Stabilizing lattice gauge theories through simplified local pseudo generators

The postulate of gauge invariance in nature does not lend itself directly to implementations of lattice gauge theories in modern setups of quantum synthetic matter. Unavoidable gauge-breaking errors in such devices require gauge invariance to be enforced for faithful quantum simulation of gauge-theory physics. This poses major experimental challenges, in large part due to the complexity of the gauge-symmetry generators. Here, we show that gauge invariance can be reliably stabilized by employing simplified \textit{local pseudogenerators} designed such that within the physical sector they act identically to the actual local generator. Dynamically, they give rise to emergent exact gauge theories up to timescales polynomial and even exponential in the protection strength. This obviates the need for implementing often complex multi-body full gauge symmetries, thereby further reducing experimental overhead in physical realizations. We showcase our method in the $\mathbb{Z}_2$ lattice gauge theory, and discuss experimental considerations for its realization in modern ultracold-atom setups.


I. INTRODUCTION
Gauge theories are a cornerstone of modern physics [1], describing the interactions between elementary particles as mediated by gauge bosons. They implement physical laws of nature through local constraints in space and time [2]. A paradigmatic example is Gauss's law in quantum electrodynamics, which enforces an intrinsic relation between the distribution of charged matter and the associated electromagnetic field.
We introduce the concept of the local pseudogenerator (LPG), which is designed to behave identically to the full generator within, but not necessarily outside, the target sector; see Fig. 1. This relieves significant engineering requirements, rendering the LPG with fewer-body terms than its full counterpart. As we demonstrate nu- 0 d v X E P j F K 3 w 4 i Z U q i P V N / T 6 Q 0 1 H o c + q Y z p D j U i 9 5 U / M 9 r J x h c e i m X c Y I g 2 X x R k A g b I 3 s a g d 3 n C h i K s S G U K W 5 u t d m Q K s r Q B F U w I b i L L y + T R q X s n p U r t + e l 6 l U W R 5 4 c k W N y S l x y Q a r k h t R I n T C i y D N 5 J W / W o / V i v V s f 8 9 a c l c 0 c k j + w P n 8 A M j G S 8 Q = = < / l a t e x i t > g tar j < l a t e x i t s h a 1 _ b a s e 6 4 = " D V n P m 4 Y l H f q l T c D R y L b q u D 8 w 4 W 4 = " > A A A B 9 X i c b V B N S 8 N A E N 3 U r 1 q / q h 6 9 B I v g q S R V U P B S 8 O K x g v 2 A N i 2 b 7 a R d u 9 m E 3 Y l a Q v + H F w + K e P W / e P P f u G 1 z 0 N Y H A 4 / 3 Z p i Z 5 8 e C a 3 S c b y u 3 s r q 2 v p H f L G x t 7 + z u F f c P G j p K F I M 6 i 0 S k W j 7 V I L i E O n I U 0 I o V 0 N A X 0 P R H 1 1 O / + Q B K 8 0 j e 4 T g G L 6 Q D y Q P O K B q p O + j d d z s I T 5 g i V Z N e s e S U n R n s Z e J m p E Q y 1 H r F r 0 4 / Y k k I E p m g W r d d J 0 Y v p Q o 5 E z A p d B I N M W U j O o C 2 o Z K G o L 1 0 d v X E P j F K 3 w 4 i Z U q i P V N / T 6 Q 0 1 H o c + q Y z p D j U i 9 5 U / M 9 r J x h c e i m X c Y I g 2 X x R k A g b I 3 s a g d 3 n C h i K s S G U K W 5 u t d m Q K s r Q B F U w I b i L L y + T R q X s n p U r t + e l 6 l U W R 5 4 c k W N y S l x y Q a r k h t R I n T C i y D N 5 J W / W o / V i v V s f 8 9 a c l c 0 c k j + w P n 8 A M j G S 8 Q = = < / l a t e x i t > g j , w j < l a t e x i t s h a 1 _ b a s e 6 4 = " / s l T f b 1 W q V v p u n p c Z y 0 u W c M t m r c = " > A A A C K X i c b V D L S s N A F J 3 4 r P U V d e k m W A Q X p S R t f R Q 3 B T c u K 9 g H p C F M p p N 0 7 O T B z E Q p I b / j x l 9 x o 6 C o W 3 / E S Z q F t l 6 Y y + G c e 2 b u H C e i h A t d / 1 S W l l d W 1 9 Z L G + X N r e 2 d X X V v v 8 f D m C H c R S E N 2 c C B H F M S 4 K 4 g g u J B x D D 0 H Y r 7 z u Q q 0 / v 3 m H E S B r d i G m H L h 1 5 A X I K g k J S t t p N h f o n J P M d K 9 J p e 1 W v 1 U 9 n O m q l n 3 6 X V O b 3 V y L S W b E Y j f Z A D t l q R r r y 0 R W A U o A K K 6 t j q 6 3 A U o t j H g U A U c m 4 a e i S s B D J B E M V p e R h z H E E 0 g R 4 2 J Q y g j 7 m V 5 E u k 2 r F k R p o b M n k C o e X s b 0 c C f c 6 n v i M n f S j G f F 7 L y P 8 0 M x b u h Z W Q I I o F D t D s I T e m m g i 1 L D Z t R B h G g k 4 l g I g R u a u G x p B B J G S 4 Z R m C M f / l R d C r y 9 B q 9 Z t m p X 1 Z x F E C h + A I n A A D n I M 2 u A Y d 0 A U I P I J n 8 A b e l S f l R f l Q v m a j S 0 r h O Q B / S v n + A X m A o 6 Q = < / l a t e x i t > target sector (g tar 1 , g tar 2 , . . .) < l a t e x i t s h a 1 _ b a s e 6 4 = " 2 e 1 + d 1 + p J x R w j C 6 z 6 N V d t o p O Z 5 I = " x L e y u l H e Z Z h x t u I M Q v P G T / 5 L z W t X b r d Z O 9 0 r 1 w 1 E c c 2 S D b J I K 8 c g + q Z N j c k I a h J M 7 8 k C e y Y t z 7 z w 6 r 8 7 7 V + u E M 5 p Z J 7 / g f H w C K 3 6 n J g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " V B y z T 7 W i b o i L / n N j W y D 4 K j G 0 r 0 c = " > A A A C Y 3 i c f V F N a 9 t A E F 2 p T e I 4 a a M 4 u Z W C i A n 0 U I z k f D W E Q q C H 9 p h C H Q c s I U b r s b 3 J a i V 2 R 6 W O o j / Z W 2 + 9 5 H 9 k / X F o n d K B H R 5 v 5 u 3 M v k 0 L K Q w F w S / H f f F y b X 2 j s d n c 2 n 7 1 e s f b b V 2 b v N Q c e z y X u b 5 J w a A U C n s k S O J N o R G y V G I / v f s 0 q / e / o z Y i V 9 9 o W m C c w V i J k e B A l k q 8 + y q a X z L Q 4 z S u g k 7 w P u h 0 T 2 w 6 P a 6 j C V D 1 u U 5 u 6 4 e o M C K J C H 9 Q R a D r S I M a S / y 4 I j 4 / m g n P b Q q P F u r + f 9 S J 1 7 b z 5 u E / B + E S t N k y r h L v Z z T M e Z m h I i 7 B m E E Y F B R X o E l w i X U z K g 0 W w O 9 g j A M L F W R o 4 m q + Y e 0 f W m b o j 3 J t j y J / z v 6 p q C A z Z p q l t j M D m p j V 2 o z 8 V 2 1 Q 0 u h D X A l V l I S K L w a N S u l T 7 s 8 M 9 4 d C I y c 5 t Q C 4 F n Z X n 0 9 A A y f 7 L U 1 r Q r j 6 5 O f g u m s d 7 X S / H r c v L 5 Z 2 N N g b d s D e s Z C d s U v 2 h V 2 x H u P s t 7 P u 7 D i e 8 + h u u S 1 3 f 9 H q O k v N H v s r 3 L d P 0 M K 3 U g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " X t X 5 X W d w f q d y M h L L 4 h t j k D B C n 6 I = " > A A A C G H i c b V B N S 8 N A E N 3 U r 1 q / o h 6 9 B I t Q D 9 a k C g o i F L z 0 W M E 2 h a a G z X b b b r v 5 Y H c i l p C f 4 c W / 4 s W D I l 5 7 8 9 + 4 b X P Q 6 o O B x 3 s z z M z z I s 4 k m O a X l l t a X l l d y 6 8 X N j a 3 t n f 0 3 b 2 m D G N B a I O E P B Q t D 0 v K W U A b w I D T V i Q o 9 j 1 O b W 9 0 M / X t B y o k C 4 M 7 G E e 0 4 + N + w H q M Y F C S q 5 8 2 n Q G G p J a 6 9 n X T k b H v D o k 7 L M 1 E O 3 W H J 3 1 3 e O 8 A f Y Q E s E i P X b 1 o l s 0 Z j L / E y k g R Z a i 7 + s T p h i T 2 a Q C E Y y n b l h l B J 8 E C G O E 0 L T i x p B E m I 9 y n b U U D 7 F P Z S W a P p c a R U r p G L x S q A j B m 6 s + J B P t S j n 1 P d f o Y B n L R m 4 r / e e 0 Y e p e d h A V R D D Q g 8 0 W 9 m B s Q G t O U j C 4 T l A A f K 4 K J Y O p W g w y w w A R U l g U V g r X 4 8 l / S r J S t s 3 L l 9 r x Y v c r i y K M D d I h K y E I X q I p q q I 4 a i K A n 9 I L e 0 L v 2 r L 1 q H 9 r n v D W n Z T P 7 6 B e 0 y T e W V 6 C 5 < / l a t e x i t > VĤ W < l a t e x i t s h a 1 _ b a s e 6 4 = " p y X 4 l h + 9 o q 7 1 Z D j r < l a t e x i t s h a 1 _ b a s e 6 4 = " p y X 4 l h + 9 o q 7 1 Z D j r In the presence of gauge-breaking errors at strength λ, the target sector (g tar 1 , g tar 2 , . . .) is energetically isolated by the LPG protection VĤW , where cj ∈ R; j cj[wj(g tar j ) − g tar j ] = 0 ⇐⇒ wj(g tar j ) = g tar j , ∀j. At sufficient strength V , LPG protection induces an emergent global symmetry that coincides with the local gauge symmetry within the target sector.
merically and analytically, this approach is powerfulsuppressing even nonlocal errors up to all accessible times-and the LPG is readily implementable in modern quantum-simulation platforms, e.g., ultracold atoms and superconducting qubits.
The rest of our paper is structured as follows: In Sec. II, we outline the concept and theory of local pseudogenerators. We demonstrate the efficacy of LPG gauge protection in the (1+1)−D and (2+1)−D Z 2 lattice gauge theory in Secs. III and IV, respectively. We summarize our results and provide an outlook in Sec. V. Appendix A contains supporting numerical results and Appendix B includes our detailed analytic derivations.

II. LOCAL-PSEUDOGENERATOR GAUGE PROTECTION
In an LGT, couplings between matter and gauge fields have to follow a certain set of rules dictated by the generators of gauge symmetryĜ j in order to fulfill Gauss's law. Here, j denotes the sites of the lattice, where the matter fields are located, the gauge fields live on links in between sites, and we consider Abelian gauge symmetries. Gauge invariance is embodied in the conservation of all G j by the system HamiltonianĤ 0 : [Ĥ 0 ,Ĝ j ] = 0, ∀j. This leads to physical sectors which are characterized by conserved quantum numbers given by the eigenvalues g j ofĜ j . These in turn specify the allowed distributions of matter and the corresponding configurations of electric flux. We denote the desired target sector as the set of all states {|ψ tar } satisfyingĜ j |ψ tar = g tar j |ψ tar , ∀j. The implementation ofĤ 0 in a realistic QSM setup will lead to gauge-breaking errors λĤ err at strength λ, which couple sectors with different g j . These can be reliably suppressed using the energy-penalty term VĤ pen G = V j (Ĝ j − g tar j ) 2 at sufficiently large positive protection strength V [39]. Effectively then, VĤ pen G brings the target sector within the ground-state manifold, and any processes driving the system away from it are rendered energetically unfavorable.
Generically, VĤ pen G is experimentally very challenging to realize. Recently, however, protection terms linear inĜ j have been proposed in the form of VĤ lin G = V j c j (Ĝ j − g tar j ) [36]. If the coefficients c j are real numbers such that j c j (g j − g tar j ) = 0 if and only if g j = g tar j , ∀j, then gauge invariance can be reliably stabilized up to all accessible times [36]. Such a sequence c j has been referred to as compliant. Using such linear gauge protection may mean the difference between implementing quartic or quadratic terms, such as in the case of U (1) LGTs [36]. However, in the case of other models, such as Z 2 LGTs, (Ĝ j − g tar j ) 2 ∝Ĝ j − g tar j , witĥ G j composed of complex multi-body multi-species terms [40]. In this case, linear protection offers no advantage over its quadratic energy-penalty counterpart.
The major contribution of this work is to introduce the concept of local pseudogeneratorsŴ j (g tar j ), see Fig. 1, which must satisfy the relation Note thatŴ j (g tar j ) is dependent on g tar j and is required to act identically toĜ j only within the local target sector, but not necessarily outside it. Indeed,Ŵ j (g tar j ) and G j do not need to commute. This naturally relaxes the engineering overhead onŴ j (g tar j ), reducing its number of interacting particles per term relative toĜ j . This technical advantage is the main motivation behind the concept of LPGs. One can now employ the principle of linear gauge protection [36] using the LPG, rather than the full generatorĜ j , through the term which ensures reliably suppression of violations due to any coherent local gauge-breaking errors when the condition j c j [w j (g tar j ) − g tar j ] = 0 ⇐⇒ w j (g tar j ) = g tar j , ∀j, is satisfied (i.e., c j is compliant), where w j (g tar j ) is the eigenvalue ofŴ j (g tar j ). Nevertheless, as we will demonstrate in the following, a noncompliant sequence can still reliably stabilize gauge invariance in the case of local gauge-breaking errors up to all accessible times. Table I. Eigenvalues gj and wj of the local full generatorĜj and the local pseudogeneratorŴj, respectively, for the different possible configurations of the fields on the local constraint specified by matter site j and its neighboring links. Whenever either generator has an eigenvalue g tar j , the other does too, i.e., gj = g tar j ⇐⇒ wj = g tar j . Contrapositively, whenever either is not g tar j , neither is the other: gj = g tar j ⇐⇒ wj = g tar j , though wj and gj need not be equal in this case. In our numerical simulations, we have chosen the target sector to be g tar j = +1 (green entries), but the conclusions are unaltered for g tar j = −1 (red entries), as our method is general and independent of the particular choice of the local target sector.
Following the prescription of the LPG given in Eq. (1), a suitable LPG forĜ j of Eq. (4) iŝ We find thatŴ j (g tar j ) |ψ = g tar j |ψ , ∀j, if and only if |ψ is in the target sector; see Table I. We emphasize thatŴ j is not an actual local generator of the Z 2 gauge symmetry. In fact, [Ĥ 0 ,Ŵ j ] = 0, ∀j.
In the following, we will numerically test gauge protection based on the LPG. Without loss of generality, we will henceforth select the target gauge sector to be g tar j = +1, ∀j.  )P0. At sufficiently large V , the dynamics underĤ is reproduced byĤ adj within an error ∝ tV 2 0 L 2 /V , i.e., up to a timescale τ adj ∝ V /(V0L) 2 , with V0 an energy scale dependent on the model parameters (but not V ), as we analytically predict (see Appendix B).

A. Local and nonlocal gauge errors
We prepare our system in the staggered-matter initial state |ψ 0 in the target sector (see Appendix A for details), and quench it with the faulty gauge theorŷ is an experimentally relevant local error term inspired from the setup of Ref. [17]. The coefficients α 1,...,4 are real numbers whose relative values depend on the driving parameter in the Floquet setup used to implement H 0 ; cf. Appendix A 4 for exact expressions. Here, we normalize them such that their sum is unity in order to encapsulate the error strength in λ. We additionally include the nonlocal error term which though very unlikely to occur in typical experimental setups, is ideal to scrutinize the efficacy of the LPG protection. Note thatĤ 0 ,Ĥ 1 , andĤ nloc 1 all conserve boson number, which allows us to work within a given sector of the corresponding global U(1) symmetry. This permits in exact diagonalization (ED) system sizes of L = 6 matter sites and L = 6 gauge links (equivalent to 12 spin-1/2 degrees of freedom) in the bosonic halffilling sector. However, our method also works for errors violating both the global U(1) symmetry and the local Z 2 gauge symmetry, and also for different initial states and model-parameter values (see Appendix A for supporting results). We employ open boundary conditions for experimental relevance.
Suppression of gauge violations due to gauge-breaking terms such as those of Eqs. (6) and (7) has been shown to be effective using the "full" protection term VĤ pen . This term is complicated to implement experimentally owing toĜ j containing threebody terms; cf. Eq. (4). This is the main reason why the LPG protection (2) is ideal here, given thatŴ j includes single and two-body terms only; see Eq. (5). Indeed, the level of difficulty for implementingŴ j is lower than that of the ideal gauge theoryĤ 0 itself.
We are interested in the dynamics of local observables in the wake of the quench. In particular, we analyze the temporally averaged gauge violation and staggered boson number Figure 2(a) shows the dynamics of the gauge violation for a fixed gauge-breaking strength λ at various values of the protection strength V , as calculated through ED. At early times, the gauge violation grows ∝ λ 2 t 2 as predicted by time-dependent perturbation theory [39]. After this initial growth, we see two distinct behaviors. At small V , the gauge violation is not suppressed, but rather grows to a maximal value at late times. However, at sufficiently large V , we see that the gauge violation plateaus at a timescale ∝ 1/V to a value ∝ λ 2 /V 2 , in accordance with degenerate perturbation theory [39], up to indefinite evolution times. Indeed, adapting results on slow heating in periodically driven systems [45], LPG protection with a rational compliant sequence can be shown to stabilize gauge invariance up to times exponential in V , as we derive analytically in Appendix B 1.
The long-time gauge violation as a function of J/V is shown in Fig. 2(b). There, the two-regime behavior is clear in case of a compliant sequence. The long-time gauge violation goes from an uncontrolled-error regime at small V to a controlled-error regime at sufficiently large V , at which it scales ∝ λ 2 /V 2 . When it comes to the noncompliant sequence c j = [6(−1) j + 5]/11, however, the violation does not enter a controlled-error regime, instead remaining above a minimum value no matter how large V is. This is directly related to the nonlocal error term H nloc 1 , which creates transitions between the few gaugeinvariant sectors from which the LPG protection cannot isolate the target sector in the case of a noncompliant sequence. However, as we will show later, the noncompliant sequence is very powerful against local errors.
As derived analytically in Appendix B 2 through the quantum Zeno effect, we prove that the dynamics of local observables under the faulty theoryĤ is faithfully reproduced by an adjusted gauge theoryĤ adj = H 0 + λP 0 (Ĥ 1 +Ĥ nloc 1 )P 0 , whereP 0 is the projector onto the target sector. This occurs up to an error upper bound ∝ tV 2 0 L 2 /V , yielding a timescale τ adj ∝ V /(V 0 L) 2 , where V 0 is an energy constant depending on the microscopic parameters λ/J and h/J. We find numerically that this is indeed the case for the staggered boson number under LPG protection with a compliant sequence as shown in Fig. 2(c). In the inset, the error in the dynamics under the faulty theoryĤ with respect toĤ adj grows linearly in time and is suppressed ∝ 1/V . It is to be noted here that althoughĤ adj is generally different from the ideal gauge theoryĤ 0 , it nevertheless has an exact local gauge symmetry.

B. Experimentally relevant local gauge errors
We now demonstrate the efficacy of LPG protection with an experimentally feasible periodic noncompliant sequence c j , in the case of the local gauge-breaking terms of Eq. (6). The faulty theory is now described byĤ =Ĥ 0 + λĤ 1 + V jŴ j [6(−1) j + 5]/11, and we  quench again the staggered-matter initial state |ψ 0 .
The dynamics of the gauge violation in Fig. 3(a) demonstrates reliable stabilization of gauge invariance with a plateau ∝ λ 2 /V 2 beginning at t ∝ 1/V and persisting over indefinite times at large enough V . Indeed, the transition from an uncontrolled to a controllederror regime displayed in Fig. 3(b) occurs already at small values of V ∼ 5J, which is readily accessible in quantum-simulation setups [17,20,26]. The dynamics ofn stag in Fig. 3(c) is faithfully reproduced by the adjusted gauge theoryĤ 0 + λP 0Ĥ1P0 up to the timescale τ adj ∝ V /(V 0 L) 2 , with an error growing linearly in time and exhibiting a suppression ∝ 1/V , as predicted analytically in Appendix B 2.
Within state-of-the-art quantum-simulation setups, it is possible to set λ ∼ 0.1J and V /λ ∼ O(3 − 28) [17,20,26]. Restricting our dynamics within experimentally feasible evolution times t 100/J, we find in Fig. 4 that the staggered boson occupation is reliably reproduced by the adjusted gauge theory for V /J = 2 with demonstrates that LPG protection gives rise to an adjusted gauge theoryĤ0 + λP0Ĥ1P0 during all experimentally relevant evolution times already at V = 2J and λ = 0.1J, well within the accessible parameter range of state-of-the-art QSM devices.
λ/J = 0.1, i.e., well within the range of experimentally accessible parameters. This bodes well for ongoing efforts to stabilize local symmetries in quantum simulations of LGTs.
It is worth mentioning that in the (1 + 1)−D Z 2 LGT, the LPG term given in Eq. (5) is comprised of a singlebody term, which is straightforward to realize in QSM setups, and of a two-body term, which can be reliably engineered using density-density interactions that, for e.g., naturally arise in ultracold-atom setups, where they are readily tuned using Feshbach resonances [46], or in Rydberg arrays through dipole-dipole interactions [47].
We now show that the LPG protection scheme is not limited to strictly one-dimensional settings. To this end we consider a minimal Z 2 LGT on a small triangular lattice shown in Fig. 5, and described by the Hamiltonian with the constraint that there is only a single link at the common edge of the plaquettes P, i.e.,τ . Gauge invariance is encoded by two types of generators. The first isĜ j at a local constraint residing in only one plaquette and denoted by the matter site j and its neighboring links, which is identical to its counterpart in (1+1)−D. The second isĜ l,j at a local constraint shared by two plaquettes with eigenvalues g l,j = ±1, defined at a local constraint denoted by the matter site l and its neighboring link on one plaquette and the matter site j and its neighboring link on the second, along with the neighboring link common to both plaquettes. For clarity, we list them here explicitly:  To construct LPG terms with only up to two-body interactions for this system, we make a general ansatz for W j . This ansatz only contains couplings betweenτ x and n associated with a given vertex and treats all Z 2 electric field terms on equal footing. Allowing for arbitrary interaction strengths and requiring the eigenenergies of the constructed interaction term to collapse in a given target gauge sector yields possible solutions for the form ofŴ j .
Experimentally relevant local gauge-breaking errors for this model have been determined to be of the form with β 1 = 0.06 and β 2 = β 3 = β 4 = 0.01 [48], although we have checked that our qualitative picture remains the same for other values of β 1...4 . Furthermore, in order to further scrutinize the LPG protection in (2 + 1)−D, we have also included the experimentally very unlikely nonlocal error The LPG protection term used to suppress gauge violations due to these errors is described by with the noncompliant sequence c j ∈ {−1, 2, −3, 5}/5. As we will see, for this 2D geometry, even a noncompliant sequence renders LPG protection powerful enough to suppress such extreme nonlocal gauge-breaking errors. We prepare our initial state |ψ 0 in the target sector g tar 1 = g tar 6 = −1 and g tar 2,4 = g tar 3,5 = +1 (see Fig. 5), and quench with the faulty gauge theoryĤ =Ĥ 0 + λ(Ĥ 1 + H nloc 1 ) + VĤ W for λ/J = 0.01 and h/J = 0.54, although we have checked that our qualitative conclusions hold for other values of these parameters. We show the dynamics of the temporally averaged gauge violation, Eq. (8a), for several values of V (see legend) in Fig. 6(a). Remarkably, we see at sufficiently large V a suppression of the gauge violation, which enters a plateau at the timescale ∝ 1/V with a value ∝ λ 2 /V 2 even when the sequence is noncompliant and the gauge-breaking error includes strongly nonlocal terms. We have not been able to find a noncompliant sequence that achieves this for the (1 + 1)−D model; we speculate that in higher dimensions the higher connectivity may further restrict how gauge violations spread [49,50]. A scan of the long-time gauge violation as a function of J/V also shows two distinct regimes. For small enough V , the violation cannot be directly related to the value of V , and falls into an uncontrollederror regime. At sufficiently large V , we find that the gauge violation enters a controlled-error regime and be- Finally, we look in Fig. 6(c) at the temporally averaged absolute electric field where L = 5 is the number of links on the triangular lattice of Fig. 5. The qualitative picture is the same as for our other results, with LPG protection giving rise to an adjusted gauge theoryĤ adj =Ĥ 0 + λP 0 (Ĥ 1 +Ĥ nloc 1 )P 0 that faithfully reproduces the dynamics of E under the faulty gauge theory within an error upper bound ∝ tV 2 0 L 2 /V , i.e., up to a timescale τ adj = V /(V 0 L) 2 . The inset shows how the deviation of the dynamics under the faulty theory relative to that under the adjusted gauge theory scales ∝ 1/V and grows linearly in time, which is within our analytic predictions (see Sec. B 2).  LGT on a triangular lattice with gauge-breaking termsĤerr =Ĥ1 +Ĥ nloc 1 given in Eqs. (12) and (13). LPG protection with a noncompliant sequence, see Eq. (14), is used to stabilize gauge invariance. (a) Gauge-violation dynamics at sufficiently large V settles into a plateau ∝ λ 2 /V 2 that begins at a timescale ∝ 1/V and lasts up to all accessible times in ED. It is remarkable that this occurs despite the LPG-protection sequence being noncompliant, which seems unable to protect against such extremely nonlocal errors in (1 + 1)−D, see Fig. 2

(b). (b)
A two-regime picture emerges, with an uncontrolled long-time violation at small enough values of V , while at sufficiently large values of V the long-time violation enters a regime of controlled error ∝ λ 2 /V 2 . (c) LPG protection gives rise to the adjusted gauge theoryĤ adj =Ĥ0 + λP0ĤerrP0, which faithfully reproduces the dynamics of the electric field under the faulty theory up to a timescale τ adj ∝ V /(V0L) 2 . As predicted analytically, the corresponding error grows linear in time and is suppressed as ∝ 1/V .

V. SUMMARY AND OUTLOOK
We have introduced the concept of simplified local pseudogenerators (LPGs) that behave within the target sector identically to the actual generators of the gauge symmetry. This greatly simplifies experimental requirements compared to the implementation of the full generator to stabilize gauge invariance, as by construction the pseudogenerator has fewer particles per term than its full counterpart. We have demonstrated the efficacy of LPG protection in one and two spatial dimensions even under the severe case of nonlocal errors with support over the entire lattice, where it stabilized gauge invariance up to all accessible times in ED. We have also provided analytic predictions supporting these findings, and predicting the emergence of an adjusted gauge theory up to timescales polynomial in the LPG protection strength. Furthermore, we have shown that LPG protection provides robust stability of gauge invariance within experimentally accessible parameter regimes in current quantum simulators, which means LPGs should be a viable tool that can already be employed in such devices.
Even though we have focused in the main results on the Z 2 LGT, which has a discrete spectrum, we emphasize that LPG protection is general and can be employed for other Abelian gauge theories in any dimension. An immediate future direction arising from our work is extending LPG protection to non-Abelian LGTs, where the concept of linear protection does not work in general [38] specifically because the local generators do not commute. It would be interesting to investigate whether commuting LPGs can be contrived that act within the target sector as the actual generators of the non-Abelian gauge symmetry.

ACKNOWLEDGMENTS
We are grateful to Haifeng Lang for stimulating discussions, and for a meticulous reading of and valuable comments on our manuscript. This work is part of and supported by Provincia Autonoma di Trento, the ERC Start  In this Appendix, we provide numerical results supporting the conclusions of the main text for the (1+1)−D Z 2 LGT, by showcasing the efficacy of LPG protection compared to "full" energy-penalty protection, and by demonstrating its robustness to various initial conditions, model parameters, nonperturbative errors, and also to nonlocal gauge-breaking terms that simultaneously violate the global U(1) symmetry of boson-number conservation. Comparing LPG protection to full protection in the case of experimentally relevant local errors. The LPG protection sequence here is noncompliant, with cj = [6(−1) j + 5]/11, but as shown in the main text, this is sufficient to protect against local gauge errors. Here we have chosen λ/J = 0.01 and h/J = 0.54, but we have checked that our results hold for other values of these parameters. Even though the full protection shows slightly better protection quantitatively, the LPG protection exhibits similar qualitative performance, with the transition from an uncontrolled-error to a controlled-error ∝ λ 2 /V 2 regime occurring at V 5J compared to V 3J for the full protection. In the case of LPG protection, certain resonances between the target sector and other gauge-invariant sectors are not fully controlled at certain values of V within the controlled-error regime (see small "spike" at J/V ≈ 3.7 × 10 −3 ), but nevertheless they are still reliably suppressed.

Comparison with full gauge protection
It is interesting to compare the performance of the LPG protection of Eq. (2) with a noncompliant sequence to that of the full protection VĤ pen G = V j (1 − G j ), where here the target sector is chosen to be g tar j = +1 as in the main text. For this purpose, we scan the "infinite"-time gauge violation ε ∞ = 1 − lim t→∞ L j=1 ψ(t)|Ĝ j |ψ(t) /L as a function of J/V under LPG protection with a noncompliant sequence and under full protection, in the presence of the experimentally relevant local errors given in Eq. (6). In our ED calculations, "infinite" time is chosen to be numerically anywhere between t/J = 10 5 − 10 12 , as the result is qualitatively independent of the value of t 10 5 /J. The results are shown in Fig. 7, where the LPG protection ex-hibits qualitatively similar efficacy to the full protection. Indeed, in both cases we see a clear transition from an uncontrolled-error to a controlled-error regime where the steady-state value of the gauge violation scales ∝ λ 2 /V 2 . We have chosen here the experimentally feasible noncompliant sequence c j = [6(−1) j + 5]/11 for the LPG protection. Unlike the case of a compliant sequence, this does not isolate the target sector from all other gaugeinvariant sectors. This leads to imperfections at a few values of V in the behavior of the infinite-time violation within the controlled-error regime, albeit the suppression of the violation is still remarkably reliable at these values as well. As such, this is quite encouraging news for ongoing experiments that the LPG protection with the experimentally feasible noncompliant periodic sequence can perform qualitatively as well as the full protection.   . Initial states used in our ED calculations. Circles represent matter sites, where red circles denote single hard-core boson occupation and white circles are empty matter sites. The yellow arrows on the links between matter sites denote the eigenvalue of the electric field τ x j,j+1 as ±1 when pointing right (left). All three initial states are in the target sector g tar j = +1, ∀j. In the main text, we have focused on |ψ0 , although LPG protection offers reliable stabilization of gauge invariance independently of the initial state, as shown in Fig. 9(a).

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In the main text, we have focused on the staggeredmatter initial state |ψ 0 shown in Fig. 8. However, LPG protection works for generic initial states within a gauge-invariant sector. In keeping with experimental relevance, we quench different initial product states |ψ 0 and |ψ 0 , shown in Fig. 8, with the faulty Hamiltonian H =Ĥ 0 + λĤ 1 + V jŴ j [6(−1) j + 5]/11. We look at the long-time gauge violation as a function of J/V , which is displayed in Fig. 9(a). The conclusion is qualitatively and, more or less, quantitatively the same between the three considered initial states, with a clear transition from an uncontrolled-error regime at small enough V , to a controlled-error ∝ λ 2 /V 2 regime at sufficiently large V .
This robustness to initial conditions is also present when it comes to different values of the model parameters. Fixing λ/J = 0.01, and quenching |ψ 0 witĥ H =Ĥ 0 +λĤ 1 +V jŴ j [6(−1) j +5]/11, we find that the long-time violation exhibits the same qualitative transition between uncontrolled and controlled error ∝ λ 2 /V 2 as a function of J/V regardless of the value of h/J, as shown in Fig. 9(b). Note that the (1 + 1)−D Z 2 LGT has a phase transition from a deconfined phase at h/J = 0 to a confined phase at h/J > 0 [40], but LPG protection works efficiently in either phase, at least for the system sizes considered.
In the main text, we have focused on perturbative errors (λ/J < 1), but LPG protection works also for nonperturbative errors, as demonstrated in Fig. 9(c).
Here we again quench |ψ 0 withĤ =Ĥ 0 + λĤ 1 + V jŴ j [6(−1) j +5]/11, and plot the infinite-time gauge violation as a function of J/V for various values of λ/J, including the nonperturbative regime λ = J. The qualitative behavior of a transition between an uncontrollederror regime for small enough V to one with a controlled violation ∝ λ 2 /V 2 at sufficiently large V persists regardless of the value of λ. Naturally, the larger λ is, = ξ=±1 j 1 + ξτ z j,j+1 1 + ξ âj +â † j , which also violates boson-number conservation. Here we restrict to a system of only L = 4 matter sites to reduce numerical overhead in light of the large evolution times we access. The initial state is the staggeredmatter product state |ψ0 of Fig. 8 but with only L = 4 matter sites and L = 4 gauge links. The qualitative picture is identical to that of Fig. 2. the larger the value of the minimal V required to be in the controlled-error regime. However, we note that typical error strengths in modern QSM setups are usually λ/J < 1 [17].

Results with gauge-breaking errors that do not conserve boson number
In the main text, we have rigorously tested the LPG protection with a compliant sequence against local and nonlocal errors, where both conserve boson number, as does the ideal gauge theoryĤ 0 . Even thoughĤ 1 naturally hosts a global U(1) symmetry as derived in Ref. [17], we have chosenĤ nloc 1 in Eq. (7) to also conserve boson number in order to reduce numerical overhead and reach L = 6 matter sites in our ED calculations within the bosonic half-filling sector for the large evolution times we access. However, our conclusions are independent of whether or not the global U(1) symmetry associated with boson-number conservation is preserved. We test this assertion by modifying the nonlocal gauge-breaking error into the form which restricts us numerically to L = 4 matter sites.
We quench the corresponding staggered-matter initial state |ψ 0 by the faulty theoryĤ =Ĥ 0 + λ(Ĥ 1 + H nloc 1 ) + V j c jŴj with the compliant sequence c j ∈ {−115, 116, −118, 122}/122. Even though we set λ/J = 0.01 and h/J = 0.54, we have checked that our results hold for other values of these parameters. In Fig. 10(a), we show the ensuing dynamics of the gauge violation. The qualitative behavior is identical to the case of bosonnumber-conserving nonlocal errors discussed in the main text. The gauge violation grows initially ∝ λ 2 t 2 , in agreement with time-dependent perturbation theory [39], before settling into a plateau. The latter shows no direct relation to the protection strength when V is too small. However, at sufficiently large V , the violation plateau begins at a timescale ∝ 1/V and takes on a value ∝ λ 2 /V 2 . This behavior is further confirmed in Fig. 10(b), which shows the long-time violation as a function of J/V . At values of V that are too small, the violation is uncontrolled, whereas at sufficiently large V , the long-time violation is controlled and scales ∝ λ 2 /V 2 . Note once again how the noncompliant sequence c j = [6(−1) j + 5]/11 is not sufficient to achieve reliable gauge invariance in the case of nonlocal errors here, regardless of how large V is. Instead, it seems to always remain above a certain minimal value.
The dynamics of the staggered boson number, Eq. (8b), is shown in Fig. 10(c), and the qualitative picture is the same as that of a nonlocal error that conserves boson number, see Fig. 2(c). Indeed, we find that an adjusted gauge theoryĤ adj =Ĥ 0 + λP 0 (Ĥ 1 +Ĥ nloc 1 )P 0 faithfully reproduces the dynamics of the local observable up to a timescale τ adj ∝ V /(V 0 L) 2 , with an error that is suppressed ∝ 1/V and grows linearly in time (see inset of Fig. 10(c)), as predicted analytically (see Sec. B 2).

Local-error coefficients αm
The coefficients α 1...4 of the local error term, Eq. (6), in the 1D Z 2 LGT are inspired from an extended version of building-block errors arising in the construction of the effective Floquet Hamiltonian in the experiment of Ref. [17]. Explicitly, they read where J q (χ) is the Bessel function of the first kind and order q, and the variable χ is a dimensionless driving parameter that is set to the experimentally relevant [17] value χ = 1.84 for the related results of this work, although we have checked that our qualitative picture is independent of the choice of χ. We have also used K(χ) as a nonzero factor enforcing and that, in addition to the compliance condition, we fulfill the following two conditions: Once these conditions are satisfied, then starting in any initial state |ψ 0 within the target gauge sector g j = g tar j , ∀j, will give rise to dynamics where the gauge violation remains bounded from above as up to a timescale τ ren ∝ V −1 0 eṼ /V0 , whereĜ = j (Ĝ j − g tar j ) 2 /L is the gauge-violation operator, and K is a model-parameter-dependent term, but which is independent ofṼ and system size. Details of this proof in the context of gauge protection have been outlined in Ref. [36]. The latter work deals specifically with a protection term linear in the full gen-eratorĜ j with a rational compliant sequence. However, since the LPG protectionṼĤ pro = VĤ W satisfies the condition of compliance, and as Eqs. (B7) are also satisfied, the derivation of Ref. [36] applies in full also here, and, as such, we refer the interested reader there for its details.
Nevertheless, a few comments are in order. Even though the timescale τ ren ∝ V −1 0 eṼ /V0 may not appear directly volume-dependent, a larger V is required with larger system size L in order to achieve a given level of reliability. This becomes clear when looking at Eq. (B3). As mentioned, c j form a compliant sequence of rational numbers normalized such that max j {|c j |} = 1.
Let us call f j the set of smallest integers such that f j / max m {|f m |} = c j . As such, we can rewritẽ meaning thatṼ = V / max j {|f j |} is sufficient to make the spectrum ofĤ pro integer. Assuming that a given value ofṼ brings about a certain level of gauge-error suppression, a larger system size will lead to a larger max j {|f j |}, meaning that V has to become larger in order to retain the same value ofṼ . Naturally, this becomes intractable in the thermodynamic limit. However, we also see in our ED calculations that even the noncompliant sequence, which does not grow with system size, achieves reliable protection for local errors up to indefinite times, even though we cannot analytically predict this. The nonlocal errors we have considered in this work are very drastic, and only such errors require the compliant sequence. Another point worth mentioning is that our analytic arguments for the compliant sequence strictly only apply for local errors, and extreme nonlocal errors with support over the whole lattice in the thermodynamic limit are not within the operator algebras we have defined. However, as we see in our numerical results, LPG protection with a compliant sequence still suppresses gauge violations up to indefinite times even in the presence of such extreme errors on a finite system, and this is within the ARHH framework but cannot be guaranteed in the thermodynamic limit. Furthermore, LPG protection with a noncompliant sequence, which does not fulfill all the conditions of the ARHH formalism, still offers stable gauge invariance up to indefinite times when gauge-breaking errors are local. This cannot be guaranteed by the ARHH framework, but it is not ruled out either. Indeed, this formalism gives a guaranteed minimal (worst-case scenario) timescale exponential in V up to which gauge invariance is stabilized in the presence of errors with a finite spatial support (that does not grow with system size) given that the compliance condition and Eqs. (B7) are satisfied, but it does not forbid stable gauge invariance when any of these conditions are not strictly met.
Finally, it is to be noted that obtaining a closed form of the renormalized gauge theory is generically difficult. Moreover, we cannot numerically test how faithfully such a renormalized gauge theory reproduces the LPG-protected dynamics under the faulty theory, as this would require reaching exponentially long times within systems in the thermodynamic limit, for which no general techniques exist.

Adjusted gauge theory
It is useful for ongoing experiments to be able to have an exact form of an emergent gauge theory in the wake of a quench with the faulty gauge theoryĤ = H 0 + λĤ err + VĤ W , whereĤ err =Ĥ 1 + ηĤ nloc 1 with η = 0 or 1. One can show through the quantum Zeno effect (QZE) [52][53][54][55] in the case of LPG protection with a compliant or suitably chosen noncompliant sequence at sufficiently large protection strength V , that an adjusted gauge theoryĤ adj =Ĥ 0 + λP 0ĤerrP0 arises up to a timescale τ adj ∝ V /(V 0 L) 2 [36]. Specifically, at sufficiently large V the dynamics underĤ is restricted to the "decoherence-free" subspace ofĤ W . In the large-V limit, the time-evolution operator reads [52][53][54][55] lim V →∞ e −iĤt = e −i[VĤ W + mP m (Ĥ0+λĤerr)Pm]t , (B10) up to a residual additive term ∝ V 2 0 L 2 t/V . We now consider the conditions for which the QZE can promise reliable stabilization of gauge invariance in the dynamics up to the resulting timescale τ adj ∝ V /(V 0 L) 2 .

a.ĤW is nondegenerate
In this case, gauge invariance is stable for a genericĤ err so long as the coefficients c j are sufficiently incommen-surate. In other words, given any two pseudo superselection sectors w = (w 1 , w 2 , . . .) and w = (w 1 , w 2 , . . .) of W j , ∀j, then the sequence must satisfy j c j (w j − w j ) = 0. This condition is readily satisfied when c j is a sequence of random or irrational numbers, for example.
We note here that a pseudo superselection sector w ofĤ W is not necessarily gauge-invariant except when it coincides with the target sector, i.e., when w = g tar = (g tar 1 , g tar 2 , . . .).

b.ĤW is degenerate
In the case the termĤ 0 + λĤ err does not lift the degeneracy ofĤ W in first-order perturbation theory, then we can utilize that P m Ĥ 0 + λĤ err P m = w,w ∈DmP where D m is the set of all pseudo superselection sectors w ofŴ j such thatĤ W |ψ = m |ψ , ∀ |ψ ∈ w, andP w is the projector onto the pseudo superselection sector w. Since we prepare our initial state in the target sector w = (g tar 1 , g tar 2 , . . .), gauge-noninvariant processes driving the dynamics out of this sector will be suppressed in the time evolution for large V , because different sectors do not couple in the QZE regime as evidenced in Eq. (B12), and this is precisely because second-order perturbation theory is beyond the timescale of QZE protection.
As mentioned, LPG protection can be shown to stabilize gauge invariance for an adequately chosen, yet not necessarily compliant, sequence c j through an effective QZE behavior up to a residual additive term ∝ t(V 0 L) 2 /V . In particular, the latter can be formulated as e −iĤt − e −i[VĤ W + mP m(Ĥ0 +λĤerr)Pm]t ≤ Q ∝ tV 2 0 L 2 V . (B13) Projecting onto the target sector, this becomes where here we have utilized the fact that in the target sector, where we initialize our system,Ĥ 0 andP 0Ĥ0P0 drive identical dynamics, and so the adjusted gauge the-oryĤ adj =Ĥ 0 + λP 0ĤerrP0 has naturally appeared in our formalism. It is to be noted, however, that the adjusted gauge theory can also be derived through the formalism of constrained quantum dynamics in the case of full protection [56,57]. As we will show in the following, the inequality (B14) translates to the dynamics of a local observableÔ under the faulty theory being gauge-invariant up to an error upper bound ∝ t(V 0 L) 2 /V . The dynamics of a local ob-servableÔ under the faulty theoryĤ deviates from that under the adjusted gauge theory as ψ(t)| e iĤtÔ e −iĤt − e iĤ adj tÔ e −iĤ adj t |ψ(t) ≤ P 0 e iĤtÔ e −iĤt − e iĤ adj tÔ e −iĤ adj t P 0 = 1 2 P 0 e iĤt − e iĤ adj t Ô e −iĤt + e iĤtÔ e −iĤt − e −iĤ adj t + e iĤt − e iĤ adj t Ô e −iĤ adj t + e iĤ adj tÔ e −iĤt − e −iĤ adj t P 0 ≤Q ∝ 2 tV 2 0 L 2 V .
As such, we have proven that the adjusted gauge theorŷ H adj faithfully reproduces the dynamics of a local observ-ableÔ under the faulty theoryĤ with large V up to a timescale τ adj ∝ V /(V 0 L) 2 . This is very promising for ongoing QSM setups implementing LGTs, since it means that an emergent exact gauge theory can still be derived in closed form and realized experimentally, allowing for a controlled assessment of the fidelity of the realization.