# Reliability of lattice gauge theories in the thermodynamic limit

Maarten Van Damme  Department of Physics and Astronomy, University of Ghent,
Krijgslaan 281, 9000 Gent, Belgium  Haifeng Lang  INO-CNR BEC Center and
Department of Physics, University of Trento, Via Sommarive 14, I-38123 Trento,
Italy  Theoretical Chemistry, Institute of Physical Chemistry, Heidelberg
University, Im Neuenheimer Feld 229, 69120 Heidelberg, Germany  Philipp Hauke
INO-CNR BEC Center and Department of Physics, University of Trento, Via
Sommarive 14, I-38123 Trento, Italy  Jad C. Halimeh  INO-CNR BEC Center and
Department of Physics, University of Trento, Via Sommarive 14, I-38123 Trento,
Italy

(August 27, 2024)

###### Abstract

Although gauge invariance is a postulate in fundamental theories of nature
such as quantum electrodynamics, in quantum-simulation implementations of
gauge theories it is compromised by experimental imperfections. In a recent
work [Halimeh and Hauke, Phys. Rev. Lett. 125, 030503 (2020)], it has been
shown in finite-size spin-$1/2$ quantum link lattice gauge theories that upon
introducing an energy-penalty term of sufficiently large strength $V$, unitary
gauge-breaking errors at strength $\lambda$ are suppressed
$\propto\lambda^{2}/V^{2}$ up to all accessible evolution times. Here, we show
numerically that this result extends to quantum link models in the
thermodynamic limit and with larger spin-$S$. As we show analytically, the
dynamics at short times is described by an adjusted gauge theory up to a
timescale that is at earliest $\tau_{\text{adj}}\propto\sqrt{V/V_{0}^{3}}$,
with $V_{0}$ an energy factor. Moreover, our analytics predicts that a
renormalized gauge theory dominates at intermediate times up to a timescale
$\tau_{\text{ren}}\propto\exp(V/V_{0})/V_{0}$. In both emergent gauge
theories, $V$ is volume-independent and scales at worst $\sim S^{2}$.
Furthermore, we numerically demonstrate that robust gauge invariance is also
retained through a single-body gauge-protection term, which is experimentally
straightforward to implement in ultracold-atom setups and NISQ devices.

###### Contents

  1. I Introduction
  2. II Model
  3. III Quench dynamics
    1. III.1 Full protection
    2. III.2 Linear protection
  4. IV Summary of emergent gauge theories and timescales
    1. IV.1 Adjusted gauge theory
    2. IV.2 Renormalized gauge theory
  5. V Conclusion
  6. A Supporting numerical results
    1. A.1 Results for electric field
      1. A.1.1 Full protection
      2. A.1.2 Linear protection
    2. A.2 Temporally nonaveraged gauge violation
    3. A.3 Convergence with bond dimension in iMPS
  7. B Abanin-De Roeck-Ho-Huveneers (ARHH) method
  8. C Constrained quantum dynamics
  9. D Quantum Zeno effect

##  I Introduction

Recent years have witnessed impressive progress in the level of control and
precision achieved in synthetic quantum matter.Bloch  _et al._ (2008);
Lewenstein  _et al._ (2012); Blatt and Roos (2012); Hauke  _et al._ (2012) In
addition to allowing for the exploration of exotic phenomena such as many-body
localization,Schreiber  _et al._ (2015); Choi  _et al._ (2016); Smith  _et
al._ (2016) the Kibble-Zurek mechanism,Xu  _et al._ (2014); Anquez  _et al._
(2016); Clark  _et al._ (2016); Cui  _et al._ (2016); Keesling  _et al._
(2019) dynamical phase transitions,Jurcevic  _et al._ (2017); Zhang  _et al._
(2017); Fläschner  _et al._ (2018) prethermalization,Gring  _et al._ (2012);
Langen  _et al._ (2015); Neyenhuis  _et al._ (2017) and many-body
dephasing,Kaplan  _et al._ (2020) this technological advancement has
facilitated the realization of complex multi-species systems such as lattice
gauge theories. Martinez  _et al._ (2016); Bernien  _et al._ (2017); Dai  _et
al._ (2017); Klco  _et al._ (2018); Kokail  _et al._ (2019); Schweizer  _et
al._ (2019); Görg  _et al._ (2019); Mil  _et al._ (2020); Brower  _et al._
(2020); Klco  _et al._ (2020); Yang  _et al._ (2020) Not only can this ability
enable a possible foray into questions of high-energy physics in inexpensive
low-energy tabletop quantum simulators,Wiese (2013); Zohar  _et al._ (2015);
Dalmonte and Montangero (2016); Bañuls  _et al._ (2019) it also sets the
grounds for a standard experimental benchmark for the latter. Indeed, even
though in nature gauge invariance is a postulate, such as Gauss’s law in
quantum electrodynamics (QED), it is not guaranteed in generic ultracold-atom
implementations of lattice gauge theories (excepting for realizations that use
manifestly gauge-invariant mappings to reduce the number of degrees of
freedomZohar  _et al._ (2015); Muschik  _et al._ (2017); Surace  _et al._
(2020)). This potential detriment is due to experimental errors avoidable only
by way of unrealistic fine-tuning in the experimental parameters. Amidst the
current intense drive in both academia and industry to build reliable quantum
simulators, gauge-theory implementations therefore promise to serve as a
measure of experimental control and precision regarding how robustly gauge
invariance can be enforced.

Figure 1: (Color online). Energy protection in lattice gauge theories. The
initial gauge-invariant state is quenched by the “faulty” gauge theory
$H=H_{0}+\lambda H_{1}+H_{\text{pro}}$; the ideal gauge theory $H_{0}$ is the
spin-$S$ $\mathrm{U}(1)$ quantum link model given in Eq. (1), the gauge-
breaking term $\lambda H_{1}$ describes experimentally relevant local errors
given in Eq. (3), and $H_{\text{pro}}$ is an “energy penalty” used to suppress
gauge violations and implemented using either full protection
$H_{\text{pro}}=VH_{G}=V\sum_{j}G_{j}^{2}$ or linear protection
$H_{\text{pro}}=V\tilde{H}_{G}=V\sum_{j}c_{j}G_{j}$, with $c_{j}$ a properly
chosen sequence of normalized rational numbers. $G_{j}$ is the generator of
Gauss’s law at the local constraint comprised of matter site $j$ and its
adjacent links $(j-1,j)$ and $(j,j+1)$, as given in Eq. (2). Since $H_{0}$ is
gauge-invariant, it satisfies $[H_{0},G_{j}]=0,\forall j$. The initial state
is chosen in the physical sector $G_{j}\ket{\psi_{0}}=0,\forall j$, and such
that the matter fields are empty
$\sigma_{j}^{z}\ket{\psi_{0}}=-1\ket{\psi_{0}},\,\forall j$ and with staggered
polarization on the links
$s_{j,j+1}^{z}\ket{\psi_{0}}=(-1)^{j+1}S\ket{\psi_{0}}$. Here,
$\sigma_{j}^{z}$ is the Pauli matrix denoting matter occupation at site $j$
and $s_{j,j+1}^{z}$ is the spin-$S$ matrix depicting the electric field at
link $(j,j+1)$. We have checked that the conclusions of our work are
independent of the choice of initial state.

A convenient framework for the realization of lattice gauge theories is given
by quantum link modelsWiese (2013); Chandrasekharan and Wiese (1997) (QLMs).
In this framework, matter fields (such as electrons and positrons in QED) are
represented by fermionic degrees of freedom located at sites of a lattice,
while the gauge degree of freedom (such as the electric field) is represented
by spins of length $S$ located at the links connecting neighboring lattice
sites. One way of protecting gauge symmetry has been proposed in the form of
energy penalites.Zohar and Reznik (2011); Banerjee  _et al._ (2012); Zohar
_et al._ (2013); Hauke  _et al._ (2013); Kühn  _et al._ (2014); Kuno  _et al._
(2015); Dutta  _et al._ (2017); Kuno  _et al._ (2017); Dehkharghani  _et al._
(2017); Barros  _et al._ (2019); Halimeh and Hauke (2020); Lamm  _et al._
(2020); Halimeh  _et al._ (2020) Reliable gauge invariance in the dynamics of
a quantum link spin-$1/2$ lattice gauge theory $H_{0}$ in the presence of
gauge-breaking terms $\lambda H_{1}$ has been demonstrated numerically in
finite systems through the introduction of full and linear protection terms
$VH_{G}=V\sum_{j}G_{j}^{2}$ and $V\tilde{H}_{G}=V\sum_{j}c_{j}G_{j}$,
respectively. These energetically isolate the target gauge sector in Hilbert
space,Halimeh and Hauke (2020); Halimeh  _et al._ (2020) where $G_{j}$ is the
generator of Gauss’s law at lattice site $j$ and $c_{j}$ is a sequence of
appropriately chosen rational numbers (see Sec. III.2); cf. Fig. 1. The
resulting system has been shown to be analytically connected to the ideal
(error-free) theory for a sufficiently large protection- to error-strength
ratio $V/\lambda$ for all accessible times in exact diagonalization.Halimeh
and Hauke (2020) At long times, the gauge violation settles into a steady-
state value $\propto\lambda^{2}/V^{2}$ in this controlled-error regime.

Nevertheless, two interesting questions remain unsettled. How will the gauge
protection scheme work in the thermodynamic limit where energetic overlap
between different gauge sectors may become unavoidable? And how well does the
protection work at larger link spin length $S$? In this work, we show through
infinite matrix product state calculations (iMPS), which work directly in the
thermodynamic limit, that at sufficiently large volume-independent protection
strength $V$ the gauge violation remains reliably suppressed
$\propto\lambda^{2}/V^{2}$ up to all accessible evolution times for both full
and linear gauge protection. Further, we demonstrate analytically that $V$
scales at worst $\sim S^{2}$ under full protection $VH_{G}$. Even more,
specific realizations can perform decisively better than this rigorous bound.
For example, in the scenario considered in this work, the gauge violation is
suppressed further with $S$ for fixed values of $\lambda$ and $V$ under both
full and linear protection. Moreover, we analytically prove in the case of
full protection at sufficiently large volume-independent $V$ that an adjusted
version of the ideal gauge theory arises and persists until a timescale
$\tau_{\text{adj}}\propto\sqrt{V/V_{0}^{3}}$, where the energy scale $V_{0}$
is roughly given by a linear sum of $\\{\lambda,g^{2}aS^{2},\mu,J\\}$, which
means $V_{0}\sim S^{2}$ in the worst scenario. This complements earlier
findingsHalimeh  _et al._ (2020) for an adjusted gauge theory in the case of
linear protection up to a timescale $\tilde{\tau}_{\text{adj}}\propto
V/(V_{0}L)^{2}$.

In the subsequent stages, a renormalized gauge theory dominates lasting up to
a timescale $\tau_{\text{ren}}\propto\exp(V/V_{0})/V_{0}$ for both full and
linear protection, with the latter only in the case of a compliant sequence
(no such emergent theory exists in the case of a noncompliant sequence; see
Sec. III.2 and Ref. Halimeh  _et al._ , 2020). Only beyond this exponentially
large timescale, the energy protection can no longer guarantee reliable gauge
invariance. It is to be noted, however, that in the case of linear protection
with a compliant sequence, $V$ will have to be increased with system size to
maintain a given level of gauge fidelity, because the spacing in the compliant
sequence $c_{j}$ decreases with system size (see Sec. III.2).

The rest of the paper is organized as follows: We present our main model in
Sec. II, where the ideal gauge theory is represented by the one-dimensional
spin-$S$ $\mathrm{U}(1)$ quantum link model in the presence of experimentally
relevant gauge-breaking terms. In Sec. III, we present our numerical and
analytic results on quench dynamics of various observables while including
either full (Sec. III.1) or linear (Sec. III.2) gauge-protection terms to
suppress gauge violations. In Sec. IV, we summarize the different emergent
gauge theories and corresponding timescales. We conclude and discuss future
directions in Sec. V. We complement the main part of the paper with several
Appendices. Supporting numerical results are found in Appendix A. The
analytics of our paper are found in Appendix B for the Abanin-De Roeck-Ho-
Huveneers (ARHH) method,Abanin  _et al._ (2017) Appendix C for constrained
quantum dynamics,Gong  _et al._ (2020a, b) and Appendix D for the quantum Zeno
effect.Facchi and Pascazio (2002); Facchi  _et al._ (2004, 2009); Burgarth
_et al._ (2019)

##  II Model

To put our discussion on a formal footing, we consider the paradigmatic one-
dimensional spin-$S$ $\mathrm{U}(1)$ quantum link model (QLM) described by the
HamiltonianHauke  _et al._ (2013); Yang  _et al._ (2016); Kasper  _et al._
(2017)

$\displaystyle H_{0}=$
$\displaystyle\,\sum_{j=1}^{L}\bigg{[}\frac{-J}{2a\sqrt{S(S+1)}}\big{(}\sigma^{-}_{j}s^{+}_{j,j+1}\sigma^{-}_{j,j+1}+\text{H.c.}\big{)}$
$\displaystyle+\frac{\mu}{2}\sigma^{z}_{j}+\frac{g^{2}a}{2}\big{(}s^{z}_{j,j+1}\big{)}^{2}\bigg{]}\,.$
(1)

The Pauli matrices $\sigma_{j}$ represent the matter fields on matter site $j$
with rest mass $\mu$, where $\sigma^{z}_{j}$ denotes their occupation and
$\sigma^{\pm}_{j}$ are the creation and annihilation operators, respectively.
$L$ is the total number of matter sites. The spin-$S$ matrices
$s^{x,z}_{j,j+1}$ represent the gauge and electric fields, respectively, with
electric-field-ladder operators $s^{\pm}_{j,j+1}$, on the link $(j,j+1)$. The
lattice spacing and gauge coupling are denoted by $a$ and $g$, respectively.
The term $\propto J$, which couples the matter and gauge fields, describes the
creation or annihilation of an ‘electron-positron’ pair and the concomitant
flipping in the intermediate electric field to satisfy Gauss’s law, the
generator of which is

$\displaystyle
G_{j}=\frac{(-1)^{j}}{2}\Big{[}2\big{(}s^{z}_{j-1,j}+s^{z}_{j,j+1}\big{)}+\sigma^{z}_{j}+1\Big{]}.$
(2)

We consider the physical (or target) sector as that consisting of the states
$\ket{\psi}$ such that $G_{j}\ket{\psi}=0,\,\forall j$, where Gauss’s law is
satisfied. The $\mathrm{U}(1)$ QLM is gauge-invariant, i.e.,
$[H_{0},G_{j}]=0,\,\forall j$.

In a realistic experiment without infinite fine-tuning, however, there will be
unavoidable coherent errors that violate Gauss’s law. Even if just
perturbative, such errors are crucial to understand and, if possible, control,
in order to have a reliable gauge-theory implementation. An experimentally
relevant local error term is

$\displaystyle\lambda
H_{1}=\lambda\sum_{j=1}^{L}\bigg{[}\sigma^{-}_{j}\sigma^{-}_{j+1}+\sigma^{+}_{j}\sigma^{+}_{j+1}+\frac{2s^{x}_{j,j+1}}{\sqrt{S(S+1)}}\bigg{]},$
(3)

of strength $\lambda$, which describes unassisted creation or annihilation of
‘electron-positron’ pairs and unassisted electric-field flipping, either of
which will break gauge invariance and ultimately undermine a gauge-theory
realization.Mil  _et al._ (2020)

##  III Quench dynamics

Several proposals have been put forward to protect against gauge-breaking
errors such as those in Eq. (3),Zohar and Reznik (2011); Banerjee  _et al._
(2012); Zohar  _et al._ (2013); Hauke  _et al._ (2013); Kühn  _et al._ (2014);
Kuno  _et al._ (2015); Dutta  _et al._ (2017); Kuno  _et al._ (2017);
Dehkharghani  _et al._ (2017); Barros  _et al._ (2019); Halimeh and Hauke
(2020); Lamm  _et al._ (2020); Halimeh  _et al._ (2020) with one
straightforward method that has received a lot of interest being an energy
term that penalizes processes away from the physical sector
$G_{j}\ket{\psi}=0,\,\forall j$. In the following, we will study quench
dynamics in such a scenario. A system initially prepared in a gauge-invariant
state $\ket{\psi_{0}}$ in the target sector ($G_{j}\ket{\psi_{0}}=0,\forall
j$) is subsequently quenched by a “faulty” gauge-theory implementation

$\displaystyle H=H_{0}+\lambda H_{1}+H_{\text{pro}},$ (4)

where $H_{\text{pro}}$ is the gauge-protection term whose purpose is to
suppress gauge violations arising due to $\lambda H_{1}$. We will employ two
variants of this gauge-protection term: the full protectionHalimeh and Hauke
(2020) $H_{\rm pro}=VH_{G}=V\sum_{j}G_{j}^{2}$ (see Sec. III.1), and the
linear protectionHalimeh  _et al._ (2020) $H_{\rm
pro}=V\tilde{H}_{G}=V\sum_{j}c_{j}G_{j}$, where $c_{j}$ is a sequence of
rational numbers that we will discuss further in Sec. III.2. In the following,
we will provide our numerical results calculated in ED and iMPS. For the
latter, we find that our most demanding calculations achieve convergence at a
bond dimension $\mathcal{D}=200$ and a time-step of $\Delta t=0.005/J$. In our
ED implementations, we employ periodic boundary conditions, leading to a
system of size $2L$, with $L$ matter sites and $L$ corresponding links. We
note, however, that removing the periodic boundary conditions does not
qualitatively alter the ED results.Halimeh and Hauke (2020)

###  III.1 Full protection

A natural way to implement the energy penalty is in the form

$\displaystyle VH_{G}=V\sum_{j}G_{j}^{2},$ (5)

with $V$ the protection strength. We shall refer to this energy-penalty term
as full gauge protection.

It has been shown through exact diagonalization (ED) in finite systemsHalimeh
and Hauke (2020) that upon quenching a gauge-invariant initial state in the
physical sector with $H=H_{0}+\lambda H_{1}+VH_{G}$, the long-time gauge
violation falls in one of two regimes: an uncontrolled-error regime for
sufficiently large $\lambda/V$, where the ideal gauge-theory dynamics can no
longer be analytically retrieved, or a controlled-error regime for
sufficiently small $\lambda/V$ where the gauge violation at long times is
suppressed $\propto\lambda^{2}/V^{2}$, and the gauge-theory dynamics
perturbatively connects to that of the ideal case. The aim of this section is
to analyze the persistence of this protective power through numerical
calculations in the thermodynamic limit and through additional analytic
bounds.

Figure 2: (Color online). Gauge violation as given in Eq. (6) as function of
gauge-spin length $S$ [(a) $S=1/2$, (b) $S=1$, (c) $S=3/2$, and (d) $S=2$],
computed in the thermodynamic limit using iMPS. The initial state, outlined in
Fig. 1, is quenched with Hamiltonian $H_{0}+\lambda H_{1}+VH_{G}$, i.e., using
the “full” gauge protection given in Eq. (5). Errors, generated by the gauge-
breaking term $\lambda H_{1}$ of Eq. (3), which may be present in typical
ultracold-atom implementations of the $\mathrm{U}(1)$ quantum link model
$H_{0}$ given in Eq. (1), remain reliably suppressed
$\propto\lambda^{2}/V^{2}$ for the evolution times accessible in iMPS,
regardless of $S$. The plateau $\propto\lambda^{2}/V^{2}$ occurs at the
timescale $\propto 1/V$ at which the protection term begins to dominate. Here,
we have set $a=1/2$, $g=\sqrt{J}$, $\lambda=0.1J$, and $\mu=0.1J$, and scanned
over the protection-strength values $V/J=0,\,1,\,10,\,10^{2},\,10^{3}$,
although our conclusions are valid also for other values.

Using the iMPS technique based on the time-dependent variational principle
(TDVP),Haegeman  _et al._ (2011, 2016); Vanderstraeten  _et al._ (2019) we
calculate the quench dynamics under the Hamiltonian $H$ of Eq. (4) starting in
the initial state $\ket{\psi_{0}}$ shown in Fig. 1. For our results, we have
chosen $a=1/2$, $g=\sqrt{J}$, $\lambda=0.1J$, and $\mu=0.1J$, although we have
checked that our conclusions remain valid for different initial states and
parameter values.

The ensuing dynamics of the temporally averaged gauge violation

$\displaystyle\varepsilon(t)=\frac{1}{Lt}\int_{0}^{t}ds\,\sum_{j=1}^{L}\bra{\psi_{0}}e^{iHs}G_{j}^{2}e^{-iHs}\ket{\psi_{0}},$
(6)

is shown in Fig. 2 for various lengths $S$ of the link spin. The violation
grows $\propto\lambda^{2}t^{2}$ at short times, in agreement with time-
dependent perturbation theory (TDPT).Halimeh and Hauke (2020) Whereas in the
unprotected ($V=0$) case the gauge violation grows rapidly, at sufficiently
large $V$ it is suppressed $\propto\lambda^{2}/V^{2}$ at long times, settling
into a plateau beginning at a timescale $t\propto 1/V$ at sufficiently large
$V$. This is the timescale at which the protection term begins to dominate.
The gauge violation then remains at this plateau throughout all the accessible
evolution times in iMPS. We see that for fixed $\lambda$ and $V$, the gauge-
violation plateau actually slightly decreases with increasing $S$, although
this may be specific to our error term in Eq. (3). Indeed, the protection
strength $V$ can be shown analytically to scale $\sim S^{2}$ in the worst
case; cf. Appendices B and C. Nevertheless, the apparent small dependence of
$V$ on $S$ in the case of experimentally relevant local gauge-breaking errors
such as those of Eq. (3) is very encouraging for ongoing experiments seeking
to approach the Kogut-Susskind limit (lattice QED) by achieving larger link
spin lengths in gauge-theory implementations.Mil  _et al._ (2020)

Figure 3: (Color online). Finite-size behavior of the gauge violation at
$\lambda=0.1J$ and $V=10J$, for the link spin lengths (a) $S=1/2$ and (b)
$S=1$. The gauge violation for finite-size chains is larger than its
counterpart in the thermodynamic limit (see legends). As can be seen in the
insets, convergence relative to system size is very fast, with the gauge
violation for $L=4$ matter sites already approaching the results in the
thermodynamic limit $L\to\infty$. We have checked that this behavior persists
for other values of $\lambda$ and $V$. The dynamics in the thermodynamic limit
is calculated in iMPS, while for finite systems in ED.

To analyze the finite-size behavior of the gauge violation, we show in Fig.
3(a,b) for $S=1/2$ and $1$, respectively, at $\lambda=0.1J$ and $V=10J$, the
gauge-violation dynamics at several finite values of $L$, calculated in ED,
along with the thermodynamic limit $L\to\infty$, calculated in iMPS.
Convergence to the latter is very fast, with deviations from the iMPS results
becoming insignificant already with only $L=4$ matter sites (see insets). This
is remarkable especially from an experimental perspective since state-of-the-
art quantum-simulator implementations of lattice gauge theories are of the
order of a few dozen matter sites,Yang  _et al._ (2020) and our results
indicate that this already captures the thermodynamic limit in the behavior of
the gauge violation.

Our iMPS results show that the gauge violation in the thermodynamic limit will
still quickly settle into a plateau at sufficiently large $V/\lambda$, but
they do not tell us much about whether such a plateau will last indefinitely
given the inaccessibility of longer evolution times in iMPS. However, it has
been analytically proven that in the case of a controlled suppression of the
gauge violation in the presence of local gauge-breaking terms as those in Eq.
(3), full protection at a volume-independent strength $V$ will give rise to a
renormalized gauge theory, where the gauge violation is upper bounded
$\propto\lambda/V$ and which lasts up to a timescale
$\tau_{\text{ren}}\propto\exp(V/V_{0})/V_{0}$;Abanin  _et al._ (2017); Halimeh
_et al._ (2020)cf. Appendix B. As such, we cannot rule out that the gauge
violation will leave the plateau $\propto\lambda^{2}/V^{2}$ appearing in Fig.
2 and rise to some larger value with upper bound $\propto\lambda/V$ at longer
evolution times $t<\tau_{\text{ren}}$ that are not accessible to iMPS.
Nevertheless, as the iMPS data in Figs. 2 and 3 show, for times that are
accessible to current quantum-simulation experimentsGörg  _et al._ (2019);
Schweizer  _et al._ (2019); Mil  _et al._ (2020); Yang  _et al._ (2020) the
gauge violation in the thermodynamic limit remains suppressed
$\propto\lambda^{2}/V^{2}$, same as in the case of finite systems.Halimeh and
Hauke (2020) Indeed, for example for the experimentally feasibleHalimeh and
Hauke (2020) parameter values $\lambda=0.1J$ and $V=10J$, which lie in the
controlled-error regime, the largest evolution times achieved in iMPS are
$t_{\text{max}}\approx 10/J$. Setting $J\sim 20-30\,\mathrm{Hz}$, which is
within the typical range of values used in current ultracold-atom realizations
of lattice gauge theories,Yang  _et al._ (2020) leads to $t_{\text{max}}\sim
50-80\,\mathrm{ms}$. This is a typical lifetime of modern large-size lattice-
gauge-theory implementations in ultracold-atom setups,Yang  _et al._ (2020)
which means that our conclusions from iMPS can be readily checked in such
experiments.

Figure 4: (Color online). Deviation of particle density from ideal dynamics
due to gauge-breaking terms. Plotted is the temporally averaged particle
density for gauge-breaking strength $\lambda/J=0$ (ideal theory; red line),
and $\lambda/J=0.1$ with full protection at various strengths $V$ (lines in
shades of blue) for link spin lengths (a) $S=1/2$, (b) $S=1$, (c) $S=3/2$, and
(d) $S=2$. The deviation from the ideal theory is rapidly suppressed as $V$
increases, at least up to the evolution times accessible by iMPS. With
increasing $V$, the short-time scaling diminishes from $\propto t^{2}$ (the
theoretical upper bound) to a milder scaling $\propto t$, at a timescale
$\propto 1/V$ when the protection term dominates the dynamics.

The timescales over which the gauge symmetry can be protected are worthy of a
more detailed discussion. The ideal gauge theory $H_{0}$ is perturbed by the
gauge-breaking term $\lambda H_{1}$, against which we protect with a strong
$VH_{G}$. This scenario enables us to demonstrate the existence of an emergent
gauge theory that is perturbatively connected to $H_{0}$. In particular, the
frameworks of constrained quantum dynamicsGong  _et al._ (2020b, a) and the
Abanin-De Roeck-Ho-Huveneers (ARHH) methodAbanin  _et al._ (2017) prove to
yield useful insight.

First, let us define the adjusted gauge theory
$H_{\mathrm{adj}}=H_{0}+\lambda\mathcal{P}_{0}H_{1}\mathcal{P}_{0}$, which
preserves gauge invariance exactly and includes those terms of $\lambda H_{1}$
that act purely within the physical sector, denoted by the projector
$\mathcal{P}_{0}$. Now, after quenching a gauge-invariant initial state in the
target sector with $H=H_{0}+\lambda H_{1}+VH_{G}$, the large scale $V$
restricts the dynamics of local observables to the one generated by
$H_{\mathrm{adj}}$. Within the framework of constrained quantum dynamics, Gong
_et al._ (2020b, a) one can show the deviation for the local observable $O$,
to be bounded from above as

$\displaystyle\big{\lvert}\langle
e^{iHt}Oe^{-iHt}-e^{iH_{\text{adj}}t}Oe^{-iH_{\text{adj}}t}\rangle\big{\rvert}\leq\Delta_{\text{adj}},$
(7)

with $\Delta_{\text{adj}}\sim t^{2}V_{0}^{3}/V$. This polynomial error bound
is thus ensured up to a fractional timescale
$\tau_{\text{adj}}\propto\sqrt{V/V_{0}^{3}}$. An important feature of this
bound is its volume independence, providing an analytic proof complementary to
Ref. Halimeh  _et al._ , 2020 that energy protection of gauge symmetry can
work in the thermodynamic limit. The exact form of $\Delta_{\text{adj}}$ and
other derivational details are included in Appendix C.

This error bound also restricts the deviation of Gauss’s law from the ideal
value of $0$. As can be seen in Fig. 2, concrete physical scenarios can remain
considerably below this upper bound. In fact, we observe a timescale $\propto
1/V<\tau_{\text{adj}}$ at which the gauge violation plateaus at a constant
value $\propto\lambda^{2}/V^{2}$, the value that can also be derived in
degenerate perturbation theory.Halimeh and Hauke (2020) The timescale $\propto
1/V$ is when the protection term $H_{\text{pro}}$ starts to dominate, and
therefore the initial gauge violation growth $\propto\lambda^{2}t^{2}$ due to
the error term as calculated in TDPTHalimeh and Hauke (2020) is suppressed.
The gauge violation then remains at that value to times larger than
$\tau_{\text{adj}}$. This favorable behavior can be explained within the ARHH
framework.Abanin  _et al._ (2017) Namely, one can show the existence of a
renormalized gauge theory Halimeh  _et al._ (2020) that exists up to a
timescale $\tau_{\text{ren}}\propto\exp(V/V_{0})/V_{0}$. This framework bounds
the gauge violation within $\lambda/V$, as derived in Ref. Halimeh  _et al._ ,
2020 and also summarized in Appendix B.

Figure 5: (Color online). Finite-size behavior of the particle density under
full gauge protection. Parameters are $\lambda=0.1J$, $V=10J$, and quantum
link spin length (a) $S=1/2$ and (b) $S=1$. As in the case of the gauge
violation in Fig. 3, the result quickly approaches its value in the
thermodynamic limit already at small system sizes. Our results hold for other
values of the microscopic parameters. The dynamics in the thermodynamic limit
is calculated in iMPS, while for finite systems in ED.

As these discussions illustrate, there exist powerful frameworks to derive
analytic bounds for the gauge violation, which realistic situations can even
significantly undershoot.

It is instructive to look at other local observables where the timescale
$\propto 1/V$ does not prominently appear as it does in the gauge violation,
and where we can therefore more clearly see the features of the adjusted gauge
theory $H_{\text{adj}}=H_{0}+\lambda\mathcal{P}_{0}H_{1}\mathcal{P}_{0}$.
Indeed, even though at times longer than the timescale $\propto 1/V$ the
protection term dominates, its effect in general local observables may not be
as apparent as in the plateauing behavior in the gauge violation of Fig. 2.
This in turn would allow us to better estimate how well the dynamics of the
faulty theory $H$ reproduces the one of the adjusted gauge theory. The
particular gauge-breaking error described by Eq. (3) is characterized by
$\mathcal{P}_{0}H_{1}\mathcal{P}_{0}=0$, and so the adjusted gauge theory is
$H_{0}$ itself. As such, we expect local observables after the quench with the
faulty theory $H$ to follow the ideal-theory dynamics for sufficiently large
$V$, up to an error that according to Eq. (7) is at most $\propto
t^{2}V_{0}^{3}/V$. To check this expectation, we calculate in iMPS the
dynamics of the temporally averaged particle density

$\displaystyle n$
$\displaystyle=\frac{1}{2}+\frac{1}{2Lt}\int_{0}^{t}ds\,\sum_{j=1}^{L}\bra{\psi_{0}}e^{iHs}\sigma_{j}^{z}e^{-iHs}\ket{\psi_{0}},$
(8)

which we present in the main text, in addition to the temporally averaged
absolute value of the normalized (by spin length $S$) electric field

$\displaystyle E$
$\displaystyle=\frac{1}{SLt}\int_{0}^{t}ds\,\Big{\lvert}\sum_{j=1}^{L}(-1)^{j}\bra{\psi_{0}}e^{iHs}s_{j}^{z}e^{-iHs}\ket{\psi_{0}}\Big{\rvert},$
(9)

which we relegate to Appendix A.1. The same conclusions are drawn from both
these local observables.

Figure 4 shows the time evolution of the temporally averaged particle density
at gauge-breaking strength $\lambda=0.1J$ and various protection strengths $V$
for several values of the link spin length $S$, and compares the result to the
corresponding ideal gauge-theory dynamics propagated by only $H_{0}$. Once in
the controlled-error regime ($V\gtrsim 10J$), the particle density agrees well
with that of the ideal theory ($\lambda=V=0$; thick red line) up to an error
upper bound $\propto t^{2}V_{0}^{3}/V$ at short times, in support of our
analytic predictions. Indeed, as seen in the insets, the deviation
$\lvert\Delta n\rvert$ between the particle density due to a quench by Eq. (4)
and that due to the ideal theory $H_{0}$ is suppressed at later times as $1/V$
while growing slower than $t^{2}$, i.e., substantially more benign than our
analytically derived error upper bound. In priniple, this comparison can also
be carried out with the renormalized gauge theory as a reference for the error
upper bound. However, the form of the Hamiltonian for the renormalized gauge
theory is in general not known, rendering such a comparison unfeasible.

The finite-size behavior of the particle density is shown in Fig. 5 for two
values of the link spin length $S=1/2$ and $1$, and the quench parameters
$\lambda=0.1J$ and $V=10J$, although we have checked that our conclusions hold
for other values of the microscopic parameters. Similar to the case of the
gauge violation in Fig. 3, the finite-size behavior of the particle density
calculated in ED approaches already at a few matter sites the dynamics in the
thermodynamic limit within the evolution times accessible in iMPS. This is
particularly encouraging news from an experimental perspective, because it
shows that modern ultracold-atom implementations of lattice gauge theories,
which now reach a few dozens of matter sites,Yang  _et al._ (2020) should be
able to faithfully achieve the thermodynamic behavior of gauge-theory dynamics
at least up to timescales $\propto\sqrt{V/V_{0}^{3}}$.

Finally, it is worth mentioning that the timescale $\propto 1/V$, though not
evident in the dynamics of the particle density itself, does actually appear
in the deviation of the particle density from its dynamics under the ideal
theory, as can be seen in the insets of Fig. 4.

Figure 6: (Color online). Same as Fig. 2 but with the energy-protection term
given by the single-body Hamiltonian $V\tilde{H}_{G}=V\sum_{j}c_{j}G_{j}$ of
Eq. (10) with $c_{j}=(-1)^{j+1}$ rather than the full protection Hamiltonian
$H_{G}$ of Eq. (5). Even with a noncompliant sequence, the linear protection
works remarkably well, and yields results qualitatively identical to those
under full protection for the evolution times accessible in iMPS, including
the gauge-violation plateau $\propto\lambda^{2}/V^{2}$ that occurs at the
timescale $\propto 1/V$.

Figure 7: (Color online). Same as Fig. 3 but with the energy-protection term
given by the single-body Hamiltonian $\tilde{H}_{G}$ of Eq. (10) with
$c_{j}=(-1)^{j+1}$ rather than the ‘full’ protection Hamiltonian $H_{G}$ of
Eq. (5). Also in this case of linear protection with a noncompliant sequence,
convergence to the thermodynamic limit is very fast over the evolution times
accessible to iMPS.

###  III.2 Linear protection

Recently, it has also been shown that reliable gauge protection can be
achieved using single-body energy termsHalimeh  _et al._ (2020) of the form

$\displaystyle V\tilde{H}_{G}=V\sum_{j}c_{j}G_{j},$ (10)

with properly chosen normalized rational compliant coefficients
$c_{j}\in[-1,1]$ that satisfy the condition $\sum_{j}c_{j}g_{j}=0$ if and only
if $g_{j}=0,\,\forall j$. Here, the ‘local charges’ $g_{j}$ are the
eigenvalues of the generators $G_{j}$ of Gauss’s law.

The analysis from the constrained quantum dynamics discussed in Appendix C
does not apply here, since it requires that the protected target space be the
ground state of $H_{\rm pro}$, which is not satisfied for the linear
protection. In contrast, the analysis based on the quantum Zeno effectFacchi
and Pascazio (2002); Facchi  _et al._ (2004, 2009); Burgarth  _et al._ (2019)
as discussed in Appendix D can be adapted to this scenario. As derived within
this formalism, linear protection with a compliant or even noncompliant
sequence leads also to an adjusted gauge theory
$H_{0}+\lambda\mathcal{P}_{0}H_{1}\mathcal{P}_{0}$ for local observables up to
a system size-dependent error

$\displaystyle\big{\lvert}\langle
e^{iHt}Oe^{-iHt}-e^{iH_{\text{adj}}t}Oe^{-iH_{\text{adj}}t}\rangle\big{\rvert}\leq\tilde{\Delta}_{\text{adj}},$
(11)

with $\tilde{\Delta}_{\text{adj}}\sim tV_{0}^{2}L^{2}/V$. The specific form of
$\tilde{\Delta}_{\text{adj}}$ is given in Appendix D. This bound is therefore
ensured up to the timescale $\tilde{\tau}_{\text{adj}}\propto V/(V_{0}L)^{2}$.
A renormalized gauge theory dominates at later times, but only for a compliant
sequence, and persists up to an exponential timescale
$\tau_{\text{ren}}\propto\exp(V/V_{0})/V_{0}$, which is similar to its
counterpart in the case of full protection; cf. Appendix B. Unlike
$\tilde{\tau}_{\text{adj}}$, the timescale $\tau_{\text{ren}}$ does not
explicitly depend on the system size $L$, but the reliability of dynamics
nevertheless does. As mentioned above, the compliant sequence $c_{j}$ is a
normalized set of rational numbers, such that the largest $|c_{j}|$ is unity:
As $L$ increases, this means that the spacing between the different $c_{j}$
will get smaller on average. For a fixed value of $\lambda$, this in turn
implies that $V$ must be increased to achieve the same level of provable
reliability in the gauge invariance up to a given evolution time.

The compliance of $c_{j}$ is a sufficient albeit not a necessary condition
when the gauge-breaking errors are local. Indeed, in Ref. Halimeh  _et al._ ,
2020 it is demonstrated numerically for finite systems that for specific error
terms even noncompliant sequences such as $c_{j}=(-1)^{j+1}$ can still
suppress the gauge violation $\propto\lambda^{2}/V^{2}$ up to indefinite
times. Furthermore, such a noncompliant sequence can be shownFacchi and
Pascazio (2002); Facchi  _et al._ (2004, 2009); Burgarth  _et al._ (2019);
Halimeh  _et al._ (2020) to still give rise to at least the adjusted gauge
theory $H_{0}+\lambda\mathcal{P}_{0}H_{1}\mathcal{P}_{0}$ up to the timescale
$\tilde{\tau}_{\text{adj}}\propto V/(V_{0}L)^{2}$, just like its compliant
counterpart, albeit no analytic proof of a renormalized gauge theory up to a
timescale exponential in $V$ exists.

A strong and essential feature of iMPS is translation invariance, and as such
a compliant sequence is not optimal in an iMPS implementation as it breaks
this symmetry. In contrast, a sequence such as $c_{j}=(-1)^{j+1}$ preserves
translation invariance down to a unit cell of two adjacent matter sites. Using
this feature, we now quench the target-sector initial state $\ket{\psi_{0}}$
of Fig. 1 by the Hamiltonian $H=H_{0}+\lambda H_{1}+V\tilde{H}_{G}$, with
$\tilde{H}_{G}=\sum_{j}(-1)^{j+1}G_{j}$. The resulting dynamics of the gauge
violation is plotted in Fig. 6 for various values of the link spin length $S$.
Qualitatively, the results are identical to those of Fig. 2. The gauge
violation initially grows $\propto\lambda^{2}t^{2}$, in agreement with TDPT,
before plateauing at a timescale $\propto 1/V$ upon which the protection term
dominates. The plateau value at sufficiently large $V$ is
$\propto\lambda^{2}/V^{2}$, as expected from degenerate perturbation
theory.Halimeh and Hauke (2020) As in the case of full protection, linear
protection affords reliable gauge invariance at any link spin length $S$
within the accessible evolution times in iMPS, even with the noncompliant
sequence $c_{j}=(-1)^{j+1}$. This is remarkable as it shows that even in the
thermodynamic limit there is an experimentally feasible single-body protection
scheme with the simple coefficients $\pm 1$ that allows for well-controlled
gauge-theory dynamics at least up to times that are relevant for current
experiments. It is interesting to note that we do not find a deterioration of
the reliability of gauge invariance when increasing the link spin length $S$.
On the contrary, with our particular experimentally relevant error given in
Eq. (3), gauge invariance seems to become even more reliable with larger $S$
for a fixed value of $\lambda$ and $V$.

Figure 8: (Color online). Same as Fig. 4 but using the linear-protection term
with a noncompliant sequence, $V\tilde{H}_{G}=V\sum_{j}(-1)^{j+1}G_{j}$,
rather than full protection. The dynamics under the faulty gauge theory
$H=H_{0}+\lambda H_{1}+V\tilde{H}_{G}$ reliably approaches the ideal-theory
dynamics under $H_{0}$ (which is also the adjusted gauge theory in the case of
our gauge-breaking term) for $V\gtrsim 10$. The insets show the deviation
$\lvert\Delta n\rvert$ of the faulty dynamics from its ideal counterpart,
where the error initially grows at $\propto t^{2}$, but then slower ($\propto
t$) at later times, with this transition occurring at a timescale $\propto
1/V$ at which the protection term begins to dominate. This is milder than the
upper error bound with respect to the adjusted gauge theory (see main text and
Appendix D).

Figure 9: (Color online). Same as Fig. 5, but where gauge invariance is
supported through the linear-protection term with a noncompliant sequence,
$V\tilde{H}_{G}=V\sum_{j}(-1)^{j+1}G_{j}$. Also in this case, convergence to
the thermodynamic limit occurs at just a few matter sites during the evolution
times accessible in iMPS.

We now analyze the finite-size behavior of the gauge violation in the case of
the linear protection term $V\tilde{H}_{G}=V\sum_{j}(-1)^{j+1}G_{j}$, as shown
in Fig. 7(a,b) for link spin lengths $S=1/2$ and $1$, respectively. The same
qualitative conclusions from the full-protection case (see Fig. 5) hold in
that the dynamics of finite systems calculated through ED quickly converge to
the dynamics in the thermodynamic limit, which is calculated in iMPS. Indeed,
for both spin lengths considered, already the result for $L=4$ matter sites
almost coincide with that of the thermodynamic limit. This behavior is
intriguing because in the case of linear protection the system-size dependence
of the timescale $\tilde{\tau}_{\text{adj}}\propto V/(JL)^{2}$ for the
adjusted gauge theory may lead one to expect a slow approach to the
thermodynamic limit $L\to\infty$. Our numerical results indicate that, quite
to the contrary, this approach is as fast as in the case of full protection,
at least up to the accessible times in iMPS and for the local error presented
in the article.

It is instructive to also study the performance of linear protection with a
noncompliant sequence for local observables such as the particle density. The
results are shown in Fig. 8 for various values of the link spin length $S$,
where we compare the time evolution of the particle density upon quenching
$\ket{\psi_{0}}$ with $H=H_{0}+\lambda H_{1}+V\tilde{H}_{G}$ to that after
quenching $\ket{\psi_{0}}$ with the adjusted gauge theory
$H_{\text{adj}}=H_{0}+\lambda\mathcal{P}_{0}H_{1}\mathcal{P}_{0}=H_{0}$. As in
the case of full protection, the protected dynamics quickly approaches its
ideal-theory counterpart with larger $V$ for all values of $S$ considered. In
the insets, we also see an initial growth $\propto t^{2}$ in the deviation
$\lvert\Delta n\rvert$ that transitions to a growth $\propto t$ at later times
for a fixed $V$. The time of this transition decreases with $V$ just as in the
case of full protection. Actually, it seems that this transition timescale
$\propto 1/V$, upon which the protection term begins to dominate. However, in
contrast to the case of full protection, we do not find a suppression by $V$
in the deviation. Indeed, the adjusted gauge theory is valid up to an error
upper bound $\propto tV_{0}^{2}L^{2}/V$, which is $L$-dependent, unlike in the
case of full protection. How precisely the error grows within this bound may
therefore be influenced by the system size in different ways depending on the
value of $V$ and on the observable considered. In the case of full protection
(see Sec. III.1 and Appendix B), where the error upper bound $\propto
t^{2}V_{0}^{3}/V$ is fully independent of system size, $L$ does not seem to
play a role in the deviation growth at early times. Nevertheless, it is
remarkable that up to even intermediate evolution times the dynamics under the
faulty gauge theory $H$ is still in excellent agreement with that of the
adjusted gauge theory, as shown in Fig. 8. This behavior far exceeds the
expectations laid out in our analytics, and suggests that experimentally
relevant local errors are quite reliably controlled even under linear
protection with noncompliant sequences.

Finally, we study the finite-size behavior of the particle density in Fig.
9(a,b) for link spin lengths $S=1/2$ and $1$, respectively. Qualitatively
similar to the case of full protection, the dynamics calculated in ED for
finite $L$ quickly approach the dynamics in the thermodynamic limit
$L\to\infty$ calculated in iMPS. So not only does linear protection with a
simple noncompliant sequence $c_{j}=(-1)^{j+1}$ provide reliable gauge
invariance up to experimentally relevant times, it also faithfully reproduces
observable dynamics even in the thermodynamic limit. This is despite the fact
that our analytic derivations of a volume-dependent timescale
$\tilde{\tau}_{\text{adj}}\propto V/(V_{0}L)^{2}$ in the case of linear
protection predict that agreement with the adjusted gauge theory ($H_{0}$ in
our case) may not happen in the thermodynamic limit in a worst-case scenario.
Once again, this affirms the suggestion already extracted from Fig. 8 that
local errors such as those of Eq. (3) are faithfully controlled by linear
protection even with a simple sequence $c_{j}=(-1)^{j+1}$. In Appendix A.1.2,
we provide similar results for the electric field, which support the above
conclusions.

For nonlocal error terms the noncompliant sequence is in general not
sufficient.Halimeh  _et al._ (2020) However, typical experimental errors have
a large degree of locality, so we expect to obtain similarly good performance
in most relevant situations, potentially by slightly increasing the unit cell
of the noncompliant sequence.

##  IV Summary of emergent gauge theories and timescales

We summarize here all the timescales and emergent gauge theories discussed in
Sec. III, and their dependence on the kind of protection used (i.e., full or
linear).

###  IV.1 Adjusted gauge theory

Independent of whether full protection $VH_{G}=V\sum_{j}G_{j}^{2}$ or linear
protection $V\tilde{H}_{G}=V\sum_{j}c_{j}G_{j}$ (with a compliant or
noncompliant sequence $c_{j}$) is used to protect against gauge-breaking
errors $H_{1}$, an adjusted gauge theory
$H_{\text{adj}}=H_{0}+\lambda\mathcal{P}_{0}H_{1}\mathcal{P}_{0}$ emerges. It
is valid up to a given timescale, during which the error in the dynamics of a
local observable is controllably bounded from above with respect to the
adjusted gauge theory. In the case of full protection, this upper bound is
$\propto t^{2}V_{0}^{3}/V$, leading to the timescale of
$\tau_{\text{adj}}\propto\sqrt{V/V_{0}^{3}}$. Derivational details are
provided in Appendix C. In the case of linear protection, the upper bound is
$\propto tV_{0}^{2}L^{2}V$, yielding the timescale
$\tilde{\tau}_{\text{adj}}\propto V/(V_{0}L)^{2}$. Derivational details are
provided in Appendix D. In both cases, the energy term $V_{0}$ is roughly a
linear sum of $\\{\lambda,g^{2}aS^{2},\mu,J\\}$, and its exact form is given
in Eq. (14) in Appendix B.

As such, we find that the dynamics is related to an adjusted gauge theory up
to a fractional (in $V$) timescale that is volume-independent in the case of
full protection, but which depends on system size in the case of linear
protection. However, our numerical results in Sec. III.2 indicate that system
size can play a trivial role in the case of linear protection even when the
sequence $c_{j}$ is noncompliant. This is evidenced by the excellent agreement
of the dynamics of the particle density under the faulty theory
$H=H_{0}+\lambda H_{1}+V\sum_{j}(-1)^{j+1}G_{j}$ and that under the adjusted
theory in the thermodynamic limit, as shown in Fig. 8. Furthermore, the
finite-size behavior of the particle density in Fig. 9 also indicates that
convergence to the thermodynamic limit is fast in this case (results for the
electric field in Appendix A.1.2 yield the same conclusion). We can therefore
conclude that in the case of experimentally relevant local gauge-breaking
terms as those of Eq. (3), even linear protection with a simple noncompliant
sequence $c_{j}=(-1)^{j+1}$ can reliably reproduce the adjusted gauge theory
dynamics in the thermodynamic limit.

###  IV.2 Renormalized gauge theory

In the case of full protection or linear protection with a compliant sequence,
a renormalized gauge theory emerges lasting up to an exponential timescale
$\tau_{\text{ren}}\propto\exp(V/V_{0})/V_{0}$. The derivation of this
renormalized gauge theory is based on the ARHH framework,Abanin  _et al._
(2017) and has recently been adapted to gauge theories in Ref. Halimeh  _et
al._ , 2020. Even though $\tau_{\text{ren}}$ is not explicitly volume-
dependent, a given level of gauge invariance at a fixed error strength
$\lambda$ will require $V$ to increase with system size in the case of linear
protection with a compliant sequence but not in the case of full protection.
Derivational details of the renormalized gauge theory is presented in Appendix
B. The same degree of rigorously provable reliability cannot be achieved for
linear protection with noncompliant sequences, as in that case gauge breakings
to high order could become resonant and thus spoil the gauge symmetry at
fractionally large times (although our numerical results suggest a very
favorable behavior even in this case).

##  V Conclusion

We have presented infinite matrix product state calculations for quench
dynamics in the paradigmatic spin-$S$ $\mathrm{U}(1)$ quantum link model in
the presence of experimentally relevant local gauge-breaking errors, against
which we protect with either full or linear protection terms. We have found
that reliable gauge invariance is achieved in the thermodynamic limit for all
accessible evolution times.

In the case of full protection, our numerical results support our analytic
predictions that quench dynamics under a faulty gauge theory is reproduced by
an adjusted gauge theory. Analytic calculations demonstrate this adjusted
gauge theory to hold up to a volume-independent error bound that grows
proportional to the inverse of the protection strength. Remarkably, our
numerical results show that for the local gauge-breaking terms considered the
actual deviation of local observables grows well below this bound. This allows
the dynamics under the faulty theory at sufficiently large protection strength
to agree very well with that under the adjusted gauge theory for all times
accessible to our simulations. Exact diagonalization results at just a few
matter sites converge to those in the thermodynamic limit calculated through
infinite matrix product state techniques, indicating a wide-reaching volume-
independence of full protection.

In the case of linear protection with a noncompliant sequence, an adjusted
gauge theory also emerges, though analytics predicts an error bound that is
inversely proportional to the protection strength and grows as the square of
system size. Nevertheless, our numerical results indicate a much milder
behavior even in the thermodynamic limit, with the dynamics under the faulty
theory at sufficiently large protection strength agreeing very well with that
of the adjusted gauge theory up to all accessible evolution times. In fact, we
see that linear protection with a simple alternating noncompliant sequence
$\pm 1$ fares qualitatively just as well as full protection. Additionally, we
have also shown through exact diagonalization that the dynamics converges
quickly to the thermodynamic limit, even at just a few matter sites,
regardless of the link spin length. This bodes well for experimental efforts
attempting to realize the QED limit of lattice gauge theories on intermediate-
scale quantum devices, because our findings support the conclusion that for
experimentally relevant local gauge-breaking errors, reliable gauge invariance
can be achieved in the thermodynamic limit for large link spin length
throughout evolution times typical of modern setups. Even more, depending on
the error form, this can be possible with linear-protection sequences as
simple as a local chemical potential that alternates as $\pm 1$ on odd and
even matter sites.

It is worth mentioning that in this work we have only considered a $(1+1)-$D
Abelian gauge theory. It would be interesting to extend these studies to non-
Abelian gauge theories where local generators do not commute. Nevertheless,
investigations on $(1+1)-$D Abelian gauge theories are highly relevant for
ongoing experiments.Görg  _et al._ (2019); Schweizer  _et al._ (2019); Mil
_et al._ (2020); Yang  _et al._ (2020) Moreover, the conclusions of our work,
namely that gauge invariance in the quench dynamics of a faulty gauge theory
can be reliably achieved in the thermodynamic limit through energy protection,
complement the findings of Ref. Damme  _et al._ , 2020 in equilibrium for the
same $(1+1)-$D Abelian gauge theory (see also Ref. Borla  _et al._ , 2021 for
similar conclusion in equilibrium for a gauged Kitaev chain). There, full
protection of sufficiently large strength produces a renormalized gauge
theory. This behavior is possible because the full protection term lends mass
to the Higgs boson with which the gauge-breaking errors couple.Poppitz and
Shang (2008); Kuno  _et al._ (2015); Bazavov  _et al._ (2015) Another current
frontier for gauge quantum simulations is presented by higher
dimensions.Paulson  _et al._ (2020); Ott  _et al._ (2020) The resilience and
protection against gauge breaking errors in quantum link models and other
lattice gauge theories in $(2+1)-$D and $(3+1)-$D, where a massless photon
exists, is thus another interesting avenue for the future studies.

## Acknowledgments

MVD and HL contributed equally to this work. JCH is grateful to Fred
Jendrzejewski, Guoxian Su, and Bing Yang for discussions related to
experimental aspects of this work. We acknowledge support by Provincia
Autonoma di Trento, the ERC Starting Grant StrEnQTh (Project-ID 804305), Q@TN
— Quantum Science and Technology in Trento, ERC grants No 715861 (ERQUAF) and
647905 (QUTE), and from Research Foundation Flanders (FWO) via grant GOE1520N.

##  Appendix A Supporting numerical results

Figure 10: (Color online). Deviation of electric field from ideal dynamics due
to gauge-breaking terms. Plotted is the temporally averaged absolute value of
the electric field (normalized by spin length $S$) for gauge-breaking strength
$\lambda/J=0$ (ideal dynamics; red line) and $\lambda/J=0.1$ with full
protection $VH_{G}=V\sum_{j}G_{j}^{2}$ at various strengths $V$ (faulty gauge
theory; shades of blue) for link spin lengths (a) $S=1/2$, (b) $S=1$, (c)
$S=3/2$, and (d) $S=2$. The deviation from the ideal theory is rapidly
suppressed as $V$ increases, at least up to the evolution times accessible by
iMPS. With increasing $V$, the short-time scaling diminishes from $\propto
t^{2}$ (the theoretical upper bound) to a milder scaling $\propto t$ at a
timescale $\propto 1/V$, after which the protection term begins to dominate.

Figure 11: (Color online). Finite-size behavior of the electric field under
full gauge protection. Parameters are $\lambda=0.1J$, $V=10J$, and quantum
link spin length (a) $S=1/2$ and (b) $S=1$. As in the case of the gauge
violation in Fig. 3 and particle density in Fig. 5, the result quickly
approaches its value in the thermodynamic limit already at a few matter sites.

###  A.1 Results for electric field

In the main text, we have provided numerical results in both ED and iMPS for
the gauge violation and particle density in the case of both full and linear
protection. Here, we supplement these results by showing their counterparts
for the electric field normalized by the link spin length $S$, as given in Eq.
(9). Conclusions are qualitatively the same as those reported in the main
text.

####  A.1.1 Full protection

The temporally averaged dynamics of the electric field under the faulty theory
$H=H_{0}+\lambda H_{1}+VH_{G}$ is shown in Fig. 10, at a fixed gauge-breaking
error strength $\lambda$ and for various values of the link spin length $S$
and the protection strength $V$. As in the case of the particle density, see
Eq. (8), the dynamics agrees very well with that under the adjusted gauge
theory
$H_{\text{adj}}=H_{0}+\lambda\mathcal{P}_{0}H_{1}\mathcal{P}_{0}=H_{0}$. As
the insets in Fig. 10 show, the error grows within the analytically derived
upper bound of $t^{2}V_{0}^{3}/V$ (details of the derivation can be found in
Appendix C) at sufficiently large $V$ and for all link spin lengths $S$
considered.

The finite-size behavior of the electric field, displayed in Fig. 11(a,b) for
$S=1/2$ and $1$, respectively, shows quick convergence to the thermodynamic
limit already at a few matter sites, similarly to the cases of the gauge
violation (Fig. 3) and particle density (Fig. 5).

####  A.1.2 Linear protection

Similarly to its counterpart for the particle density (Fig. 8), the dynamics
of the electric field in the case of linear protection with a noncompliant
sequence, $V\tilde{H}_{G}=V\sum_{j}(-1)^{j+1}G_{j}$, shows great agreement
with that under the adjusted gauge theory $H_{\text{adj}}=H_{0}$ for all
accessible times; see Fig. 12. This is particularly surprising because the
error upper bound for the adjusted gauge theory in the case of linear
protection is $\propto tV_{0}^{2}L^{2}/V$, which becomes infinite in the
thermodynamic limit. Nevertheless, our iMPS results indicate that this upper
bound is extremely conservative, at least for the scenario considered here.
Furthermore, the finite-size behavior of the electric field in this case also
shows fast convergence to the thermodynamic limit for all accessible evolution
times in iMPS, as exhibited in Fig. 13. This is especially encouraging for
ongoing experimental efforts to realize reliable lattice gauge theories,
because a single-body protection term such as
$V\tilde{H}_{G}=V\sum_{j}(-1)^{j+1}G_{j}$ is straightforward to engineer,
e.g., through an optical superlattice in cold-atom experiments.Yang  _et al._
(2020)

Figure 12: (Color online). Same as Fig. 10 but using a linear protection term
$V\tilde{H}_{G}=V\sum_{j}c_{j}G_{j}$ with the noncompliant sequence
$c_{j}=(-1)^{j+1}$. Same as in the case of the particle density in Fig. 8 of
the main text, the dynamics of the electric field under the faulty gauge
theory $H=H_{0}+\lambda H_{1}+V\tilde{H}_{G}$ reliably approaches the ideal-
theory dynamics under $H_{0}$ (which is also the adjusted gauge theory) for
$V\gtrsim 10$. The insets show the deviation $\lvert\Delta E\rvert$ of the
faulty dynamics from its ideal counterpart, where the error initially grows at
$\propto t^{2}$, but then slower ($\propto t$) at later times, with this
transition occurring at the timescale $\propto 1/V$ as then the protection
term begins to dominate. This is milder than the upper error bound with
respect to the adjusted gauge theory (see main text and Appendix D).

Figure 13: (Color online). Same as Fig. 11 but for linear protection with a
noncompliant sequence, $V\tilde{H}_{G}=V\sum_{j}(-1)^{j+1}G_{j}$. As in the
case of the gauge violation in Fig. 7 and particle denstiy in Fig. 9 for this
protection scheme, the dynamics quickly approaches the thermodynamic limit
already at a few matter sites.

###  A.2 Temporally nonaveraged gauge violation

For completeness, here we present the behavior of the gauge violation without
temporal averaging, given by

$\displaystyle\varepsilon_{\text{raw}}(t)=\frac{1}{L}\sum_{j}\bra{\psi_{0}}e^{iHt}G_{j}^{2}e^{-iHt}\ket{\psi_{0}}.$
(12)

The results are presented in Fig. 14 for the case of a quench on the initial
state $\ket{\psi_{0}}$ (see Fig. 1) by the faulty Hamiltonian $H=H_{0}+\lambda
H_{1}+V\sum_{j}(-1)^{j+1}G_{j}$ at different system sizes $2L$ ($L$ matter
sites and $L$ links given that we use periodic boundary conditions in our ED
calculations). The link spin length is $S=1/2$, while the error and protection
strengths are set to $\lambda=0.1J$ and $V=10J$, respectively. As can be seen,
temporal fluctuations of $\varepsilon_{\text{raw}}(t)$ decrease with system
size at small values of $L$, but already at about $L=4$ matter sites the
fluctuations behave very similarly to the thermodynamic limit. Results at
finite values of $L$ are computed in ED, while those in the thermodynamic
limit are calculated through iMPS. We have checked that the behavior seen in
Fig. 14 also applies to other cases, including full protection, different
initial states, and other various values of the microscopic parameters.

Figure 14: (Color online). Same as Fig. 7(a), but instead of showing the
temporally averaged gauge violation $\varepsilon(t)$ given in Eq. (6), we plot
the raw gauge violation
$\varepsilon_{\text{raw}}(t)=\sum_{j}\bra{\psi_{0}}e^{iHt}G_{j}^{2}e^{-iHt}\ket{\psi_{0}}/L$.
At small system sizes, the fluctuations in the signal decrease and converge
fast to the thermodynamic limit. Figure 15: (Color online). Behavior of the
gauge violation, Eq. (6), with respect to iMPS bond dimension at fixed time-
step $\Delta t=0.005/J$, in the case of a quench by the faulty Hamiltonian
with linear protection $V\tilde{H}_{G}=V\sum_{j}(-1)^{j}G_{j}$, at error
strength $\lambda=0.1J$, protection strength $V=100J$, and for link spin
length $S=3/2$ (see legend). As we increase the bond dimension, the plateau
$\propto\lambda^{2}/V^{2}$ persists longer in evolution time, until the result
diverges (see inset). The conclusions of this plot remain the same under full
protection, and also for other initial states and different values of the
microscopic parameters.

###  A.3 Convergence with bond dimension in iMPS

In the iMPS results we have shown thus far, we only included data up to
evolution times where convergence with respect to time-step and bond dimension
is achieved. For our most stringent calculations, this occurs for a time-step
$\Delta t=0.005/J$ and bond dimension $\mathcal{D}=200$. However, it is
interesting to look at the behavior of the gauge violation relative to
$\mathcal{D}$. For this purpose, we again quench the initial state
$\ket{\psi_{0}}$ (see Fig. 1) by the faulty Hamiltonian $H=H_{0}+\lambda
H_{1}+V\sum_{j}(-1)^{j+1}G_{j}$. The link spin length is $S=3/2$, while the
error and protection strengths are set to $\lambda=0.1J$ and $V=100J$,
respectively. The time-step is fixed to $\Delta t=0.005/J$. We show in Fig. 15
the ensuing gauge violation, Eq. (6), at several values of the bond dimension
$\mathcal{D}$ (see legend). After reaching the plateau
$\propto\lambda^{2}/V^{2}$, we find that as the bond dimension is increased,
the plateau persists to longer evolution times before the result diverges and
can no longer be trusted. This behavior is not restricted to the case
considered in Fig. 15, but holds also when full protection is used, or the
initial state or microscopic parameters are different.

##  Appendix B Abanin-De Roeck-Ho-Huveneers (ARHH) method

For the sake of completeness and self-containment, here we briefly review the
gauge protection for the full and compliant linear protection terms,
introduced in Ref. Halimeh  _et al._ , 2020 and based on the ARHH
framework.Abanin  _et al._ (2017)

For convenience, we define $H_{\rm bare}=H_{0}+\lambda H_{1}$. We also define
$\mathcal{P}_{n}$ as the projection operator onto the eigenstates of ${H}_{G}$
with eigenvalue $n$. The Hamiltonian $H_{\rm bare}$ can now be decomposed into
two parts: $H_{\rm diag}=\sum_{n}\mathcal{P}_{n}H_{\rm
bare}\mathcal{P}_{n}=H_{0}+\lambda\sum_{n}\mathcal{P}_{n}H_{1}\mathcal{P}_{n}$,
and $H_{\rm ndiag}=H-H_{\rm diag}-VH_{G}$, where $H=H_{0}+\lambda
H_{1}+VH_{G}$.

The full protection term $V{H}_{G}=V\sum_{j}G_{j}^{2}$ protects the gauge-
invariant dynamics in the faulty gauge theory $H$ for sufficiently large
$V$.Halimeh and Hauke (2020); Halimeh  _et al._ (2020); Abanin  _et al._
(2017) In the $\mathrm{U}(1)$ gauge theory given in Eq. (1), the spectrum of
${H}_{G}$ is composed of integers, based on the form of the corresponding
generators $G_{j}$ of Gauss’s law; cf. Eq. (2). Specifically, the ground-state
manifold with $n=0$ of ${H}_{G}$ is the physical sector we want to protect.
The projector onto the ground-state manifold is denoted as $\mathcal{P}_{0}$.

The ARHH method is based on the norm of potentials, which permits a
formulation that is volume-independent and applicable in the thermodynamic
limit. Define $\Lambda$ as a finite subset of the lattice $\mathbb{Z}^{d}$, in
$d$-dimensional space. The algebra of bounded operators acting on the total
Hilbert space $\mathcal{H}_{\Lambda}$ equipped with the standard operator norm
is denoted as $\mathcal{B}_{\Lambda}$. The subalgebra is defined as
$\mathcal{B}_{A}\subset\mathcal{B}_{\Lambda}$ of operators of the form
$O_{A}\otimes\mathcal{I}_{\Lambda\backslash A}$ with $A\subset\Lambda$.
Generally, any local operator $X$ can be decomposed (in a nonunique way) as
$X=\sum_{A\in\mathcal{P}_{c}(\Lambda)}X_{A}$ where $X_{A}\in\mathcal{B}_{A}$
and $\mathcal{P}_{c}(\Lambda)$ is the set of finite, connected (by adjacency)
subsets of $\Lambda$. We call the collection of $X_{A}$ a ‘potential’. The
family of norms on potentials with a rate parameter $\kappa\geq 0$, which
gives operators with different spatial support different weights, is defined
as

$\lvert\lvert
X\rvert\rvert_{\kappa}\coloneqq\sup\limits_{x\in\Lambda}\sum_{A\in\mathcal{P}_{c}(\Lambda):A\ni
x}e^{\kappa\lvert A\rvert}\lvert\lvert X_{A}\rvert\rvert.$ (13)

The supremum in this definition picks the lattice site $x$ with the largest
sum of weighted norms of the operators $X_{A}$ that have support on $x$.

With the above definitions, the following can be derived. Given the full
protection term $H_{G}$ with integer spectrum, assume there exists a
$\kappa_{0}$ and relevant energy scale

$\displaystyle V_{0}\coloneqq\frac{54\pi}{\kappa_{0}^{2}}(\lvert\lvert H_{\rm
diag}\rvert\rvert_{\kappa_{0}}+2\lvert\lvert H_{\rm
ndiag}\rvert\rvert_{\kappa_{0}}).$ (14)

Then, if $V$ satisfies the conditions

$\displaystyle V$ $\displaystyle\geq\frac{9\pi\lvert\lvert H_{\rm
ndiag}\rvert\rvert_{\kappa_{0}}}{\kappa_{0}},$ (15a) $\displaystyle n_{*}$
$\displaystyle\coloneqq\bigg{\lfloor}\frac{V}{V_{0}(1+\ln V-\ln
V_{0})^{3}}\bigg{\rfloor}-2\geq 1,$ (15b)

there exists a quasilocal unitary operator $Y$ satisfying

$\displaystyle YHY^{\dagger}$ $\displaystyle=VH_{G}+H^{\prime}$
$\displaystyle=VH_{G}+H^{\prime}_{\rm diag}+H^{\prime}_{\rm ndiag},$ (16)

where

$\displaystyle H^{\prime}$ $\displaystyle=YHY^{\dagger}-VH_{G},$ (17a)
$\displaystyle H^{\prime}_{\rm diag}$
$\displaystyle=\sum_{n}\mathcal{P}_{n}H^{\prime}\mathcal{P}_{n},$ (17b)
$\displaystyle H^{\prime}_{\rm ndiag}$
$\displaystyle=H^{\prime}-H^{\prime}_{\rm diag},$ (17c)
$\displaystyle\lvert\lvert H^{\prime}_{\rm diag}-H_{\rm
diag}\rvert\rvert_{\kappa_{n_{*}}}$ $\displaystyle\leq CV_{0}/V,$ (17d)
$\displaystyle\lvert\lvert H^{\prime}_{\rm
ndiag}\rvert\rvert_{\kappa_{n_{*}}}$ $\displaystyle\leq(2/3)^{n_{*}}||H_{\rm
ndiag}||_{\kappa_{0}},$ (17e) $\displaystyle\kappa_{n_{*}}$
$\displaystyle\coloneqq\frac{\kappa_{0}}{1+\log(1+n_{*})},$ (17f)

with $C$ a constant.

For arbitrary local operator $O$ and up to an exponentially large time $t$ on
the scale $e^{kn_{*}}/V_{0}$, we have

$\displaystyle\big{\lvert}\big{\lvert}U(t)^{\dagger}OU(t)-e^{it(VH_{G}+H^{\prime}_{\rm
diag})}Oe^{-it(VH_{G}+H^{\prime}_{\rm diag})}\big{\rvert}\big{\rvert}$
$\displaystyle\leq\frac{K(O)}{V},$ (18)

where $U(t)=e^{-iHt}$ is the time-evolution operator,
$0<k<(d+1)^{-1}\ln{(3/2)}$, and $K(O)$ is $V$\- and volume-independent but
model parameter-dependent term.

In the operator norm sense, the dynamics of all local observables as generated
by the effective Hamiltonian $H^{\prime}_{\rm diag}$ is perturbatively close
(in $V_{0}/V$) to $H_{\rm diag}$. As discussed in Ref. Halimeh  _et al._ ,
2020, we cannot expect $H^{\prime}_{\rm diag}$ to be a gauge-invariant
Hamiltonian. However, if we prepare the initial state in the physical target
space and focus on the dynamics of expectation values, the above operator norm
bound can be translated to

$\displaystyle\big{\lvert}\langle U^{\dagger}(t)OU(t)-e^{iH^{\prime}_{\rm
eff}t}Oe^{-iH^{\prime}_{\rm eff}t}\rangle\big{\rvert}\leq\frac{K(O)}{V},$ (19)

where $H^{\prime}_{\rm eff}=\mathcal{P}_{0}H^{\prime}\mathcal{P}_{0}$. In
contrast to $H^{\prime}_{\rm diag}$, $H^{\prime}_{\rm eff}$ is a gauge-
invariant Hamiltonian. Up to now, we obtain an effective renormalized gauge
Hamiltonian $H^{\prime}_{\rm eff}=\mathcal{P}_{0}H^{\prime}\mathcal{P}_{0}$,
which governs all the dynamics of expectation values of local observables $O$
with an error bounded by $K(O)/V$ up to a timescale $e^{kn_{\star}}/V_{0}$.
This is what we refer to in the main text as the renormalized timescale, which
can be written as $\tau_{\text{ren}}\propto\exp(V/V_{0})/V_{0}$ when $V$ is
sufficiently large. It is worth mentioning that this timescale also exists in
the case of linear protection with a compliant sequence $c_{j}$ (for
derivational details see Ref. Halimeh  _et al._ , 2020). However, in that case
$V$ has to be increased with system size $L$ in order to achieve a given level
of gauge-invariance reliability. This is because the larger $L$ is, the
smaller is the spacing in the normalized rational coefficients $c_{j}$ of the
compliant sequence. As such, even though the renormalized timescale itself is
not explicitly volume-dependent, the reliability is not in the case of linear
protection with a compliant sequence. This is not a problem in the case of
full protection, and therefore the reliability is volume-independent.

Let us now look at how $V$ scales with the link spin length $S$. The exact
form of $H^{\prime}_{\rm eff}$ is in general difficult to obtain.
Nevertheless, this framework permits us to make general scaling statements. In
particular, the required protection strength is volume-independent and
determined by Eqs. (15a) and (15b). For a perturbative error term $\lambda
H_{1}$ and sufficiently large $S$, Eq. (15b) determines the minimal protection
strength as $V_{\mathrm{min}}\sim V_{0}$. In this case, $V_{0}$ roughly scales
as the maximum of the norm of the local terms of $H_{\rm bare}$, which in our
model is $g^{2}a(s^{z}_{j,j+1})^{2}/2\sim S^{2}$. Therefore, in the worst case
the minimal protection strength $V_{\mathrm{min}}$ scales as $S^{2}$ when $S$
is sufficiently large.

##  Appendix C Constrained quantum dynamics

As mentioned in Appendix B treating the ARHH method, the dynamics of local
observables can be protected up to an exponential timescale. However, the fact
that the renormalized effective Hamiltonian is in general unknown makes that
method somewhat inconvenient for practical applications to concrete systems,
particularly when knowledge of an emergent theory is desired to quantify a
controlled error. Using a simpler framework also in the context of full
protection, we can additionally prove that the dynamics is generated by an
adjusted gauge theory described by the Hamiltonian
$H_{\text{adj}}=H_{0}+\lambda\mathcal{P}_{0}H_{1}\mathcal{P}_{0}$ up to a
timescale fractional in the protection strength $V$ and is still volume-
independent and scales as $S^{2}$ when $S$ is sufficiently large. For the
physical error given by Eq. (3), this adjusted gauge theory Hamiltonian is
exactly $H_{0}$ since $\mathcal{P}_{0}H_{1}\mathcal{P}_{0}=0$ in this case.
This section adapts recent work treating a universal bound on the constrained
quantum dynamics.Gong  _et al._ (2020b, a)

The ingredients of the derivations of this bound are:

  1. 1.

The Hamiltonian $H$ and observable $O$ are local in the sense of the
potentials introduced in Appendix B, which is identical to the locality
requirement of the ARHH method.

  2. 2.

All terms in the decomposition of $H_{\rm
pro}=\sum_{A\in\mathcal{P}_{c}(\Lambda)}H_{{\rm pro},A}$ commute with one
other, where the collection of $H_{{\rm pro},A}$ is a potential.

  3. 3.

The ground-state manifold of $H_{\rm pro}$ is frustration-free, i.e., the
ground state also minimizes the local energies $H_{{\rm pro},A}$ everywhere
for all $A$.

  4. 4.

There exists a sufficiently large energy gap $\Delta_{0}$ between the ground
state and excited states of $H_{\rm pro}$. By sufficiently large, we mean here
that $V$ scales at least $\sim\lvert\lvert H_{\rm
bare}\rvert\rvert_{\kappa=0}$.

If all these four conditions are satisfied, the following bound holds:

$\displaystyle\big{\lvert}\big{\lvert}\mathcal{P}_{0}\big{[}U(t)^{\dagger}OU(t)-e^{it(H_{\rm
pro}+\mathcal{P}_{0}H_{\rm bare}\mathcal{P}_{0})}Oe^{-it(H_{\rm
pro}+\mathcal{P}_{0}H_{\rm
bare}\mathcal{P}_{0})}\big{]}\mathcal{P}_{0}\big{\rvert}\big{\rvert}\leq\frac{\lvert\lvert
H_{\rm bare}\rvert\rvert_{\kappa=0}}{\Delta_{0}}p(t),$ (20)

where $p(t)$ is a polynomial in $||H_{\rm bare}||_{\kappa=0}t$ with degree
$d+1$ and (at most) order-one coefficients.

It is easy to check that the model $H=H_{0}+\lambda H_{1}+VH_{G}$ in this work
satisfies all these necessary conditions. The decomposition of $H_{\rm
pro}=VH_{G}$ can be chosen as $H_{{\rm pro},A}=VG_{j}^{2}$ and the energy gap
of $H_{\rm pro}$ is $V$. To apply the bound to $H$, we immediately obtain that
the dynamics of the arbitrary local observable $O$ is generated by the
adjusted Hamiltonian
$H_{\text{adj}}=H_{0}+\lambda\mathcal{P}_{0}H_{1}\mathcal{P}_{0}$ with an
error bounded by $\lvert\lvert H_{0}+\lambda H_{1}\rvert\rvert_{\kappa=0}/V$
up to a timescale fractional in $V$, which is volume-independent,

$\displaystyle\big{\lvert}\langle U^{\dagger}(t)OU(t)-e^{iH_{\rm
adj}t}Oe^{-iH_{\rm adj}t}\rangle\big{\rvert}$ $\displaystyle\leq\frac{||H_{\rm
bare}||_{\kappa=0}}{V}p(t)\sim\frac{t^{2}V_{0}^{3}}{V},$ (21)

In the above expression, we have already used $V_{0}$ instead of $||H_{\rm
bare}||_{\kappa=0}$ since they are roughly of the same scale in magnitude. In
the large $S$ limit, as discussed in the Appendix B, $||H_{0}+\lambda
H_{1}||_{\kappa=0}$ scales as $S^{2}$, which means the protection strength is
volume-independent and scales as $S^{2}$ in the worst case.

##  Appendix D Quantum Zeno effect

In Ref. Halimeh  _et al._ , 2020, it has been shown that linear protection
with the noncompliant sequence of coefficents $c_{j}=(-1)^{j+1}$ can protect
the evolution operator $\exp(-iHt)$ of the spin-$1/2$ $\mathrm{U}(1)$ gauge-
theory dynamics because of the quantum Zeno effect,Facchi and Pascazio (2002);
Facchi  _et al._ (2004, 2009); Burgarth  _et al._ (2019)

$\displaystyle\big{\lvert}\big{\lvert}e^{-iHt}-e^{-i(H_{\rm adj}+H_{\rm
pro})t}\big{\rvert}\big{\rvert}\leq\frac{q(t)}{V},$ (22)

where $q(t)$ is a polynomial in $t(V_{0,\rm qze}L)^{2}$ of degree $1$ and a
coefficient of order $1$, and $V_{0,{\rm qze}}\sim V_{0}$ is the relevant
energy scale for the quantum Zeno effect. Therefore, the deviation of
observables $O$ is given by

$\displaystyle\big{\lvert}\langle U^{\dagger}(t)OU(t)-e^{iH_{\rm
adj}t}Oe^{-iH_{\rm
adj}t}\rangle\big{\rvert}\leq\big{\lvert}\big{\lvert}\langle
U^{\dagger}(t)OU(t)-e^{iH_{\rm adj}t}Oe^{-iH_{\rm
adj}t}\rangle\big{\rvert}\big{\rvert}$
$\displaystyle=\big{\lvert}\big{\lvert}\big{[}U^{\dagger}(t)-e^{iH_{\rm
adj}t}\big{]}OU(t)+\big{[}U^{\dagger}(t)-e^{iH_{\rm adj}t}\big{]}Oe^{-iH_{\rm
adj}t}+U^{\dagger}(t)O\big{[}U(t)-e^{-iH_{\rm adj}t}\big{]}+e^{iH_{\rm
adj}t}O\big{[}U^{\dagger}(t)-e^{-iH_{\rm
adj}t}\big{]}\big{\rvert}\big{\rvert}$ $\displaystyle\leq
4\frac{q(t)}{V}\sim\frac{tV_{0}^{2}L^{2}}{V}.$ (23)

Hence, the gauge theory is reliable up to a timescale $t\propto V/(V_{0,{\rm
qze}}L)^{2}\sim V/(V_{0}L)^{2}$, with a controlled violation of
$\mathcal{O}(V_{0,{\rm qze}}^{2}L^{2}/V)\sim\mathcal{O}(V_{0}^{2}L^{2}/V)$.

## References

  * Bloch  _et al._ (2008) I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008).
  * Lewenstein  _et al._ (2012) M. Lewenstein, A. Sanpera, and V. Ahufinger,  _Ultracold Atoms in Optical Lattices: Simulating quantum many-body systems_ (OUP Oxford, 2012).
  * Blatt and Roos (2012) R. Blatt and C. F. Roos, Nature Physics 8, 277 (2012).
  * Hauke  _et al._ (2012) P. Hauke, F. M. Cucchietti, L. Tagliacozzo, I. Deutsch, and M. Lewenstein, Reports on Progress in Physics 75, 082401 (2012).
  * Schreiber  _et al._ (2015) M. Schreiber, S. S. Hodgman, P. Bordia, H. P. Lüschen, M. H. Fischer, R. Vosk, E. Altman, U. Schneider, and I. Bloch, Science 349, 842 (2015).
  * Choi  _et al._ (2016) J.-y. Choi, S. Hild, J. Zeiher, P. Schauß, A. Rubio-Abadal, T. Yefsah, V. Khemani, D. A. Huse, I. Bloch, and C. Gross, Science 352, 1547 (2016).
  * Smith  _et al._ (2016) J. Smith, A. Lee, P. Richerme, B. Neyenhuis, P. W. Hess, P. Hauke, M. Heyl, D. A. Huse, and C. Monroe, Nature Physics 12, 907 (2016).
  * Xu  _et al._ (2014) X.-Y. Xu, Y.-J. Han, K. Sun, J.-S. Xu, J.-S. Tang, C.-F. Li, and G.-C. Guo, Phys. Rev. Lett. 112, 035701 (2014).
  * Anquez  _et al._ (2016) M. Anquez, B. A. Robbins, H. M. Bharath, M. Boguslawski, T. M. Hoang, and M. S. Chapman, Phys. Rev. Lett. 116, 155301 (2016).
  * Clark  _et al._ (2016) L. W. Clark, L. Feng, and C. Chin, Science 354, 606 (2016).
  * Cui  _et al._ (2016) J.-M. Cui, Y.-F. Huang, Z. Wang, D.-Y. Cao, J. Wang, W.-M. Lv, L. Luo, A. del Campo, Y.-J. Han, C.-F. Li, and G.-C. Guo, Scientific Reports 6, 33381 (2016).
  * Keesling  _et al._ (2019) A. Keesling, A. Omran, H. Levine, H. Bernien, H. Pichler, S. Choi, R. Samajdar, S. Schwartz, P. Silvi, S. Sachdev, P. Zoller, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, Nature 568, 207 (2019).
  * Jurcevic  _et al._ (2017) P. Jurcevic, H. Shen, P. Hauke, C. Maier, T. Brydges, C. Hempel, B. P. Lanyon, M. Heyl, R. Blatt, and C. F. Roos, Phys. Rev. Lett. 119, 080501 (2017).
  * Zhang  _et al._ (2017) J. Zhang, G. Pagano, P. W. Hess, A. Kyprianidis, P. Becker, H. Kaplan, A. V. Gorshkov, Z.-X. Gong, and C. Monroe, Nature 551, 601 (2017).
  * Fläschner  _et al._ (2018) N. Fläschner, D. Vogel, M. Tarnowski, B. S. Rem, D.-S. Lühmann, M. Heyl, J. C. Budich, L. Mathey, K. Sengstock, and C. Weitenberg, Nature Physics 14, 265 (2018).
  * Gring  _et al._ (2012) M. Gring, M. Kuhnert, T. Langen, T. Kitagawa, B. Rauer, M. Schreitl, I. Mazets, D. A. Smith, E. Demler, and J. Schmiedmayer, Science 337, 1318 (2012).
  * Langen  _et al._ (2015) T. Langen, S. Erne, R. Geiger, B. Rauer, T. Schweigler, M. Kuhnert, W. Rohringer, I. E. Mazets, T. Gasenzer, and J. Schmiedmayer, Science 348, 207 (2015).
  * Neyenhuis  _et al._ (2017) B. Neyenhuis, J. Zhang, P. W. Hess, J. Smith, A. C. Lee, P. Richerme, Z.-X. Gong, A. V. Gorshkov, and C. Monroe, Science Advances 3 (2017), 10.1126/sciadv.1700672.
  * Kaplan  _et al._ (2020) H. B. Kaplan, L. Guo, W. L. Tan, A. De, F. Marquardt, G. Pagano, and C. Monroe, Phys. Rev. Lett. 125, 120605 (2020).
  * Martinez  _et al._ (2016) E. A. Martinez, C. A. Muschik, P. Schindler, D. Nigg, A. Erhard, M. Heyl, P. Hauke, M. Dalmonte, T. Monz, P. Zoller, and R. Blatt, Nature 534, 516 (2016).
  * Bernien  _et al._ (2017) H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, Nature 551, 579 (2017).
  * Dai  _et al._ (2017) H.-N. Dai, B. Yang, A. Reingruber, H. Sun, X.-F. Xu, Y.-A. Chen, Z.-S. Yuan, and J.-W. Pan, Nature Physics 13, 1195 (2017).
  * 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, Phys. Rev. A 98, 032331 (2018).
  * Kokail  _et al._ (2019) C. Kokail, C. Maier, R. van Bijnen, T. Brydges, M. K. Joshi, P. Jurcevic, C. A. Muschik, P. Silvi, R. Blatt, C. F. Roos, and P. Zoller, Nature 569, 355 (2019).
  * Schweizer  _et al._ (2019) C. Schweizer, F. Grusdt, M. Berngruber, L. Barbiero, E. Demler, N. Goldman, I. Bloch, and M. Aidelsburger, Nature Physics (2019), 10.1038/s41567-019-0649-7.
  * Görg  _et al._ (2019) F. Görg, K. Sandholzer, J. Minguzzi, R. Desbuquois, M. Messer, and T. Esslinger, Nature Physics 15, 1161 (2019).
  * Mil  _et al._ (2020) A. Mil, T. V. Zache, A. Hegde, A. Xia, R. P. Bhatt, M. K. Oberthaler, P. Hauke, J. Berges, and F. Jendrzejewski, Science 367, 1128 (2020).
  * Brower  _et al._ (2020) R. C. Brower, D. Berenstein, and H. Kawai, “Lattice gauge theory for a quantum computer,” (2020), arXiv:2002.10028 [hep-lat] .
  * Klco  _et al._ (2020) N. Klco, M. J. Savage, and J. R. Stryker, Phys. Rev. D 101, 074512 (2020).
  * Yang  _et al._ (2020) B. Yang, H. Sun, R. Ott, H.-Y. Wang, T. V. Zache, J. C. Halimeh, Z.-S. Yuan, P. Hauke, and J.-W. Pan, Nature 587, 392 (2020).
  * Wiese (2013) U.-J. Wiese, Annalen der Physik 525, 777 (2013).
  * Zohar  _et al._ (2015) E. Zohar, J. I. Cirac, and B. Reznik, Reports on Progress in Physics 79, 014401 (2015).
  * Dalmonte and Montangero (2016) M. Dalmonte and S. Montangero, Contemporary Physics 57, 388 (2016), https://doi.org/10.1080/00107514.2016.1151199 .
  * Bañuls  _et al._ (2019) M. C. Bañuls, R. Blatt, J. Catani, A. Celi, J. I. Cirac, M. Dalmonte, L. Fallani, K. Jansen, M. Lewenstein, S. Montangero, C. A. Muschik, B. Reznik, E. Rico, L. Tagliacozzo, K. Van Acoleyen, F. Verstraete, U.-J. Wiese, M. Wingate, J. Zakrzewski, and P. Zoller, ArXiv e-prints (2019), arXiv:1911.00003 [quant-ph] .
  * Muschik  _et al._ (2017) C. Muschik, M. Heyl, E. Martinez, T. Monz, P. Schindler, B. Vogell, M. Dalmonte, P. Hauke, R. Blatt, and P. Zoller, New Journal of Physics 19, 103020 (2017).
  * Surace  _et al._ (2020) F. M. Surace, P. P. Mazza, G. Giudici, A. Lerose, A. Gambassi, and M. Dalmonte, Phys. Rev. X 10, 021041 (2020).
  * Chandrasekharan and Wiese (1997) S. Chandrasekharan and U.-J. Wiese, Nuclear Physics B 492, 455 (1997).
  * Zohar and Reznik (2011) E. Zohar and B. Reznik, Phys. Rev. Lett. 107, 275301 (2011).
  * Banerjee  _et al._ (2012) D. Banerjee, M. Dalmonte, M. Müller, E. Rico, P. Stebler, U.-J. Wiese, and P. Zoller, Phys. Rev. Lett. 109, 175302 (2012).
  * Zohar  _et al._ (2013) E. Zohar, J. I. Cirac, and B. Reznik, Phys. Rev. A 88, 023617 (2013).
  * Hauke  _et al._ (2013) P. Hauke, D. Marcos, M. Dalmonte, and P. Zoller, Phys. Rev. X 3, 041018 (2013).
  * Kühn  _et al._ (2014) S. Kühn, J. I. Cirac, and M.-C. Bañuls, Phys. Rev. A 90, 042305 (2014).
  * Kuno  _et al._ (2015) Y. Kuno, K. Kasamatsu, Y. Takahashi, I. Ichinose, and T. Matsui, New Journal of Physics 17, 063005 (2015).
  * Dutta  _et al._ (2017) O. Dutta, L. Tagliacozzo, M. Lewenstein, and J. Zakrzewski, Phys. Rev. A 95, 053608 (2017).
  * Kuno  _et al._ (2017) Y. Kuno, S. Sakane, K. Kasamatsu, I. Ichinose, and T. Matsui, Phys. Rev. D 95, 094507 (2017).
  * Dehkharghani  _et al._ (2017) A. S. Dehkharghani, E. Rico, N. T. Zinner, and A. Negretti, Phys. Rev. A 96, 043611 (2017).
  * Barros  _et al._ (2019) J. C. P. Barros, M. Burrello, and A. Trombettoni, ArXiv e-prints (2019), arXiv:1911.06022 [quant-ph] .
  * Halimeh and Hauke (2020) J. C. Halimeh and P. Hauke, Phys. Rev. Lett. 125, 030503 (2020).
  * Lamm  _et al._ (2020) H. Lamm, S. Lawrence, and Y. Yamauchi,  (2020), arXiv:2005.12688 [quant-ph] .
  * Halimeh  _et al._ (2020) J. C. Halimeh, H. Lang, J. Mildenberger, Z. Jiang, and P. Hauke, (2020), arXiv:2007.00668 [quant-ph] .
  * Abanin  _et al._ (2017) D. Abanin, W. De Roeck, W. W. Ho, and F. Huveneers, Communications in Mathematical Physics 354, 809 (2017).
  * Gong  _et al._ (2020a) Z. Gong, N. Yoshioka, N. Shibata, and R. Hamazaki, Physical Review Letters 124, 210606 (2020a).
  * Gong  _et al._ (2020b) Z. Gong, N. Yoshioka, N. Shibata, and R. Hamazaki, Physical Review A 101, 052122 (2020b).
  * Facchi and Pascazio (2002) P. Facchi and S. Pascazio, Physical review letters 89, 080401 (2002).
  * Facchi  _et al._ (2004) P. Facchi, D. Lidar, and S. Pascazio, Physical Review A 69, 032314 (2004).
  * Facchi  _et al._ (2009) P. Facchi, G. Marmo, and S. Pascazio, in  _Journal of Physics: Conference Series_ , Vol. 196 (IOP Publishing, 2009) p. 012017.
  * Burgarth  _et al._ (2019) D. Burgarth, P. Facchi, H. Nakazato, S. Pascazio, and K. Yuasa, Quantum 3, 152 (2019).
  * Yang  _et al._ (2016) D. Yang, G. S. Giri, M. Johanning, C. Wunderlich, P. Zoller, and P. Hauke, Phys. Rev. A 94, 052321 (2016).
  * Kasper  _et al._ (2017) V. Kasper, F. Hebenstreit, F. Jendrzejewski, M. K. Oberthaler, and J. Berges, New Journal of Physics 19, 023030 (2017).
  * Haegeman  _et al._ (2011) J. Haegeman, J. I. Cirac, T. J. Osborne, I. Pižorn, H. Verschelde, and F. Verstraete, Phys. Rev. Lett. 107, 070601 (2011).
  * Haegeman  _et al._ (2016) J. Haegeman, C. Lubich, I. Oseledets, B. Vandereycken, and F. Verstraete, Phys. Rev. B 94, 165116 (2016).
  * Vanderstraeten  _et al._ (2019) L. Vanderstraeten, J. Haegeman, and F. Verstraete, SciPost Phys. Lect. Notes , 7 (2019).
  * Damme  _et al._ (2020) M. V. Damme, J. C. Halimeh, and P. Hauke, (2020), arXiv:2010.07338 [cond-mat.quant-gas] .
  * Borla  _et al._ (2021) U. Borla, R. Verresen, J. Shah, and S. Moroz, “Gauging the kitaev chain,” (2021), arXiv:2010.00607 [cond-mat.str-el] .
  * Poppitz and Shang (2008) E. Poppitz and Y. Shang, International Journal of Modern Physics A 23, 4545 (2008), https://doi.org/10.1142/S0217751X08041281 .
  * Bazavov  _et al._ (2015) A. Bazavov, Y. Meurice, S.-W. Tsai, J. Unmuth-Yockey, and J. Zhang, Phys. Rev. D 92, 076003 (2015).
  * Paulson  _et al._ (2020) D. Paulson, L. Dellantonio, J. F. Haase, A. Celi, A. Kan, A. Jena, C. Kokail, R. van Bijnen, K. Jansen, P. Zoller, and C. A. Muschik, (2020), arXiv:2008.09252 [quant-ph] .
  * Ott  _et al._ (2020) R. Ott, T. V. Zache, F. Jendrzejewski, and J. Berges, (2020), arXiv:2012.10432 [cond-mat.quant-gas] .