
in IV and discuss the viability of the scheme in terms of its implementation.

##  II A suitable framework to study dynamics: LSH framework

In this work, we focus on a particular framework, namely the loop-string-
hadron (LSH) framework that has been developed for SU(2) Raychowdhury and
Stryker (2020a) and SU(3) Kadam  _et al._ (2022) gauge theories. The SU(2)
framework has been studied extensively and found to be advantageous over other
available frameworks to describe the same physics. The key feature of LSH
formalism is that it is constructed to be manifestly gauge invariant. The
Hilbert space is characterized by on-site quantum numbers, which correspond to
gauge invariant and physical loop-string-hadron excitations at each site, and
their dynamics are described by a Hamiltonian, which contains combinations of
occupation number operator and ladder operators for the associated loop-
string-hadron excitations.

However, encountering non-local excitations is unavoidable in a gauge theory,
which is true for the LSH framework. The notion of non-locality in the LSH
framework arises from the fact that the gauge invariant configurations defined
on a lattice, equivalent to Wilson loops or strings in the theory, are not a
tensor product of on-site LSH states belonging to the on-site LSH Hilbert
spaces, but instead are local LSH states weaved via Abelian constraints on
each link. The LSH Hamiltonian commutes with the Abelian constraint and the
spectrum and dynamics match that of the original Kogut-Susskind Hamiltonian
for a particular theory. In this work, we emphasize a couple of interesting
and useful conclusions regarding using LSH for Hamiltonian simulation that can
be applicable and beneficial to use in appropriate setups.

###  II.1 LSH basis

The LSH basis is defined locally at each lattice site which is characterized
by a set of integer-valued quantum numbers, namely the loop and string quantum
numbers. Loop quantum numbers are bosonic implying the Hilbert space to be of
infinite dimension, while the string quantum numbers are fermionic. However,
for practical computation one needs to impose a finite cut-off on the bosonic
quantum number so that the Hilbert space is of finite dimension.

For SU(2) gauge theory, the LSH basis states at a site $r$ in $1+1$-d is given
by

$|n_{l},n_{i},n_{o}\rangle_{r}$

, where $n_{l}\in\\{0,\infty\\}$ is the loop quantum number and
$n_{i},n_{o}\in\\{0,1\\}$ are fermionic quantum numbers. Imposing a cut-off
$\Lambda$, one would obtain an onsite Hilbert space of dimension
$2^{2}(\Lambda+1)$.

For the case of SU(3) gauge theory defined on a one-dimensional spatial
lattice, the on-site Hilbert space is characterized by two loop and three
string quantum numbers as

$\displaystyle|n_{P},n_{Q},\nu_{\underline{1}},\nu_{0},\nu_{1}\rangle_{r},$
(2)

where, $n_{P},n_{Q}\in\\{0,\infty\\}$ and
$\nu_{\underline{1}},\nu_{0},\nu_{1}\in\\{0,1\\}$. Pictorial representations
of the LSH states are given in Fig. 1. For a bosonic cut-off $\Lambda$, the
dimension of the on-site LSH Hilbert space is $2^{3}(\Lambda+1)^{2}.$

Figure 1: On-site LSH state for (a) SU(2) and (b) SU(3) gauge theories in
$1+1$ dimension. Note that the fermionic excitations are denoted as circles,
which can be either empty or filled, denoting the corresponding fermionic
occupation numbers to be $0$ and $1$, respectively. For SU(2), flux lines are
undirected, while for SU(3) being directed, they can flow in two different
directions. Within LSH framework, the fermionic occupation numbers are
associated with incoming and outgoing strings in different directions.

###  II.2 LSH Hamiltonian

Following the original Kogut-Susskind construction of the Hamiltonian Kogut
and Susskind (1975), the Hamiltonian in the LSH framework also consists of
electric, mass, fermionic hopping, and magnetic field terms. Considering the
simplest case of one spatial dimension, the magnetic term is not present, but
the Hamiltonian (scaled to make it dimensionless) is written as:

$\displaystyle H=H_{E}+\mu H_{m}+xH_{I}$ (3)

where $\mu$ and $x$ are dimensionless parameters that depend upon the lattice
spacing, coupling, and mass of the fermions. The strong coupling limit of
lattice gauge theory is always interesting as the theory is confined in this
regime, defined by $x\rightarrow 0$, keeping $\mu$ to be finite. The LSH basis
described before is a strong coupling eigenbasis, i.e., in the LSH basis, the
matrix representation of $H_{E}$ and $H_{m}$ are both diagonal. Also note
that, both the terms $H_{E}$ and $H_{m}$ are ultra-local in the strong
coupling basis, so contributions from all the lattice sites are added to make
the electric energy and mass of the entire system.

In the LSH framework, one can define on-site occupation number operators for
the LSH basis both for the SU(2) and SU(3) case, and the onsite contribution
of electric and mass term is obtained as a function of occupation numbers of
the LSH excitations at that site:

$\displaystyle\hat{n}_{l}|\\{n_{l}\\},\\{n_{f}\\}\rangle$ $\displaystyle=$
$\displaystyle n_{l}|\\{n_{l}\\},\\{n_{f}\\}\rangle$ (4)
$\displaystyle\hat{n}_{f}|\\{n_{l}\\},\\{n_{f}\\}\rangle$ $\displaystyle=$
$\displaystyle n_{f}|\\{n_{l}\\},\\{n_{f}\\}\rangle$ (5)

where $\\{n_{l}\\}$ denotes the set of bosonic quantum numbers, i.e. $n_{l}$
for SU(2) and $n_{P}~{}\&~{}n_{Q}$ for SU(3). Similarly, The LSH Hamiltonian
is given in Appendix A both for SU(2) and SU(3). Interestingly, the mass part
of Hamiltonian is effectively proportional to the total occupation number of
fermions at each site. The electric part, on the other hand, counts the flux
lines that originate or enter a site. A closer inspection of Fig. 1 reveals
that non-zero occupation of the fermionic LSH modes often comes with flux
lines attached to it, and hence, the fermionic LSH excitations also contribute
to the on-site electric energy. The detailed expressions are given in the
Appendix A.

For a $1+1$ dimensional theory, the only term responsible for dynamics is the
hopping or ‘matter-gauge interaction’ term $H_{I}$. The construction of the
interaction Hamiltonian $H_{I}$ is nontrivial, compared to its SU(2)
counterpart as evident in the detail construction given in Kadam  _et al._
(2022). This is due to the fact that matter content at a SU(3) site contains
much more detailed inner structure than just being present as string end
operators for the SU(2) case. This can be qualitatively appreciated from FIG.
1. To finalize a representation of $H_{I}$ in LSH basis, we consider the
following set of normalized ladder operators, which acting on a on-site LSH
state, changes its excitations by $\pm 1$ unit.

$\displaystyle\Gamma^{\dagger}_{L}|\\{n_{l}\\},\\{n_{f}\\}\rangle$
$\displaystyle=$ $\displaystyle|\\{n_{l}|n_{L}+1\\},\\{n_{f}\\}\rangle$ (6)
$\displaystyle\chi^{\dagger}_{F}|\\{n_{l}\\},\\{n_{f}\\}\rangle$
$\displaystyle=$
$\displaystyle(1-\delta_{1,n_{F}})|\\{n_{l}\\},\\{n_{f}|n_{F}+1\\}\rangle$ (7)
$\displaystyle\Gamma_{L}|\\{n_{l}\\},\\{n_{f}\\}\rangle$ $\displaystyle=$
$\displaystyle|\\{n_{l}|n_{L}-1\\},\\{n_{f}\\}\rangle$ (8)
$\displaystyle\chi_{F}|\\{n_{l}\\},\\{n_{f}\\}\rangle$ $\displaystyle=$
$\displaystyle(1-\delta_{0,n_{F}})|\\{n_{l}\\},\\{n_{f}|n_{F}-1\\}\rangle$ (9)

The notation $\\{n_{l}|n_{L}\pm 1\\}$ and $\\{n_{f}|n_{F}\pm 1\\}$ within the
ket denotes that only the specific bosonic $(L)$ or fermionic $(F)$ LSH
quantum number has changed, while keeping all other in the set fixed.

The following definition captures the inner structure of gauge invariant
dynamics for this theory:

$\displaystyle H_{I}$ $\displaystyle=$
$\displaystyle\sum_{r}\Bigg{[}\sum_{F=\underline{1},0,1}\chi_{F}^{\dagger}(r)\Big{[}\hat{O}_{1}^{(F)}(\\{\hat{n}_{l}(r),\hat{n}_{f}(r)\\})\times\hat{O}_{2}^{(F)}(\\{\hat{n}_{l}(r+1),\hat{n}_{f}(r+1)\\})\Big{]}\chi_{F}(r+1)\Bigg{]}+\mathrm{h.c.}$
(10)

The exact structure of
$\hat{O}_{1/2}^{F}(\\{\hat{n}_{l}(r),\hat{n}_{f}(r)\\})$ for
$F=\underline{1},0,1$ are listed in the Appendix A, equations (58)-(56).

###  II.3 Gauss law constraints in LSH framework

The physical implication of gauge invariance is in some non-local structures,
and the same is reflected in the non-local Wilson loop and string basis as the
gauge invariant basis of gauge theories. However, as listed in FIG, the LSH
framework is an intermediate step in building on-site gauge invariant Hilbert
space. 1. Connecting the on-site LSH Hilbert space to the space of Wilson
loops and strings requires continuity of flux lines (for each type of directed
flux lines for SU(3)) across neighboring lattice sites. This is true for
arbitrary dimensions as well.

Analyzing the $1+1$-dimensional SU(3) theory, let us focus on-site $r$, which
is connected to links along direction $1$ and $\underline{1}$. Careful
inspection of directed flux lines in FIG. 1 yields:

$\displaystyle P_{1}(r)$ $\displaystyle=$ $\displaystyle
n_{P}(r)+\nu_{1}(r)\Big{(}1-\nu_{0}(r)\Big{)}$ (11) $\displaystyle Q_{1}(r)$
$\displaystyle=$ $\displaystyle
n_{Q}(r)+\nu_{0}(r)\Big{(}1-\nu_{\underline{1}}(r)\Big{)}$ (12)

Where $P_{1}(r)$ (and $Q_{1}(r)$ ) denote the number of flux lines going from
left to right (and right to left) respectively on the link along $1$ direction
connected to site $r$.

The same on the link in direction $\underline{1}$ connected to site $r$ is
given by:

$\displaystyle P_{\underline{1}}(r)$ $\displaystyle=$ $\displaystyle
n_{P}(r)+\nu_{0}(r)\Big{(}1-\nu_{1}(r)\Big{)}$ (13) $\displaystyle
Q_{\underline{1}}(r)$ $\displaystyle=$ $\displaystyle
n_{Q}(r)+\nu_{\underline{1}}(r)\Big{(}1-\nu_{0}(r)\Big{)}$ (14)

Continuity of flux lines or the local LSH states correspond to Wilson loop and
string states of the gauge theory demands for the constraints:

$\displaystyle P_{1}(r)=P_{\underline{1}}(r+1)$ (15) $\displaystyle
Q_{1}(r)=Q_{\underline{1}}(r+1)$ (16)

This implies that the space of Wilson loops and strings are not a direct
product of LSH states but rather a projected subspace that satisfies a pair of
local Abelian Gauss-law (AGL) constraints given in (15,16).

###  II.4 Global symmetries of the LSH framework in $1+1$ dimension

For a SU(3) gauge theory on one-dimensional staggered lattice, the global
symmetry remains to be SU(3) which can be generated by $8$ charges:

$\displaystyle Q^{a}$ $\displaystyle=$
$\displaystyle\sum_{r=0,2,..}\Psi^{\dagger}_{\alpha}(r)\left(\frac{\lambda^{a}}{2}\right)^{\alpha}_{\beta}\Psi^{\beta}(r)$
(17)
$\displaystyle-\sum_{r=1,3,..}\Psi_{\alpha}(r)\left(\frac{\lambda^{*a}}{2}\right)^{\alpha}_{\beta}\Psi^{\dagger\beta}(r)$

where, $\Psi_{\alpha}(r)$ are fermionic triplets (anti-triplets) (fundamental
irreps of SU(3)) located at even (odd) sites $r$ and $\lambda^{a}$ are Gell-
Mann matrices for $a=1,2,...,8$. A global state defined on the lattice can
thus be characterized by global SU(3) casimirs $|p,q,I,M,Y\rangle$ of an SU(3)
irrep using the standard notation, where $p,q$ is analogous to $j$ quantum
number (Casimir) of SU(2) irreps and the set $I,M,Y$ denote the magnetic
quantum numbers analogous to $m$ of an SU(2) irrep denoted as $|j,m\rangle$.
Naturally, analyzing the global symmetries of the SU(3) gauge theory is more
involved than the SU(2) theory when using the irrep-basis or Kogut-Susskind
formalism of the theory. However, global symmetries play a crucial role by
providing a block diagonal structure of the Hamiltonian and analyzing the
entanglement properties of gauge theories, which is important in the context
of quantum computation. The global symmetry structure of the same theory in
the LSH framework is found to be much more intuitive and can be used for more
efficient computations at ease.

The LSH framework, being a solution to all the non-Abelian constraints, is
still constrained by the local Abelian Gauss law. As a consequence, the global
symmetries for LSH also become Abelian. For $1+1$-d theory coupled to
staggered matter fields, as depicted in (10), the interaction term manifestly
preserves the global occupation numbers of the three individual gauge
invariant fermionic or string modes:

$\displaystyle q_{f}=\sum_{r}\nu_{f}(r)$ (18)

for $f=\underline{1},0,1$. Alternately, one can also define the linear
combinations of $q_{\underline{1}},q_{0},q_{1}$ as Kadam  _et al._ (2022):

$\displaystyle\mathcal{F}$ $\displaystyle=$ $\displaystyle
q_{\underline{1}}+q_{0}+q_{1}$ $\displaystyle\mathcal{P}$ $\displaystyle=$
$\displaystyle q_{1}-q_{0}$ $\displaystyle\mathcal{Q}$ $\displaystyle=$
$\displaystyle q_{0}-q_{\underline{1}}$

Note that $\mathcal{F}$ gives the naive total fermionic occupation number for
the system, while $\mathcal{P}$ and $\mathcal{Q}$ give the imbalance between
the incoming and outgoing flux of the lattice individually for leftward and
rightward flux lines. The Hamiltonian matrix written in the LSH basis is block
diagonalized, where each block is characterized by unique set of global
quantum numbers $(\mathcal{F},\mathcal{P},\mathcal{Q})$. An ideal Hamiltonian
simulation is expected to obey the block diagonal structure of the theory.

A definite benefit of using the LSH framework is the link between these global
symmetries and the remnant local Abelian constraints in 1+1 dimension. As
illustrated in the case of SU(2) in Mathew and Raychowdhury (2022), the
particular structure of the interaction Hamiltonian for a $1+1$d theory with
manifest global symmetries is a consequence of the interaction Hamiltonian
being Abelian gauge invariant as well. However, a similar connection is
nontrivial to demonstrate for the case of SU(3) via analytic expression as the
string end operators are now of three types and involve an intermediate or
$0^{th}$ mode rather than only corresponding to incoming and outgoing string.
However, the current work establishes the connection between AGL and global
symmetries for SU(3), demonstrated numerically in the next section.

We first consider situations where the bit-flip or other errors present in
quantum hardware can lead to deviation from this ideal gauge invariant LSH
dynamics. Using two different prescriptions, we provide solutions for gauge
protection and numerically establish their validity. These solutions are
useful in tensor network simulation, analog simulation protocols, and digital
quantum circuits.

##  III Symmetry protection in erroneous dynamics

We first consider a model where the global symmetries mentioned in (III.1) are
violated. Note that, the first global symmetry characterized by $\mathcal{F}$,
is easy to preserve, as that refers to only a perfect state preparation with a
fixed total number of qubits given by $\mathcal{F}$. Even while preserving the
value of $\mathcal{F}$, the following interaction term violates
$\mathcal{P},\mathcal{Q}$, modelling bit-flip error that is most common to
occur:

$\displaystyle H^{\prime}_{I}$ $\displaystyle=$
$\displaystyle\sum_{r}\sum_{F,F^{\prime}=\underline{1},0,1}\Bigg{[}(x\delta_{F,F^{\prime}}+x^{\prime}(1-\delta_{F,F^{\prime}}))\Big{[}\hat{O}_{1}^{(F)}(\\{\hat{n}_{l}(r),\hat{n}_{f}(r)\\})\Big{]}\Big{[}\hat{O}_{2}^{(F^{\prime})}(\\{\hat{n}_{l}(r+1),\hat{n}_{f}(r+1)\\})\Big{]}\chi_{F^{\prime}}(r+1)\Bigg{]}$
(20) $\displaystyle+\mathrm{h.c.}$

Note that only a set of sub-terms of (20), i.e. only for $F=F^{\prime}$ cases
in the sum, gives the original interaction Hamiltonian (10). One can now write
a total Hamiltonian composed of the erroneous interaction as

$\displaystyle H_{err}=H_{E}+H_{M}+H^{\prime}_{I}$ (21)

It is manifest from (21) that, $H^{\prime}_{I}$ violates the global
symmetries. We numerically demonstrate in FIG. 2 that time evolution with the
erroneous Hamiltonian is not confined to the AGL satisfying sector of the LSH
Hilbert space.

Figure 2: The first row contains the dynamics of expectation values of the
global charges $\mathcal{P}$ and $\mathcal{Q}$ and the second row contains the
same for the six AGLs across three links evolving under the (erroneous)
Hamiltonian given in equation (21). Each column corresponds to the evolution
of an initial state in a particular global symmetry sector denoted by
$(\mathcal{P},\mathcal{Q})$. Here, $\langle AGL^{1}\rangle_{r,r+1},\langle
AGL^{2}\rangle_{r,r+1}$ corresponds to equation (15),(16) respectively. The
plots demonstrate that none of the global and local symmetries are preserved
by (21) as the dynamics are not coinciding with reference lines (red dotted
lines) for each.

All our numerical calculations are performed for a 4-staggered site lattice
with $x=x^{\prime}=1$ to ensure maximal mixing of the three fermionic modes
and the mass parameter $\mu=1$. We also impose a cutoff on the Bosonic quantum
numbers $n_{P},n_{Q}=2$. The QuSpin Package Weinberg and Bukov (2017, 2019) is
used to construct the Hamiltonian and its subsequent time evolution. The
initial state is chosen to be the strong-coupling vacuum (unless otherwise
specified), which satisfies the AGLs across all the links and lies in the
global symmetry sector $(\mathcal{F},\mathcal{P},\mathcal{Q})=(6,0,0)$. In the
LSH quantum numbers given in (2), the initial state is given by :

$\displaystyle\ket{\Psi}_{i}$ $\displaystyle=$
$\displaystyle\ket{0,0,0,0,0}_{0}\otimes\ket{0,0,1,1,1}_{1}$
$\displaystyle\otimes$
$\displaystyle\ket{0,0,0,0,0}_{2}\otimes\ket{0,0,1,1,1}_{3}$

The above-defined state is time-evolved using the erroneous Hamiltonian
defined in equation (21) as $\ket{\Psi(t)}=e^{-iH_{err}t}\ket{\Psi}_{i}$. FIG
2 shows the averages of the AGLs across each link along with the average of
the global symmetry operators $\mathcal{P},\mathcal{Q}$ as a function of time.
The averages move away from the ideal values and signify mixing with non-AGL
satisfying states.

###  III.1 Scheme I: LSH dynamics via global symmetry protection

A scheme for dynamically protecting global symmetries (in other words,
projecting the dynamics in different super-selection sectors) of SU(3) gauge
theory coupled to dynamical matter fields in $1+1$ dimension generalizing
Mathew and Raychowdhury (2022) is presented in this section. As discussed
previously, $H^{\prime}_{I}$ violates the super selection rules or the global
symmetries of the LSH Hamiltonian. To counteract this phenomenon, we propose a
pair of protection terms that preserve the global symmetries of the SU(3) LSH
Hamiltonian, namely the $\mathcal{P}$ and $\mathcal{Q}$ operators. We propose
adding the following set of operators to the Hamiltonian:

$\displaystyle
H_{\mathcal{P}}=\Lambda\Bigg{[}\mathcal{P}-\sum_{r=0}^{N-1}\Big{[}\nu_{0}-\nu_{\underline{1}}\Big{]}\Bigg{]}$
(23) $\displaystyle
H_{\mathcal{Q}}=\Lambda\Bigg{[}\mathcal{Q}-\sum_{r=0}^{N-1}\Big{[}\nu_{1}-\nu_{0}\Big{]}\Bigg{]}$
(24)

Here, $\mathcal{P}$ and $\mathcal{Q}$ can take integer values depending on the
symmetry sector of interest. Adding these terms to the total Hamiltonian will
ensure that an initially gauge-invariant state can be protected with, at most,
a single body protection term. The total Hamiltonian will now read as:

$\displaystyle H=H_{E}+H_{M}+H^{\prime}_{I}+H_{\mathcal{P}}+H_{\mathcal{Q}}$
(25)

where $\Lambda$ is a constant term kept large enough to ensure a sufficient
gap exists between the symmetry-protected sector and the other part of the
spectrum. The validity of the scheme in confining the dynamics in each super-
selection sector is demonstrated in FIG. 3.

The numerical study presented in FIG. 2 demonstrated that the modified
interaction Hamiltonian $H^{\prime}_{I}$ does, in effect, violate all of the
Abelian Gauss laws for the lattice and drive the dynamics into the non-AGL
satisfying sector of the LSH Hilbert space. However, the numerical study
performed in this work establishes that efficient protection of all of the
global symmetries of this theory, in effect, protects each individual Abelian
Gauss law originally present in the theory.

Suitable protection strength $\Lambda$ is also capable of ensuring a
sufficient gap between the AGL satisfying and AGL violating states in the
Hilbert space. Thus, the AGL-violating states are suppressed during time
evolution, and the dominant contribution comes from the AGL-satisfying states.
As before, the testbed for the numerical demonstrations is a 4-staggered site
lattice with $x=x^{\prime}=1$, $\mu=1$, and the strong coupling vacuum as our
initial state defined in equation (III). Time evolving this initial state
under the new Hamiltonian defined in equation (25) will ensure that the AGLs
and global symmetries are restored in the limit of $\Lambda\rightarrow\infty$.
FIG 3 shows how the local and global symmetries are being restored. The error
bars are the quantum fluctuations of the operators of interest, mainly
$\\{AGL^{\\{1,2\\}}_{r,r+1},\mathcal{P},\mathcal{Q}\\}$. These operators are
temporally averaged for a short time of 0.5T, where $T$ is of the order
$10^{3}$. This is because the size of the Hilbert space for the 4-site lattice
is of the order $10^{7}$, resulting in the norm of the Hamiltonian defined in
equation (25) becoming very large. This renders it difficult to simulate for
larger time scales.

Figure 3: The figure illustrates the temporally averaged values of the global
symmetry operators $\mathcal{P},\mathcal{Q}$ and the two Abelian Gauss laws
across three links for a 4-staggered site system. For $x=x^{\prime}=1$, taking
the $\Lambda\rightarrow\infty$ limit ensures that $\mathcal{P},\mathcal{Q}$
tend to the sector the initial state belonged to along with the Abelian Gauss
laws across the three links converging to zero. The error bars are the
temporally averaged operator fluctuations $\mathcal{O}$ for each $\Lambda$
parameter run

If one prepares an initial state in the correct global symmetry sector that
also satisfies the AGLs across each link, the symmetry protection protocol
will ensure that the large $\Lambda$ limit will approximately conserve the
global symmetries and the local constraints. With the state guaranteed to
preserve the global symmetries of the theory, we can compute the dynamics of
relevant observables of the theory, namely the staggered fermion density and
the electric Hamiltonian density across the lattice, and compare them with the
gauge-invariant dynamics. This scheme also applies to initial states belonging
to different global symmetry sectors, and Appendix B contains the results for
such a choice of initial state.

###  III.2 Scheme II: LSH dynamics via local symmetry protection scheme

As it is manifest that $H^{\prime}_{I}$ violates each individual AGL, the most
straightforward way to protect the AGLs is by adding protection terms to the
erroneous Hamiltonian corresponding to each. In this section, we demonstrate
the same to be working for the erroneous Hamiltonian given in (21). In
contrast with the previous section, where we used a single body protection
term to simulate an approximate gauge theory, here the focus is on two body
protection terms which is just equivalent to imposing the square of the two
AGLs across each link. As before, we consider equation (20) to be the
erroneous interaction term.

The full Hamiltonian will now modified to include the following two terms:

$\displaystyle
H_{P}=\Lambda\sum_{r=0}^{N-1}\Bigg{[}P_{1}(r)-P_{\underline{1}}(r+1)\Bigg{]}^{2}$
(26) $\displaystyle
H_{Q}=\Lambda\sum_{r=0}^{N-1}\Bigg{[}Q_{1}(r)-Q_{\underline{1}}(r+1)\Bigg{]}^{2}$
(27)

where $P_{1},P_{\underline{1}},Q_{1},Q_{\underline{1}}$ are defined according
to equation (11-14). The newly modified Hamiltonian can now be written out as
follows:

$\displaystyle H=H_{E}+H_{M}+H^{\prime}_{I}+H_{P}+H_{Q}$ (28)

Once again, we repeat the numerical experiment of starting with the state
defined in equation (III) and measure the AGLs as a function on
$\Lambda\rightarrow\infty$. In FIG. 4, we demonstrate that this scheme with a
suitable choice of protection strength $\Lambda$ allows the dynamics to remain
confined into the AGL satisfying sector of the Hilbert space.

Figure 4: Operator average of the local AGLs and the global symmetry operators
as a function of increasing $\Lambda$. The error bars correspond to the
temporal average of the operator fluctuation.

The $\Lambda\rightarrow\infty$ limit is where an approximate gauge theory is
expected to be recovered, and as per the numerical simulation, we see that the
operator averages tend to the desired values, with the expectation value of
the AGL to be vanishing. Additionally, the temporally average operator
fluctuations are also of the order $10^{-5}$, indicating that the non-AGL
satisfying states’ contributions are becoming negligible.

Such a local Abelian protection can be applied to higher dimensions where it
is indeed essential. In this work, we do not present calculations for higher
dimensions as that become computationally heavy and the one-dimensional study
is sufficient to demonstrate the proof of principle.

The bonus finding in this one-dimensional demonstration of the local
protection scheme is that each successful protection of local symmetry also
guarantees that the dynamics are confined to the original global symmetry
sectors (as per the initial state prepared) of the LSH Hamiltonian. This is
also demonstrated in FIG. 5.

Figure 5: Super-selection rules are preserved by protecting the local
symmetries. The initial state is chosen in the $0,0$ sector and evolved with
(28).

##  IV Conclusions and outlook

The current work demonstrates liabilities of a simple scheme of protecting all
the symmetries of non-abelian gauge theories in simulating its dynamics. The
major challenge in simulating non-Abelian gauge theories is two fold. The
first non-trivial task is preparing the initial state which is global state,
yet gauge singlet at each and every lattice site. As the Hamiltonian is
expected to commute with the generators of all gauge transformations, the
dynamics ideally should remain confined in a subspace of the entire gauge
theory Hilbert space. However, in terms of erroneous quantum devices, this is
not expected to be the reality and hence the simulation scheme demands for a
symmetry protection protocol, for any available platform. Following the warm-
up exercises towards quantum simulating simpler gauge theories carried out
over more than a decade now, the community is now focussing on the SU(3) gauge
theories Ciavarella  _et al._ (2021); Atas  _et al._ (2023b); Ciavarella
(2023); Ciavarella and Bauer (2024) in $1+1$ dimension and the final aim is to
perform simulations in higher dimension. The major restriction in achieving
the goal is definitely imposing the complicated SU(3) symmetries and
protecting the same. The loop-string-hadron framework for SU(3) in Kadam  _et
al._ (2022) provides a convenient framework for Hamiltonian simulation, where
the basis states are inherently SU(3) singlet, leading to no need of imposing
any SU(3) symmetry at any point of calculating or simulating synamics. The
current protocol, takes advantage of this fact and provides a viable scheme
for symmetry protection demonstrating the advantage of using LSH framework.

$1+1$ dimensional gauge theories, although offer the first test-bed for novel
simulation protocols are often a lot easier to analyze as the gauge field is
not dynamic. However, the global symmetry still possesses a rich non-Abelian
structure implying the superselection rules to be governed by SU(3) symmetry
and its representations. The LSH framework, being fully Abelianized leads to
simpler Abelian global symmetry sectors which is far more intuitive. The
current protocol utilizes the same for protecting all the symmetries in $1+1$
dimensions. This is the major advantage of enabling one to simulate SU(3)
gauge theory without imposing either any non-abelian nor any local symmetries.
Work is in progress in utilizing this scheme for developing analog quantum
simulation protocols, which will be reported shortly. This scheme also remains
useful for multiple directions of ongoing research efforts in developing
tensor network algorithms for calculating dynamics Bañuls (2023); Belyansky
_et al._ (2024); Knaute  _et al._ (2024); Feldman  _et al._ (2024), as well as
quantum simulation of scattering of wave-packets Davoudi  _et al._ (2024);
Bennewitz  _et al._ (2024), or studying thermalization dynamics of gauge
theories Ebner  _et al._ (2024); Davoudi  _et al._ (2023); Mueller  _et al._
(2023) using quantum simulators. Works are progressing towards generalizing
these schemes to deliver building blocks for the ultimate goal of quantum-
simulating QCD, and this current work is one of those.

## Acknowledgements

We acknowledge discussions with Pubasha Shome. 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 SERB, India and the cross-discipline research fund (C1/23/185) from BITS
Pilani.

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##  Appendix A LSH SU(3) Dictionary

The LSH formulation for the SU(3) gauge group follows a similair albeit
involved constructionKadam  _et al._ (2022). The salient features are
neverthess identical to that of the SU(2) formulation. Here, we briefly review
the key ideas necessary to understand the construction of the basis states and
the Hamiltonian. On a 1+1D lattice, the SU(3) LSH Hilbert space is
characterized by five quantum numbers
$|n_{P},n_{Q};\nu_{\underline{1}},\nu_{0},\nu_{1}\rangle$. Here, $n_{P}$ and
$n_{Q}$ are bosonic quantum numbers which take values from $\\{0,\infty\\}$.
These bosonic quantum numbers represent gluonic fluxes that are oriented in
either the ($\underline{1}\rightarrow 1$) direction or
($1\rightarrow\underline{1}$). The fermionic degrees of freedom are
characterized by $\nu_{\underline{1}},\nu_{0},\nu_{1}\in\\{0,1\\}$. Since
SU(3) is a rank-2 group, the irreducible representations (irreps) are
characterized by two indicies, denoted as $P$ and $Q$ (both taking positive
integer values). At a particular site, one can associate irreps to both sides
of the site such that $P(\underline{1},r),Q(\underline{1},r)$ denotes the
indicies to the $\underline{1}$ side of site r and $P(1,r),Q(1,r)$ denotes the
indices to the $1$ side of site r. The $P,Q$ integers are related to the SU(3)
LSH quantum numbers in the following manner:

$\displaystyle P(\underline{1},r)=n_{P}(r)+\nu_{0}(r)(1-\nu_{1}(r))$ (29)
$\displaystyle
Q(\underline{1},r)=n_{Q}(r)+\nu_{\underline{1}}(r)(1-\nu_{0}(r))$ (30)
$\displaystyle P(1,r)=n_{P}(r)+\nu_{1}(r)(1-\nu_{0}(r))$ (31) $\displaystyle
Q(1,r)=n_{Q}(r)+\nu_{0}(r)(1-\nu_{\underline{1}}(r))$ (32)

The operator constructions are such that they are locally SU(3) invariant and
as before, we have local Abelian Gauss laws. The AGL’s for the SU(3) group are
as follows (expanded form of (15,16)):

$\displaystyle
n_{P}+\nu_{1}(1-\nu_{0})\Big{|}_{r}=n_{P}+\nu_{0}(1-\nu_{1})\Big{|}_{r+1}$
(33) $\displaystyle
n_{Q}+\nu_{0}(1-\nu_{\underline{1}})\Big{|}_{r}=n_{Q}+\nu_{\underline{1}}(1-\nu_{0})\Big{|}_{r+1}$
(34)

The AGLs will be used to weave together the gauge-invariant basis states, as
was done for the SU(2) construction. The relevent operator dictionary is
provided below. The diagonal operators are defined as follows:

$\displaystyle\hat{n}_{P}|n_{P},n_{Q};\nu_{\underline{1}},\nu_{0},\nu_{1}\rangle$
$\displaystyle=n_{P}|n_{P},n_{Q};\nu_{\underline{1}},\nu_{0},\nu_{1}\rangle$
(35)
$\displaystyle\hat{n}_{Q}|n_{P},n_{Q};\nu_{\underline{1}},\nu_{0},\nu_{1}\rangle$
$\displaystyle=n_{Q}|n_{P},n_{Q};\nu_{\underline{1}},\nu_{0},\nu_{1}\rangle$
(36)
$\displaystyle\hat{\nu}_{\underline{1}}|n_{P},n_{Q};\nu_{\underline{1}},\nu_{0},\nu_{1}\rangle$
$\displaystyle=\nu_{\underline{1}}|n_{P},n_{Q};\nu_{\underline{1}},\nu_{0},\nu_{1}\rangle$
(37)
$\displaystyle\hat{\nu}_{0}|n_{P},n_{Q};\nu_{\underline{1}},\nu_{0},\nu_{1}\rangle$
$\displaystyle=\nu_{0}|n_{P},n_{Q};\nu_{\underline{1}},\nu_{0},\nu_{1}\rangle$
(38)
$\displaystyle\hat{\nu}_{1}|n_{P},n_{Q};\nu_{\underline{1}},\nu_{0},\nu_{1}\rangle$
$\displaystyle=\nu_{1}|n_{P},n_{Q};\nu_{\underline{1}},\nu_{0},\nu_{1}\rangle$
(39)

The Hamiltonian contains operators which are functions of the above set of
operators but their action on the basis states are trivially defined according
to the definitions above. The bosonic ladder operators can be defined as
follows:

$\displaystyle\hat{\Gamma}^{\pm}_{P}|n_{P},n_{Q};\nu_{\underline{1}},\nu_{0},\nu_{1}\rangle$
$\displaystyle=$ $\displaystyle|n_{P}\pm
1,n_{Q};\nu_{\underline{1}},\nu_{0},\nu_{1}\rangle$ (40)
$\displaystyle\hat{\Gamma}^{\pm}_{Q}|n_{P},n_{Q};\nu_{\underline{1}},\nu_{0},\nu_{1}\rangle$
$\displaystyle=$ $\displaystyle|n_{P},n_{Q}\pm
1;\nu_{\underline{1}},\nu_{0},\nu_{1}\rangle$ (41)

while the fermionic ones are defined as:

$\displaystyle\hat{\chi}_{\underline{1}}|n_{P},n_{Q};\nu_{\underline{1}},\nu_{0},\nu_{1}\rangle$
$\displaystyle=(1-\delta_{0,\nu_{\underline{1}}})\ket{n_{P},n_{Q};\nu_{\underline{1}}-1,\nu_{0},\nu_{1}}$
(42)
$\displaystyle\hat{\chi}^{\dagger}_{\underline{1}}|n_{P},n_{Q};\nu_{\underline{1}},\nu_{0},\nu_{1}\rangle$
$\displaystyle=(1-\delta_{1,\nu_{\underline{1}}})\ket{n_{P},n_{Q};\nu_{\underline{1}}+1,\nu_{0},\nu_{1}}$
(43)
$\displaystyle\hat{\chi}_{0}|n_{P},n_{Q};\nu_{\underline{1}},\nu_{0},\nu_{1}\rangle$
$\displaystyle=(1-\delta_{0,\nu_{0}})\ket{n_{P},n_{Q};\nu_{\underline{1}},\nu_{0}-1,\nu_{1}}$
(44)
$\displaystyle\hat{\chi}^{\dagger}_{0}|n_{P},n_{Q};\nu_{\underline{1}},\nu_{0},\nu_{1}\rangle$
$\displaystyle=(1-\delta_{1,\nu_{0}})\ket{n_{P},n_{Q};\nu_{\underline{1}},\nu_{0}-1,\nu_{1}}$
(45)
$\displaystyle\hat{\chi}_{1}|n_{P},n_{Q};\nu_{\underline{1}},\nu_{0},\nu_{1}\rangle$
$\displaystyle=(1-\delta_{0,\nu_{0}})\ket{n_{P},n_{Q};\nu_{\underline{1}},\nu_{0},\nu_{1}-1}$
(46)
$\displaystyle\hat{\chi}^{\dagger}_{1}|n_{P},n_{Q};\nu_{\underline{1}},\nu_{0},\nu_{1}\rangle$
$\displaystyle=(1-\delta_{1,\nu_{0}})\ket{n_{P},n_{Q};\nu_{\underline{1}},\nu_{0}+1,\nu_{1}}$
(47)

The fermionic operators $\hat{\chi}_{f}$, where $f\in\\{\underline{1},0,1\\}$,
satisfy the canonical anti-commutation relations:

$\displaystyle\anticommutator\Big{\hat{\chi}_{f}(r)}{\hat{\chi}_{f^{\prime}}(r^{\prime})}=\anticommutator{\hat{\chi}^{\dagger}_{f}(r)}{\hat{\chi}^{\dagger}_{f^{\prime}}(r^{\prime})}=0$
(48)
$\displaystyle\anticommutator\Big{\hat{\chi}_{f}(r)}{\hat{\chi}_{f^{\prime}}(r^{\prime})}=\delta_{f,f^{\prime}}\delta_{r,r^{\prime}}$
(49)

With the Hilbert space and the action of the relevent operators on it
explicitly stated, we now define the SU(3) LSH Hamiltonian as follows:

$\displaystyle H^{SU(3)}_{LSH}=H_{E}+H_{M}+H_{I}$ (50)

The electric and mass Hamiltonian are diagonal in the LSH basis and are
defined as

$\displaystyle H_{M}$
$\displaystyle=\mu\sum_{r}(-1)^{r}(\hat{\nu}_{\underline{1}}(r)+\hat{\nu}_{0}(r)+\hat{\nu}_{1}(r))$
(51) $\displaystyle H_{E}$
$\displaystyle=\sum_{r}\Bigg{[}\frac{1}{3}\Bigg{(}\hat{P}^{2}(1,r)+\hat{Q}^{2}(1,r)+\hat{P}(1,r)\hat{Q}(1,r)\Bigg{)}$
(52) $\displaystyle+\hat{P}(1,r)+\hat{Q}(1,r)\Bigg{]}$

Finally, the matter-gauge interaction Hamiltonian is of the following form:

$\displaystyle H_{I}$ $\displaystyle=$
$\displaystyle\sum_{r=0}^{N-1}\Bigg{[}\sum_{F=\underline{1},0,1}\chi_{F}^{\dagger}(r)\Big{[}\hat{O}_{1}^{(F)}(\\{\hat{n}_{l}(r),\hat{n}_{f}(r)\\})$
(53)
$\displaystyle\times\hat{O}_{2}^{(F)}(\\{\hat{n}_{l}(r+1),\hat{n}_{f}(r+1)\\})\Big{]}\chi_{F}(r+1)\Bigg{]}$
$\displaystyle+\mathrm{h.c.}$

where, the operators $O^{F}_{1,2}$ are defined as

$\displaystyle O^{(\underline{1})}_{1}$
$\displaystyle={\left[\left(\hat{\Gamma}_{Q}\right)^{1-\hat{\nu}_{0}}\sqrt{1+\hat{\nu}_{0}/\left(\hat{n}_{Q}+1\right)}\sqrt{1+\hat{\nu}_{1}/\left(\hat{n}_{P}+\hat{n}_{Q}+2\right)}\right]}$
(54) $\displaystyle O^{(\underline{1})}_{2}$
$\displaystyle=\left[\sqrt{1-\hat{\nu}_{0}/\left(\hat{n}_{Q}+2\right)}\sqrt{1-\hat{\nu}_{1}/\left(\hat{n}_{P}+\hat{n}_{Q}+3\right)}\left(\hat{\Gamma}_{Q}\right)^{\hat{\nu}_{0}}\right]$
(55)

$\displaystyle O^{(0)}_{1}$
$\displaystyle=\left.\left[\left(\hat{\Gamma}_{P}\right)^{1-\hat{\nu}_{1}}\left(\hat{\Gamma}_{Q}^{\dagger}\right)^{\hat{\nu}_{1}}\sqrt{1+\hat{\nu}_{1}/\left(\hat{n}_{P}+1\right.}\right)\sqrt{1-\hat{\nu}_{\underline{1}}/\left(\hat{n}_{Q}+2\right)}\right]$
(56) $\displaystyle O^{(0)}_{2}$
$\displaystyle=\left.\left[\sqrt{1-\hat{\nu}_{1}/\left(\hat{n}_{P}+2\right.}\right)\sqrt{1+\hat{\nu}_{\underline{1}}/\left(\hat{n}_{Q}+1\right)}\hat{\chi}_{0}\left(\hat{\Gamma}_{P}\right)^{\hat{\nu}_{1}}\left(\hat{\Gamma}_{Q}^{\dagger}\right)^{1-\hat{\nu}_{1}}\right]$
(57)

$\displaystyle O^{(1)}_{1}$
$\displaystyle=\left[\left(\hat{\Gamma}_{P}^{\dagger}\right)^{\hat{\nu}_{0}}\sqrt{1-\hat{\nu}_{0}/\left(\hat{n}_{P}+2\right)}\sqrt{1-\hat{\nu}_{\underline{1}}/\left(\hat{n}_{P}+\hat{n}_{Q}+3\right)}\right]$
(58) $\displaystyle O^{(1)}_{2}$
$\displaystyle=\left[\sqrt{1+\hat{\nu}_{0}/\left(\hat{n}_{P}+1\right)}\sqrt{1+\hat{\nu}_{\underline{1}}/\left(\hat{n}_{P}+\hat{n}_{Q}+2\right)}\left(\hat{\Gamma}_{P}^{\dagger}\right)^{1-\hat{\nu}_{0}}\right]$
(59)

##  Appendix B Results for different initial-states

We present the results for an initial state that belongs to a different global
symmetry sector. The initial state is chosen as (belonging to the global
symmetry sector $\mathcal{P},\mathcal{Q}=1,0$):

$\displaystyle\ket{\Psi}_{ini}$
$\displaystyle=\ket{0,0,0,0,0}_{0}\otimes\ket{0,0,0,0,0}_{1}\otimes\ket{0,0,0,1,1}_{2}\otimes\ket{0,0,1,0,1}_{3}$

The parameters used for the simulation are the same, with $x=x^{\prime}=1$ and
$\mu=1$ for time evolution with the Hamiltonian given in (25). FIG. 6
illustrates the temporally averaged values of the global symmetry operators
$\mathcal{P},\mathcal{Q}$ and the two Abelian Gauss laws across three links
for a 4-staggered site system. Taking the $\Lambda\rightarrow\infty$ limit
ensures that $\mathcal{P},\mathcal{Q}$ tends to the sector the initial state
belonged to along with the Abelian Gauss laws across the three links
converging to zero. The error bars are the temporally averaged operator
fluctuations $\mathcal{O}$ for each $\Lambda$ parameter run. We skip
presenting the results of the local protection scheme for other global
symmetry sectors, which is also valid. Starting with this particular initial
state and evolving with the AGL-protected Hamiltonian as given in (28)
confines the dynamics in the same super-selection sector.

Figure 6: Demonstration of symmetry protection using global scheme for (1,0)
sector.