Disordered Lieb-Robinson bounds in one dimension

By tightening the conventional Lieb-Robinson bounds to better handle systems which lack translation invariance, we determine the extent to which"weak links"suppress operator growth in disordered one-dimensional spin chains. In particular, we prove that ballistic growth is impossible when the distribution of coupling strengths $\mu(J)$ has a sufficiently heavy tail at small $J$, and identify the correct dynamical exponent to use instead. Furthermore, through a detailed analysis of the special case in which the couplings are genuinely random and independent, we find that the standard formulation of Lieb-Robinson bounds is insufficient to capture the complexity of the dynamics -- we must distinguish between bounds which hold for all sites of the chain and bounds which hold for a subsequence of sites, and we show by explicit example that these two can have dramatically different behaviors. All the same, our result for the dynamical exponent is tight, in that we prove by counterexample that there cannot exist any Lieb-Robinson bound with a smaller exponent. We close by discussing the implications of our results, both major and minor, for numerous applications ranging from quench dynamics to the structure of ground states.


I. INTRODUCTION
Lieb-Robinson (LR) bounds, named after Ref. [1], have proven to be a valuable mathematical tool for many-body physics and quantum information theory [2][3][4][5][6][7][8][9][10][11][12][13][14]. Conceptually, they provide hard constraints on the extent to which correlations of any type can spread through a many-body lattice system (broadly termed "operator spreading"). Numerous applications, from rigorous results on many-body ground states to lower bounds on the runtime of quantum protocols, can be found in the above references.
Given their utility, LR bounds have been both generalized and specialized in multiple ways: leveraging the commutativity and graph structure of interactions [15][16][17], allowing for long-range interactions [18][19][20][21][22], and considering open systems [23][24][25][26], to name a few. However, one ingredient that has been noticeably absent is disorder, or more generally a lack of translation invariance. To be fair, the conventional bounds do allow for nontranslation-invariant interactions, but there have been no studies assessing the tightness of the resulting bounds (and we shall find that they are far from tight). Certain works have considered related topics -Refs. [27,28] study the effects of disordered local terms (albeit with uniform interactions), and Ref. [29] derives bounds for random Hamiltonians with statistical translation invariance 1 . Others have calculated bounds for specific (often free-fermion-integrable) systems [30][31][32][33]. While interesting on their own merits, none of these quite address the question with which we concern ourselves hereto what extent is operator spreading (as constrained by LR bounds) necessarily suppressed by non-translationinvariant interactions?
Non-translation-invariant systems are known to exhibit a variety of phenomena not found in their translation-invariant counterparts. Examples include spin glass phases (both static and dynamic) [34,35], localization [36,37], and Griffiths effects [38]. Disordered fermionic models have been used to help understand quantum dots and strongly correlated metals as well [39][40][41]. Of particular (and somewhat controversial) interest is the phenomenon of many-body localization (MBL) [42][43][44][45], whose existence remains under debate [46][47][48][49][50]. Given the challenges inherent in studying not only MBL but disordered quantum many-body systems in general, it is all the more important to identify rigorous constraints such as those which LR bounds supply (although to be clear, our results and LR bounds in general are not strong enough to resolve the questions surrounding MBL, as we explain in Sec. II).
In the present work, we initiate the study of nontranslation-invariant LR bounds by considering arguably the simplest (but still quite rich) situation: onedimensional chains with nearest-neighbor interactions. An essential feature of such systems is the importance of "weak links", i.e., atypically weak interactions. We develop the machinery for analyzing these systems, prove that the LR bounds thus obtained are in a certain sense optimal, and use the results to place constraints on various physical properties and processes.
Sec. II summarizes our results in conceptual terms. We have aimed to make it sufficiently self-contained so that readers primarily interested in our conclusions and willing to forgo the derivations should be able to read Sec. II on its own. Sec. III then gives precise definitions arXiv:2208.05509v1 [cond-mat.dis-nn] 10 Aug 2022 of all quantities involved in our analysis. Sec. IV derives LR bounds for general non-translation-invariant systems, and Sec. V specializes to the case of random couplings. Sec. VI lastly discusses some implications of our results, in particular related to: quench dynamics, topological order, heating rates, ground state correlations, and machine learning of local observables.

II. SUMMARY OF RESULTS
We determine the extent to which operator spreading in 1D nearest-neighbor chains is suppressed by weak links. To be precise, we consider arbitrary d-state degrees of freedom ("qudits") interacting via any Hamiltonian of the form H(t) = l H l (t), where the sum is over links of the chain and H l (t) acts on the two sites connected by link l (although our analysis in fact allows for arbitrary local terms as well 2 ). See Fig. 1. Unlike previous works, we assume that H l (t) ≤ J l where J l varies from link to link ( · denotes the operator norm throughout). The "weak" links are those on which J l is much smaller than the typical value.
For arbitrary local operators A 0 and B r supported on sites 0 and r respectively, and with A t 0 denoting the evolution of A 0 over time t, our goal is to bound the quantity [A t 0 , B r ] as tightly as possible, making use of the weak links in the set {J l }. We particularly focus on the asymptotic behavior at large r and t. The commutator [A t 0 , B r ] is a standard and useful measure of operator spreading (as we describe in Sec. III B), but it is by no means the only such measure. In fact, most of our calculations will involve a different quantity from which bounds on the commutator easily follow.
LR bounds are best viewed as applying to families of Hamiltonians. For each set of couplings {J l }, we use H J to denote the set of all Hamiltonians as described above 2 Our results actually apply to any Hamiltonian of the form H(t) = l∈Λ H l (t) + i∈Ω h i (t), for arbitrary local terms h i (t). The local terms can trivially be accounted for by first passing to the interaction picture with respect to them -the terms H l (t) in this new frame have both the same support and the same norm as originally, hence our analysis applies equally well using them. That said, the behavior of any specific system clearly can depend strongly on the local terms. The possibility of a Hamiltonian causing operator spreading at dynamical exponent z, i.e., in a time growing asymptotically as t(r) ∝ r z to reach a large distance r. Horizontal axis is the exponent α characterizing the number of weak links in the couplings {J l } -the fraction of links between sites 0 and r having J l ≤ J is assumed to go as J α at small J and large r. Vertical axis is z. Blue region is where no Hamiltonian with such α can reach any large-distance site at such z. Red region is where a Hamiltonian exists with such α that reaches every large-distance site at (or faster than) such z. The boundary between the two is given by zc(α) = max [1/α, 1]. In the special case of independent random couplings, the α > 1 portion of the boundary is included in the red region, while the α < 1 portion (shown in purple) is where a Hamiltonian exists that reaches a subsequence of sites faster than z but where no Hamiltonian can reach every site at such z. Lastly, at point (α, z) = (1, 1), no Hamiltonian can reach every site ballistically but it is unknown whether any Hamiltonian can reach some sites ballistically. consistent with {J l } (i.e., H l (t) ≤ J l for all links l at all times t). Our bounds are uniform among H ∈ H J , in the sense that they make no reference to any property of the Hamiltonian beyond the couplings {J l }. Thus while the results are extremely general, they may not be very tight for one specific system. The "tightness" of LR bounds referred to in this paper is instead the existence of some H ∈ H J which saturates the bound.
As a consequence of their generality, LR bounds have little to say regarding, e.g., the existence of MBL -the complete lack of operator spreading in strongly disordered time-independent systems [42][43][44][45]. The same applies to other phenomena involving slow dynamics under a fixed Hamiltonian, such as activated processes in spin glasses [51][52][53]. Those H ∈ H J that saturate our bounds will instead tend to be highly time-dependent Hamiltonians, specifically designed to transmit information and better viewed as quantum circuits. These are the types of systems for which LR bounds give a reasonably full picture. Our first result, which holds for any possible set of couplings {J l }, is an improvement on the conventional LR bound. Whereas the standard analysis 3 leads to the result we show that one further has where the minimization is over all subsets λ of the links between sites 0 and r (|λ| denotes the size of the subset). Even though much of what follows will be concerned with the asymptotic behavior, Eq. (2) holds for all r and t. In these equations and throughout the entire paper, we use C to denote any constant which does not depend on r or t and whose precise value is irrelevant to our conclusions. Its value will often change between expressions (and primes/subscripts will differentiate such constants within the same expression). We next derive more explicit bounds by considering the "empirical distribution" µ r (J), defined (for a given set {J l }) as the fraction of links between sites 0 and r having J l ≤ J: where δ J l ≤J ≡ 1 if J l ≤ J and 0 otherwise. We assume that µ r (J) converges as r → ∞ (in a sense defined in Sec. IV C) to a function µ(J), and that the latter behaves as a power law at small J: µ(J) ∼ µ 0 J α with α > 0. The exponent α characterizes the prevalence of weak links, with smaller α implying more weak links. Our analysis in fact applies for more general forms of µ(J) (as we discuss in Sec. IV C), but the power-law behavior is particularly convenient and representative. Note that the convergence of µ r (J) to µ(J) does not assume anything regarding the arrangement of weak links in space -many of our results will hold regardless of where the weak links are located. An essential feature of an LR bound is the shape of the "front", i.e., the spacetime curve t(r) that separates the region in which the bound is small from that in which the bound is large (and thus vacuous). The dynamical exponent z and generalized LR velocity v are defined by the asymptotic behavior vt(r) ∼ r z at large r. Keep in mind that v has units of a genuine velocity only when z = 1 -we will stick to the term "generalized velocity" 4 for z = 1. Whereas the conventional LR bound, Eq. (1), has a ballistic front (z = 1) for all α > 0, we show that the improved bound, Eq. (2), instead has The curve z c (α) is sketched in Fig. 2.
One consequence of the above is that there cannot exist any H ∈ H J for which the operator-spreading front grows with a dynamical exponent z < z c (α). In particular, it is impossible to have a ballistic front if α < 1. We have shaded this region blue in Fig. 2 and labelled it as "asymptotically unattainable on any site" -it is impossible to construct a Hamiltonian having that value of α which, at large distances, reaches any site at that value of z.
On the other hand, we also identify an H ∈ H J whose front grows faster than any z > z c (α), again requiring only that µ r (J) → µ(J) in a suitably strong sense. The Hamiltonian is rather straightforward, consisting simply of a sequential series of SWAP gates as shown in Fig. 3. Once all gates have been applied, A t 0 is supported on site r and thus will generically fail to commute (by an r-independent amount) with B r . In order to satisfy H l (t) ≤ J l , the total runtime of the circuit is proportional to r l=1 1/J l -analysis of this sum gives the behavior of the front.
We have shaded the region z > z c (α) red in Fig. 2 and labelled it as "asymptotically attainable on all sites"it is possible (and we do so) to construct a Hamiltonian having that value of α which reaches every large-distance site at that value of z. In this sense, Eq. (4) is the dynamical exponent for a given α, i.e., Eq. (4) is tight.

4
It is rather striking that our result for z c (α) agrees exactly with the value predicted on physical grounds in Ref. [54]. The results of Ref. [54] are based on a coarsegrained description of 1D disordered systems, in which it is postulated that a region l can be characterized by an effective "growth rate" Γ l setting the rate at which operators spread across the region. The authors assume that Γ l is power-law-distributed with exponent α (although they work with the probability density having exponent α − 1), and ultimately deduce precisely Eq. (4). Our results are somewhat more limited in scope 5 , since we consider microscopic weak links rather than effective weak links emerging at long wavelengths, but this nonetheless provides a rigorous foundation for many of the concepts at work in Ref. [54]. It would be of great interest to investigate whether the other phenomena discussed in Ref. [54], such as entanglement growth and transport, can be placed on similar rigorous grounds.
Returning to Fig. 2, the situation becomes much more complicated on the boundary z = z c (α), i.e., when considering dynamics on the scale set by z c (α), both for our LR bounds and for our example Hamiltonians. We demonstrate this by a detailed analysis of the special case in which the couplings are drawn independently from a literal probability distribution µ(J). After proving that µ r (J) → µ(J) in the required sense with probability 1, and thus that the portion of Fig. 2 away from z c (α) does indeed hold, we find that the generic behavior on the boundary cannot be described by a single bound -we must introduce (at least) two types of LR bounds: • "Almost-always" (a.a.) bounds are those that hold for all sites r, excepting at most a finite number of sites. In other words, there exists a distance R such that the bound holds for all r > R.
• "Infinitely-often" (i.o.) bounds are those that hold for an infinite subsequence of sites {r k } but need not hold outside of those sites. In other words, for any distance R there exists some site r > R which is subject to the bound.
One can imagine situations in which either of the above two bounds is more relevant. For example, suppose that Alice is manipulating one site of a spin chain and wants to be confident that her actions do not disturb distant regions in a certain amount of time. In this case, a.a. bounds provide the desired guarantee. On the other hand, suppose that Bob intends to transmit a signal along the spin chain. If it is important that his signal reach every site faster than a certain rate, then i.o. bounds place the heaviest restrictions on what can be achieved. 5 In a different sense, our results are actually more general than those of Ref.
[54] -we derive bounds for an individual realization of couplings, and thus are agnostic to its origin. For example, many of our results (namely those in Sec. IV) hold equally well for quasiperiodic systems as for those with quenched randomness.
In our case (still at z = z c (α) and still assuming independent random couplings), we find different behaviors depending on how α compares to 1. If α > 1, the results are straightforward: our example Hamiltonian has a ballistic front that spreads to every site with finite velocity (note that z c (α) = 1 for α > 1). We have included this portion of the boundary with the red region in Fig. 2 to indicate that it is also "asymptotically attainable on all sites" (albeit with a maximum allowed velocity).
However, we show that if α < 1, then an i.o. bound having z = z c (α) and arbitrarily small (generalized) velocity holds, while concurrently, our example Hamiltonian does reach a subsequence {r k } asymptotically faster than z c (α) (hence no such a.a. bound can hold). Both statements hold with probability 1. Thus it is impossible to have a front which reaches every site at dynamical exponent z c (α), but it is possible (and we do so) to construct a Hamiltonian which reaches a subsequence of sites at z c (α). We have drawn this portion of the boundary purple in Fig. 2 and accordingly labelled it "asymptotically attainable on some but not all sites". Keep in mind that we have proven this final statement only for the special case of independent random couplings. Nonetheless, it is a highly non-trivial example that makes clear the importance of distinguishing between a.a. and i.o. bounds.
Interestingly, the lone point (α, z) = (1, 1) is the only portion of Fig. 2 in which we have been unable to give a definite answer. An i.o. bound with vanishing velocity still holds, but the front in our example Hamiltonian is now sub-ballistic for every site. It may be that a more complicated Hamiltonian exists which does reach a subsequence {r k } at finite velocity, yet it may instead be that a more sophisticated mathematical technique can produce an a.a. bound with vanishing velocity. Further investigation is clearly warranted.
Lastly, we discuss the implications of our results for various applications. The LR bounds themselves have physical content -the statement derived here that ballistic spreading is impossible for α < 1 can be considered an application in and of itself. That said, our results have broader consequences as well. Applications can roughly be grouped into two classes: those that follow from the existence of the front, and those that follow from the "tail" (i.e., the rapid decay of the LR bound at large distances outside the front). Since we obtain a significantly altered front for α < 1, our results have a qualitative impact on the former class. However, while we do find a more complicated tail than in the conventional bound, the behavior at the largest distances turns out to be unmodified, and thus our results have only a minor impact on the latter class.
In the remainder of the paper, we make precise and prove the above statements. Sec. III establishes notation and the formalism within which we work. Sec. IV, after reviewing the conventional LR bound, derives Eq. (2) for generic non-translation-invariant systems, and then makes use of the empirical distribution µ r (J) to derive Fig. 2. Sec. V considers the case of independent random couplings {J l } in more detail, first proving that the requirements of the preceding section are met and then examining behavior on the boundary z c (α), with particular focus on the distinction between a.a. and i.o. behavior. Sec. VI lastly discusses the consequences of the above for various applications.

A. Geometry
In this work, we consider an N -site lattice in 1D, where each site hosts a d-state degree of freedom. In other words, the Hilbert space is a tensor product of N local d-dimensional Hilbert spaces. Let Ω denote the set of all N sites and Λ denote the set of all N − 1 links. Here we consider only nearest-neighbor Hamiltonians on this lattice, i.e., Hamiltonians of the form H(t) = l∈Λ H l (t), where H l (t) is supported only on the sites connected by link l (although as noted above in footnote 2, our results hold for Hamiltonians with arbitrary local terms as well). These features are illustrated in Fig. 1.
Given a set of couplings {J l } l∈Λ , let H J be the family of all nearest-neighbor Hamiltonians for which We even allow for a non-vanishing fraction of {J l } to be infinite, meaning that there is no restriction on the corresponding terms. Pick an operator of interest A 0 supported only 6 on site 0, and similarly B r on site r (the sites can be either well inside the chain or near the edges -our bounds apply regardless). For any H ∈ H J , let A t 0 be the time evolution of A 0 , i.e., the solution to 7 Even though A 0 is supported on site 0, A t 0 will (barring trivial cases) be supported throughout the entire chain 6 In fact, it will be clear from our proof technique that the support of A 0 need not be solely site 0 but can include all sites to the left of 0. Similarly, the support of Br can include all sites to the right of r. In other words, our bounds are "many-to-many". 7 Strictly speaking, Eq. (6) is not the Heisenberg equation of motion for an arbitrary time-dependent Hamiltonian H(t).
The Heisenberg-picture operator A t 0 is defined by the relation The solution can be written |Ψ t = T e −i dsH(s) |Ψ , where the operator on the right-hand side is a time-ordered exponential -earlier times appear to the right. Thus in the expression Ψ t |A 0 |Ψ t which defines A t 0 , earlier times appear outside later times. The differential equation that correctly has A t 0 as its solution is ∂sA s the commutator must be applied with later times first and earlier times last. Despite all of this, however, our analysis applies equally well with H(t − s) in place of H(s). Since the latter is less burdensome notation-wise, we shall use Eq. (6) without further comment.
for any t > 0. The purpose of LR bounds is to place a bound on the quantity that holds uniformly for all H ∈ H J . Since only the "portion" of A t 0 which acts non-trivially on site r can possibly fail to commute with B r (see Eq. (13) below), LR bounds considered as functions of r and t constrain the extent to which local operators "spread" throughout the system.
It is important to note that the Hamiltonians in H J can have arbitrary time dependence, as long as Eq. (5) is obeyed at all times. Thus it is perhaps more informative to refer to any individual H ∈ H J as a "protocol", since it can for example be a quantum circuit designed to perform a specific task. As discussed in Sec. II, this distinction sheds light on the limitations of LR bounds.
The choice to use the operator norm in Eq. (7) has long been standard, as it enters naturally in many applications [1,4,5,8]. There are situations in which alternative norms, in particular the Frobenius norm defined may be more relevant and might behave quite differently [55]. However, the operator norm is itself an upper bound on a wide family of norms including Frobenius (see App. A). Furthermore, the transfer protocol shown in Fig. 3 leads to a commutator [A t 0 , B r ] which is O(1) using any of these norms. Thus we shall exclusively consider the operator norm in this work, and the bounds obtained are automatically tight (at least regarding the dynamical exponent) for the other norms as well.

B. Basis strings
The set of Hermitian operators acting on the Hilbert space is itself a real vector space, and thus we can express any operator as a linear combination of certain basis operators. First consider a single site i and pick a Hermitian basis {X . For the entire chain, we use the tensor product basis {X (ν) }: where ν ≡ (ν 1 , · · · , ν N ). We assume (without loss of generality) that {X (νi) i } is chosen to be orthonormal with respect to the trace product, meaning that the tensor product basis is orthonormal as well: 6 We also take X (0) i to be the identity I i . Beyond this, any choice of basis will work equally well.
We shall often refer to the basis elements as "strings", and define the support of a string to be the set of sites on which it does not have the identity: An important superoperator acting on the space of Hermitian operators is that which projects onto basis strings whose supports contain site i, i.e., strings that act nontrivially on site i. We denote this superoperator by P i : Similarly, for any subset of sites ω ⊆ Ω, we define P ω to project onto basis strings which act non-trivially somewhere (not necessarily everywhere) within ω. A useful inequality (see App. A) is that for any ω and any operator O, Also note that , and so for D(r, t) in Eq. (7), we have the trivial bound In what follows, we shall focus on bounding P r A t 0 , with a bound on D(r, t) following automatically by Eq. (13).
The next important superoperator is the generator of time evolution under H(t): and so (Eq. (6)) We will also need the generator corresponding to a subset of terms in the Hamiltonian. For any subset of links λ ⊆ Λ, define Clearly Denote the evolution superoperator itself by U(t), i.e., A t 0 ≡ U(t)A 0 (and define U λ (t) analogously). We can express the action of U(t) (and U λ (t)) in terms of a timeordered exponential: where T denotes time-ordering (note the ordering in the bottom line of Eq. (17) -earlier times appear inside later times). Note that Visual interpretation of Eq. (18). ω consists of the sites in red, Ω/ω the sites in blue. λ consists of the dashed links, Λ/λ the solid links. Evolution under only H Λ/λ cannot transform a string having the identity on ω into one with a non-identity element, nor vice-versa.
Lastly, suppose λ ⊆ Λ contains every link which connects a subset of sites ω to its complement Ω/ω (see Fig. 4). It is intuitively clear that evolution under L Λ/λ (t) alone cannot transform a basis operator which acts trivially on ω into one which acts non-trivially, or vice versa. Put precisely, We give a proof of Eq. (18) in App. B.

C. Types of Lieb-Robinson bounds
To reiterate, the purpose of LR bounds is to place an upper limit on P r A t 0 (and thus D(r, t)) which applies to every H ∈ H J simultaneously. The bounds we construct will be of the form where γ > 0 and the function f (x) decays to zero as x → ∞ and remains finite (or even diverges) as x → −∞. A simple and common example is f (x) = exp [−κx] for some κ > 0. Although we shall not indicate so explicitly, note that all quantities here are functions of the couplings In the large-r and large-t limit, one identifies two important features from Eq. (19): • There is a "front" defined by vt = r z . For vt > r z , the right-hand side of Eq. (19) is large (and thus the bound is vacuous), while for vt < r z , P r A 0 (t) must be small. Thus the spacetime curve vt = r z constitutes an envelope that constrains the expansion of A t 0 . We refer to z as the "dynamical exponent" and v as the "generalized velocity" of the bound (only when z = 1 will we speak simply of the "velocity"). Of particular interest are the largest value of z and smallest value of v for which a bound as in Eq. (19) holds.
• At fixed t, there is the "tail" behavior as r → ∞, characterized by the exponent γ and the large-x behavior of f (x) (e.g., exponential or power-law). Even though P r A t 0 need never be identically zero, the tail describes how rapidly it must decay at large distances. As mentioned in Sec. II, we shall find it necessary to distinguish between different types of LR bounds, based on whether a statement such as Eq. (19) holds for all sites or merely a subsequence of sites: • We call Eq. (19) an "almost-always" (a.a.) bound if there exists an R such that it holds for all r > R. This is the sense in which all past works (to our knowledge) have derived and discussed LR bounds 8 .
• We call Eq. (19) an "infinitely-often" (i.o.) bound if for every R, it holds for some r > R. This is equivalent to saying that there exists a subsequence {r k } ∞ k=1 on which the bound holds. Fig. 5 illustrates the distinction between the two, showing a hypothetical curve [A t 0 , B r ] versus r alongside consistent a.a. and i.o. bounds. Note that a.a. bounds are automatically i.o. bounds, but not vice-versa. The distinction is likely unimportant for translation-invariant systems (although we are not aware of any works that compare the two to begin with), but it turns out to be 8 Strictly speaking, past works have derived bounds that hold for every single site (i.e., R = 0). Here we do not differentiate between "almost-always" and such "always" bounds (although one arguably could), since our focus is on asymptotic behavior. Given an a.a. bound with R > 0, one simple way to extend it to all sites is to use the trivial bound [A t 0 , Br] ≤ 2 A 0 Br for r ≤ R. If the couplings {J l } are bounded by some Jmax, another way is to use the conventional LR bound corresponding to Jmax for r ≤ R (Eq. (31) below). Since the number of sites which violate the a.a. bound is by definition finite, neither prescription changes any of the asymptotic behavior. essential in non-translation-invariant systems -as mentioned in Sec. II, there will be situations in which we can derive i.o. bounds but are unable to derive corresponding a.a. bounds (even proving that no such a.a. bound can exist).
For any link l, pass to the interaction picture with respect to all other terms in the Hamiltonian by defining The equation of motion for A t 0I (see Eqs. (15) and (17)) is from which it follows that Since we are considering a 1D chain, l is the only link that connects the sites on its right (denoted > l) to the sites on its left (denoted < l). We thus use Eq. (18) to obtain Taking the norm then gives Together with Eq. (12), we thus have and Eq. (25) becomes Taking r > 0 for concreteness and supposing that l lies between sites 0 and r, P >l A 0 = 0 and we are left with To obtain a closed-form bound on P r A t 0 , first note that P r A t (12)), then use Eq. (29) iteratively: bound P ≥r A t 0 in terms of P ≥r−1 A s 0 , then bound the latter in terms of P ≥r−2 A s 0 , and so on until reaching the origin (at which point do not introduce P ≥0 -simply use that From here, one usually takes all J l to equal a common value J. This gives (using that r! ≥ (r/e) r ) Eq. (31) is of the form in Eq. (19), with z = 1, γ = 1, and v = 4eJ: we have a ballistic (a.a.) LR bound with exponential tail and velocity of order J. One can certainly use Eq. (30) for non-translationinvariant {J l } as well. Yet it is easy to see that the result might be rather weak. Suppose that at large r, the empirical distribution of couplings (Eq. (3)) approaches a function µ(J) (in some sufficiently strong sense -we are only reasoning schematically for the moment). Then As long as log J is integrable with respect to µ(J), we still obtain a ballistic LR bound: where log 4J ≡ dµ(J) log 4J. Yet the requirement that log J be integrable is quite easy to satisfy: if µ(J) ∼ J α at small J for any α > 0, however small, then J is finite and Eq. (30) is ballistic. Clearly this is a much weaker claim than that in Fig. 2. As we show in the following subsections, taking the small-J links more seriously allows us to prove that the actual dynamics must be sub-ballistic for any α < 1.

B. Improvements via integrating out links
In words, we improve on the conventional bound by passing to a further interaction picture with respect to all "large" terms in the Hamiltonian, namely all H l (t) whose norms exceed 9 some threshold . On the one hand, the remaining terms have a larger support in this interaction picture, and we do not try to describe their structure within that support. Yet in return, every remaining term has norm less than , and so no dynamics can occur on any scale faster than −1 . The latter effect, which suppresses operator spreading, turns out to be the dominant one when there is a sufficient number of weak links. We sketch the situation in Fig. 6.
We now make this argument precise. Pick any subset of links λ ≡ {l 1 , · · · , l n } lying between 0 and r. Pass to the interaction picture with respect to H Λ/λ : As in Eq. (22), the equation of motion for A t 0 is where The transformed operator H li (t) is supported on (potentially) all sites between l i−1 and l i+1 , yet it has the same norm as H li (t), namely bounded by J li . Thus we can apply the same procedure as in Sec. IV A to the operator A t 0 , with λ in place of Λ and with H λ (t) as the 9 Hamiltonian. Following an identical derivation to that of Eq. (29), we obtain Use that P r A t 0 ≤ 2 P >ln A t 0 = 2 P >ln A t 0 to start the iteration, and A s 0 = A 0 to terminate it. Thus Since λ is arbitrary, we are free to use the subset that gives the tightest bound: where the minimum is over all subsets of links between 0 and r. Eq. (38) is our improved LR bound. Although the minimization in Eq. (38) may seem computationally expensive due to the 2 r possible λ, it can be performed efficiently. For a fixed size of λ, the optimal choice is clearly those links having the |λ| smallest values of J l . Thus one need only sort {J l } r l=1 beforehand, and the minimization amounts to simply checking the r possible values of |λ|.
C. An explicit bound in terms of the distribution of couplings We now make use of Eq. (38) to prove general results in terms of the "distribution" of couplings µ r (J), by which we mean the fraction of links between 0 and r for which J l ≤ J. Reproducing Eq. (3) (recall that δ J l ≤J ≡ 1 if J l ≤ J and 0 otherwise), From this definition, for any > 0 such that µ r ( ) = 0, we can choose λ to be the subset of links with J l ≤ and thus have the LR bound Now suppose that µ r (J) converges 10 as r → ∞ to a 10 One might be concerned by us taking r → ∞ since we are considering a finite chain of N sites. However, note that all of the bounds derived in Secs. IV and V are independent of N . Thus to be precise, we are supposing that we have an infinite sequence of couplings {J l } which obeys Eqs. (42) and (43), and we are considering the restriction of an infinite lattice to N sites. Any statement involving r should be interpreted as applying when N > r.
function µ(J) defined on [0, ∞), with the small-J ("tail") behavior 11 for some µ 0 > 0. The parameter α > 0 will play an essential role in what follows. We do need to specify the precise sense in which µ r (J) converges. It will not be sufficient to require merely that lim r→∞ µ r (J) = µ(J) at any fixed J, but also that µ r (J) behave as µ(J) on scales which decrease with increasing r. This is expressed by the following conditions, which we assume to be met: • For any β ∈ [0, 1/α) and J > 0, • For any β > 1/α, As an example of why Eqs. (42) and (43) are necessary, rather than simply the condition lim r→∞ µ r (J) = µ(J) (which is contained in Eq. (42) as the case β = 0), suppose that there is a single link on which J l = 0. Clearly there cannot be any operator spreading past link l (note that this is captured by the general bound in Eq. (38)). On the other hand, the value of a single coupling does not affect the fraction which are less than any value in the large-r limit. Thus lim r→∞ µ r (J) at any fixed value of J does not identify individual anomalously weak links, whereas Eq. (43) does. In other words, Eqs. (42) and (43) are a precise way of stating that the weakest links between sites 0 and r are also distributed in a manner behaving as µ(J) at large r.
We feel that these conditions are reasonable to expect in practice. We show in Sec. V that if each J l is chosen independently from a literal probability distribution µ(J) obeying Eq. (41), then Eqs. (42) and (43) are satisfied with probability 1 -in this sense, any sufficiently disordered set of couplings will meet our requirements. Whether Eqs. (42) and (43) hold in any specific situation obviously depends on the system under consideration, but in any case (and perhaps more importantly), one can always return to Eq. (38) if needed.
Let us first consider α ≥ 1. From Eq. (41) and the assumption that lim r→∞ µ r (J) = µ(J), there exists 12 R and > 0 such that for all r > R, µ r ( ) > C α (recall our convention that C is any r-and t-independent constant whose value may change from line to line). Thus Eq. (40) becomes (for r > R) This is simply a conventional LR bound with ballistic front and exponential tail (a.a. because it holds for all r > R), having velocity 4e/C α−1 . Now consider α < 1. Pick any β ∈ (0, 1/α), and then from Eqs. (41) and (42), there exists R such that for all r > R, µ r (r −β ) > Cr −βα . Taking = r −β in Eq. (40), we thus have that for all r > R, This is an a.a. LR bound having exponents Note that the front is sub-ballistic precisely for α < 1 (whereas it is no tighter than Eq. (44) for α ≥ 1). By taking β 1/α, we can make the dynamical exponent z arbitrarily close to 1/α. This actually implies that for any z < 1/α, we have an a.a. LR bound with arbitrarily small generalized velocity. Fixing z and for any v > 0, setting β ∈ (z, 1/α) and taking r sufficiently large (so that 4er −β < Cvr 1−βα−z ) gives Interestingly, the tightest tail corresponds to the opposite limit of β. Setting β = 0 gives the standard exponential tail (albeit only at distances r > vt for some v), whereas increasing β gives an increasingly stretched-exponential tail. The optimal choice of β depends on the specific application: one should take β 1/α if constraining the shape of the front is most important, but one should set β = 0 if constraining the tail is most important.
Note that this analysis can straightforwardly be extended to limiting distributions µ(J) which are not simple power laws, with the expected results. First of all, if µ(J) decays to 0 at small J faster than any power law (e.g., µ(J) ∼ exp [−1/J]), then a conventional LR bound as in Eq. (44) still holds. If µ(J) decays slower than any power law (e.g., µ(J) ∼ 1/ log J −1 ), and if Eq. (42) is obeyed for all β > 0, then z = ∞ in that an LR bound with infinitesimal v holds for any finite z. Lastly, our main result still applies if µ(J) scales not solely as J α but as J α p(J) for some sub-power-law function p(J)an LR bound with arbitrarily small generalized velocity holds for any z < z c (α), as in Eq. (48).

D. Tightness of the bound
As discussed in Sec. II, LR bounds should be complemented by an understanding of their tightness, ideally by constructing an explicit protocol H ∈ H J that saturates the bound. To that end, we consider the simple transfer protocol shown in Fig. 3. Denoting the total runtime of the circuit by T r , clearly P r A Tr 0 = A 0 , and so any valid LR bound must have a front which encompasses the spacetime point (r, T r ). We shall focus on the dynamical exponent -if T r = O(r z ), then no LR bound can have a dynamical exponent larger than z.
Effecting a SWAP gate for arbitrary d-dimensional local Hilbert spaces is not entirely trivial 13 , but a construction is given in Ref. [56]. For completeness, we give the relevant details in App. C. The only interactions needed (per SWAP gate) are a finite number of controlled-Z gates. In our case, since Eq. (5) must be respected, the time per controlled-Z gate across link l is O(1/J l ). Thus the total runtime is We again assume that the distribution of couplings µ r (J) satisfies Eqs. (42) and (43). As we demonstrate below, it then follows that for any z > max[1/α, 1], lim r→∞ T r r z = 0.
Let us first note that it is the same threshold exponent z c (α) ≡ max[1/α, 1] which enters into both Eq. (50) and (48). We can thus say that the dynamical exponent is z c (α) in the following sense: • For any z < z c (α), there is no H ∈ H J that can generate correlations at any sufficiently large distance r in a time of order r z .
• For any z > z c (α), we know of an explicit protocol H ∈ H J that can generate correlations at every sufficiently large distance r in a time vanishing compared to r z .
However, the behavior precisely at z = z c (α) is far more complicated and system-dependent. In particular, the distinction between a.a. and i.o. bounds becomes essential, as we demonstrate explicitly in Sec. V. Now we turn to the proof of Eq. (50). It will be convenient to define Y l ≡ 1/J l , so that T r = r l=1 Y l . Note that µ r (J), the fraction of links with J l ≤ J, is equivalently the fraction with Y l ≥ 1/J. Fix γ > 0 and define a 0 ≡ 0, a k ≡ r (k−1)γ for k ≥ 1 (writing a k instead of a k (r) for conciseness). Also define p k to be the fraction of links with Y l ∈ [a k , a k+1 ), equivalently By definition, we have the bound For p 0 and p 1 , we shall simply use that p 0 ≤ 1, p 1 ≤ 1. For p k with k ≥ 2, first note that the second term in Eq. (51) can be neglected relative to the first at large r. Thus it follows from Eqs. (41) and (42) that for any η > 0, there exists R k such that for all r > R k , Furthermore, Eq. (43) implies that if we take K to be the smallest integer greater than 1/αγ, then there exists R ∞ such that for all r > R ∞ , min r l=1 J l > r −Kγ = a −1

K+1
and therefore p k = 0 for all k ≥ K + 1. All together, we have that for r > max[R 2 , · · · , R K , R ∞ ], Eq. (52) becomes First consider α > 1. The sum in Eq. (55) is then O(1) with respect to r. Since γ is arbitrary, Eq. (50) follows for any z > 1 (namely choose γ < (z − 1)/α). Note that this conclusion also applies when µ(J) decays faster than a power law -in such a case, p k (for k ≥ 2) is even smaller than for any finite α and thus Eq. (55) remains a valid bound.
Next suppose α ≤ 1. The sum now grows no faster than O(r K(1−α)γ ), and thus The latter inequality follows because, by definition, 1/αγ < K ≤ 1 + 1/αγ. Again, since γ is arbitrary, Eq. (50) follows. Incidentally, this line of reasoning puts our discussion regarding the failure of the conventional LR bound (Eq. (32) in particular) on firmer ground. As noted above, the couplings enter into the conventional bound via the sum r l=1 log J l . We have that The sums over k again terminate at K, but now the summands go as r −kαγ log r −kγ and are dominated by small k regardless of α. More precisely, using Eq. (54) and the analogous lower bound on p k gives Since γ can be arbitrarily small, inserting into Eq. (30) gives an LR bound whose front, while not necessarily quite ballistic, cannot have a dynamical exponent larger than 1. As we have now established, that bound is far from tight.

V. DISORDERED LIEB-ROBINSON BOUNDS
As a non-trivial example of a situation in which the above results apply, here we suppose that each J l is drawn i.i.d. from a probability distribution µ(J) whose small-J behavior is given by Eq. (41). We first prove Eqs. (42) and (43), not merely in some average sense but with probability 1, using standard techniques. The results in Secs. IV C and IV D then follow.
We next consider the threshold case z = max[1/α, 1] in more detail. For α > 1, it follows immediately from the strong law of large numbers (see Refs. [57,58] for an introduction) that the transfer protocol in Sec. IV D reaches all sites ballistically. For α < 1, on the other hand, the distinction between a.a. and i.o. bounds becomes important -we derive an i.o. bound with arbitrarily small generalized velocity, implying that no protocol can reach every site in time T r = O(r 1/α ), but also show that the above transfer protocol does reach an infinite subsequence of sites in time T r = O(r 1/α ) (again with probability 1). Interestingly, α = 1 is the only point at which we cannot give a definite answer. Our i.o. bound still applies, but the transfer protocol now fails to reach any site in time T r = O(r). It could be that an a.a. bound with arbitrarily small velocity holds for α = 1, but we have not succeeded in proving so.
Finally, we discuss some straightforward extensions of the above results.

A. Convergence of the distribution
We first prove Eq. (42). This requires some tools from probability theory which can be found in textbooks on the subject [57,58] but are likely not common knowledge among physicists. Here we apply these tools with-out further comment for ease of presentation, but include a description of them in App. D for completeness.
To prove Eq. (42), pick any > 0 and β ∈ [0, 1/α), then define the event E r to be Abbreviating µ(Jr −β ) by µ for conciseness, Eq. (59) is equivalently the event that the number of couplings less than Jr −β is not between (1 − )rµ and (1 + )rµ. We can evaluate the latter directly: where the final inequality follows because the summand is maximized at n = (1 + )rµ. One can confirm that the right-hand side goes as r exp [−Cr 1−βα ] at large r, with C positive. Since 1 − βα > 0, Pr[E r ] is therefore summable and the probability of E r occurring infinitely often is zero (see App. D). The probability that this occurs for any rational or β, i.e., that µ r (Jr −β )/µ(Jr −β ) does not converge to 1 for any β < 1/α, is likewise zero. We now prove Eq. (43). Pick any M > 0 and β > 1/α, and consider the event Clearly if J l > M r −β , then J l > M s −β for all s > r. Thus the following events are equivalent: Since the couplings {J l } are independent, the probability of the right-hand side is straightforward to evaluate: Pr ∀r > R : where the sum in the lower line is convergent because βα > 1. Therefore lim R→∞ Pr ∀r > R : and Pr[min r l=1 J l ≤ M r −β i.o.] = 0. The probability for any M ∈ N or rational β, i.e., the probability that min r l=1 J l /r −β does not diverge, is likewise zero. This completes the proof of Eq. (43).

B. Fluctuations at the threshold exponent
We now take the dynamical exponent z to be the threshold value max [1/α, 1], still for the model in which all couplings {J l } are chosen i.i.d. from µ(J). We determine whether an LR bound with arbitrarily small v holds, both in the a.a. and i.o. sense. In the situations where we can provide a decisive answer (which will be all α = 1), this completes the diagram in Fig. 2.
First suppose α > 1. Then dµ(J)J −1 is finite (note that dµ(J) ∼ CJ α−1 dJ at small J), and recall that we identified a specific protocol for which the runtime is T r = C r l=1 1/J l . Since the couplings are i.i.d., the strong law of large numbers (SLLN) [57,58] gives that with probability 1, Thus the above transfer protocol reaches every sufficiently large-distance site with a non-zero velocity (namely [ dµ(J)J −1 ] −1 ). Now suppose α < 1. Return to Eq. (38), and take the subset λ to be solely the link connecting r − 1 and r: Although a rather loose bound, the probability that even the right-hand side of Eq. (67) exceeds vtr −1/α for all large r vanishes, for any v > 0. In other words, we have an i.o. bound with arbitrarily small generalized velocity: with probability 1, there exists a subsequence {r i } for which To prove Eq. (68), simply compute the probability that for all r > R, J r > vr −1/α : This establishes an i.o. bound with arbitrarily small v, but let us consider again the previous transfer protocol. One can show that with probability 1, lim inf meaning that this protocol does reach a subsequence of sites {r i } in time of order (and in fact asymptotically smaller than) r 1/α i . The proof of Eq. (70), which we have adapted from Ref.
[59], begins by noting that T r r −1/α converges in distribution to a non-negative random variable S whose support includes 0 (a fact which is wellestablished but by no means trivial, e.g., see Ref. [58]). Thus pick any > 0. We have that Pr[S ≥ ] < 1, and at the same time, we can always choose a sequence of numbers {C k } such that Pr[S ≥ C k ] < 1/k 2 → 0.
The convergence in distribution of T r r −1/α to S implies that we can construct a subsequence {r k } with the following properties: • r k+1 − r k → ∞ as k → ∞; • For sufficiently large k, • k > , where we define (for later convenience) From Eq. (71), it follows that Pr[T r k r −1/α k ≥ C k ] < 2/k 2 , and thus with probability 1, there is some K such that Since T r k < C k r 1/α k with probability 1, we have that using Eq. (72). Note that the differences T r k +1 − T r k are mutually independent, and thus the probability on the right-hand side factors: Furthermore, since r k+1 − r k → ∞, the random variable (T r k+1 − T r k )/(r k+1 − r k ) 1/α itself converges in distribution to S, and so for sufficiently large k, 14 The right-hand side is strictly less than one, meaning the infinite product in Eq. (75) evaluates to zero. Thus Pr[E k ] = 0 and therefore In other words, Eq. (70) holds with probability 1. It remains only to consider α = 1. The calculation in Eq. (69) still holds, and thus an i.o. bound with arbitrarily small velocity exists with probability 1. Yet for the transfer protocol, we now have i.e., no site is reached ballistically. Compare to Eq. (70). It may be that a more sophisticated protocol is able to reach a subsequence ballistically, or it may be that no such protocol exists. We have been unable to rule out either possibility. To prove Eq. (78), following Ref. [58], pick any M > 0 and define truncated random variables Y l ≡ min[Y l , M ] (recall that Y l ≡ 1/J l ). The expectation value of Y l , denoted Y (M ), is finite and therefore the SLLN applies. Thus with probability 1,

C. Extensions
Lastly, we discuss some straightforward extensions of the above results. These are not meant to be exhaustive, nor do we expect them to be particularly tight bounds -we only wish to point out some generalizations that can be obtained with little additional work.
Couplings with finite-range correlations: As a first example, suppose that the couplings {J l } are correlated, but that correlations exist only within a finite range ξ. By the latter, we mean that joint distributions µ (n) factor only if all couplings involved are separated by at least ξ sites, e.g., Such correlations present no difficulties -simply first pass to the interaction picture with respect to all but every ξ'th link, and then the previous analysis applies. All lengths are reduced by a factor of ξ, and thus the generalized LR velocity is increased by a factor of ξ z (which we again do not claim to be a particularly accurate estimate), but the diagram in Fig. 2

remains unmodified.
Bounds for multiple energy scales: Returning to a strictly 1D chain, suppose that J l can only take the values J and J, where 0 < 1. The couplings are still chosen independently, and the probability of J l = J is α . This is a discrete analogue to the situation from the previous subsections, for which J l could take any value greater than zero. Here, any LR bound will clearly have a ballistic front, but one can still ask how the LR velocity v compares to the two scales J and J. By taking λ in Eq. (38) to be those links with J l = J (the fraction of which approaches α at large r with probability 1), and comparing to the conventional LR bound, we find that Analogous to the previous results, v J for α < 1. Furthermore, the dependence of v on in Eq. (81) is tight -the average of 1/J l is finite for all α, namely given by (1 − α + α−1 )/J, and so the SLLN again applies, as in Eq. (66). Taking 1, the velocity of the transfer protocol is therefore Eq. (81) up to prefactors.
We can easily generalize to there being an arbitrary (finite) number of widely separated energy scales. Suppose that J l = γ k J with probability α k for k ∈ {1, · · · , K}, and J l = J otherwise. Assume 0 < γ 1 < · · · < γ K and 0 < α 1 < · · · < α K . The optimal LR velocity is now The dependence on is again tight. While simple, this result does highlight that in general, neither the largest nor the smallest energy scale necessarily determines the relevant velocity for operator growth on its own. Bounds for ladders: Consider a system such as in Fig. 7, in which sites are labelled by (i, j) with i ∈ {· · · , −1, 0, 1, · · · } and j ∈ {1, · · · , M }. The Hamiltonian is still given by a sum of terms for each link of the lattice. Interactions along vertical links are arbitrary, and interactions along horizontal links (denoted H ij (t) for the link between (i − 1, j) and (i, j)) obey H ij (t) ≤ J ij . Each J ij is again drawn independently from µ(J).
Since our analysis does not make any assumptions regarding the nature of the local Hilbert space, we can simply identify each set of sites connected vertically -{(i, j)} M j=1 for fixed i -as comprising a single "local" Hilbert space. However, the interaction between neighboring i is then j H ij (t), meaning that the coefficient J l entering into bounds such as Eq. (38) should be j J lj . The probability of j J lj ≤ J is bounded by Thus the correct exponent to use in our analysis is now M α, and in particular, z c (α) = max[1/M α, 1]. Obtaining an improvement over the conventional LR bound now requires α < 1/M , but nonetheless, we still have that z c → ∞ as α → 0. In fact, we can adapt the construction of Ref. [60] to show that this result for z c (α) is tight. We discuss this assuming two states per site (d = 2) labelled by |0 and |1 -the same protocol extends to arbitrary d simply by acting as the identity in the subspace orthogonal to |0 and |1 , and it still produces an O(1) commutator for generic operators A 0 and B r which act non-trivially on |0 and |1 .
Taking cues from Ref. [60], consider starting in the product state with a|0 + b|1 on site (i − 1, j) and |0 on all other sites of rungs i−1 and i. Since arbitrary vertical interactions are allowed, we can construct a unitary that takes this state to (a|0 i−1 + b|1 i−1 ) ⊗ (|0 i + |1 i )/ √ 2 in arbitrarily short time, where |0 i and |1 i denote the states on rung i with all sites in |0 and |1 respectively.
The states (|0 i + |1 i )/ √ 2 and (|0 i − |1 i )/ √ 2 can then be converted into |0 i and |1 i respectively, again in arbitrarily short time, using interactions solely on rung i. This procedure thus transforms the product state having a|0 + b|1 on site (i − 1, j) and |0 otherwise into the generalized GHZ state a|0 i−1 ⊗ |0 i + b|1 i−1 ⊗ |1 i . Subsequently applying the procedure in reverse, albeit with the roles of rungs i − 1 and i exchanged, then takes this generalized GHZ state into the product state having a|0 + b|1 on site (i, j) and |0 otherwise. The net effect is that the state on site (i − 1, j) has been transferred to site (i, j) in a time 2π/ j J ij (coming from Eq. (84)). Repeating the transfer sequentially from rung 0 to r, we have a protocol analogous to Fig. 3 with runtime T r = r l=1 2π/ j J ij . Thus not only does our LR bound apply to the ladder of Fig. 7, with j J lj in place of J l , but so does our analysis of the 1D transfer protocol, again using j J lj as an effective horizontal coupling. The result derived above that z c (α) = max[1/M α, 1] is therefore tight. Note that if we restrict ourselves to bounded-strength vertical interactions, but with bounds that are spatially uniform, then the dynamical exponent remains unaffected even once the time required to effect all single-rung transformations is incorporated. We leave the more complicated situation in which the vertical interactions themselves have weak links as a direction for future work.
The fact that z c (α) → 1 as M → ∞ for any α > 0 suggests that our conclusions may not extrapolate to higher dimensions (and analogously to longer-range interactions). There are far more paths connecting any two sites in higher dimensions, and it may be that transport remains ballistic for any power-law distribution of weak links. Of course, the analysis of the ladder presented here only accounts for weak links in one direction, and so the behavior of truly multi-dimensional disordered systems remains an important open question.

VI. APPLICATIONS
In Sec. IV, we derived a modified LR bound for nontranslation-invariant systems -Eq. (45) -requiring only that the empirical distribution µ r (J) converge to a function µ(J) ∼ CJ α (as formalized by Eqs. (42) and (43)). For α < 1, the modified bound gives a significant improvement over the conventional bound, and even guarantees that operator spreading is sub-ballistic. Here we consider the consequences of this result for various applications (assume α < 1 throughout).
On the one hand, the manner in which LR bounds are used in the following is quite similar from case to case. Yet we shall see that different contexts come with different caveats, some more than others, and we discuss the many open directions for future work.
Note as well that we are working with the general bound of Sec. IV rather than the more detailed results of Sec. V. In particular, implications of the "a.a."-"i.o." distinction for the following applications warrant further investigation.
A. Growth of correlations LR bounds directly place limitations on the extent to which correlations can develop following a quench. Consider the correlation function where the expectation value is in a product state |Ψ , time evolution is under a Hamiltonian H(t) ∈ H J , and the operators A 0 and B 0 are supported on sites to the left of 0 and to the right of r respectively (note that our analysis in Sec. IV applies equally well to such operators even if they are not strictly local). Thus G(0) = 0, and one would like to understand how G(t) grows in time.
The authors of Ref. [5] show that one can bound (assuming A = B = 1 for simplicity) Supposing that r is sufficiently large, we can immediately apply Eq. (45): where β can take any value in (0, 1/α). The fact that one can choose an optimal β depending on t makes Eq. (87) slightly more interesting than the conventional bound. However, care must be taken in varying β. We have shown that for any β ∈ (0, 1/α), there exists R(β) such that the bound holds for r > R(β), but we have not shown that the convergence is uniform in β (i.e., that R(β) can be made independent of β). Rather than impose an additional requirement on the convergence of µ r (J) and attempt to verify it in non-trivial situations, here we shall simply choose a finite set {β i } n i=0 with 0 ≡ β 0 < · · · < β n < 1/α (note that β = 0 simply recovers the conventional LR bound). The precise statement of Eq. (87) is that for r/2 > max[R(β 0 ), · · · , R(β n )], The behavior of Eq. (88) is shown in the dashed red line of Fig. 8. At large r, it is given by exp [−Cr + C t] until t = O(r), then by exp [−Cr 1−β1α + C t/r β1 ] until t = O(r 1+β1(1−α) ), and in general, Of course, we are free to choose as large a set {β i } as we like (although this may increase the distance required for the bound to hold). One should heuristically think of Eq. (88) as minimizing over all β ∈ (0, 1/α) for any value of r, shown as the solid red line in Fig. 8.

B. Creation of topological order
A related application is lower bounds on the time needed to create topological order. Again following Ref. [5], we say that two states |Ψ 1 and |Ψ 2 in a 1D system of size N are "topologically ordered" (relative to each other) if there exist constants c 1 ∈ (0, 1) and c 2 > 0 such that, for every observable O supported on a set with diameter c 1 N or less, In words, no "local" operator (even one supported on a non-vanishing fraction of the system) can distinguish between or couple such states.
Suppose that |Ψ 1 and |Ψ 2 are topologically ordered and have been prepared from states |Φ 1 and |Φ 2 via time evolution under a Hamiltonian H(t) ∈ H J : Ref. [5] shows that, for any operator O with diameter less than c 1 N/2, and analogously for the off-diagonal matrix elements. We again use that Note that for t N , Eq. (93) is exponentially small in N . Returning to Eq. (92), |Φ 1 and |Φ 2 then satisfy the definition of topological order. In other words, for times less than O(N ), it is impossible to prepare topologically ordered states (|Ψ 1 and |Ψ 2 ) from any states which are not themselves topologically ordered (|Φ 1 and |Φ 2 ).
The above conclusion is identical to that of the conventional case, but note that Eq. (93) in fact remains small for much longer times, until t = O(N 1/α ). The bound is no longer exponential (rather stretched-exponential), and so the definition in Eq. (90) is not strictly met, but |Φ 1 and |Φ 2 exhibit a slightly looser sense of topological order nonetheless. In this sense, we have that for nontranslation-invariant systems with α < 1, it is impossible to prepare ordered states from unordered in times less than O(N 1/α ).

C. Heating in periodically driven systems
The authors of Ref. [8] consider energy absorption in weakly driven systems, for which the Hamiltonian is of the form We shall limit ourselves to 1D, although Ref. [8] treats a general dimension. Within linear response theory [61], i.e., to leading order at small g, the energy absorption rate is proportional to σ(ω) ≡ ij σ ij (ω), where The expectation value denoted by · is taken in the initial (potentially mixed) state of the system. Strictly speaking, σ ij (ω) is a distribution and should be integrated against test functions to obtain meaningful results.
The authors first derive a bound assuming a translation-invariant constraint H l ≤ J.
There is already the potential for tightening Eq. (96) in non-translation-invariant systems, since the constants C and κ involve sums over connected paths with factors of J l , much as in Sec. IV A. However, we suspect that a more careful analysis of the path sum would not yield a significant improvement (see Eq. (32)), and so we continue to use Eq. (96). LR bounds enter into the analysis of Ref. [8] as a means of bounding |σ(ω)| by O(N ) (N being the number of sites in the system -note that naively bounding |σ(ω)| ≤ ij |σ ij (ω)| would give a bound O(N 2 )). In particular, consider two sites i and j separated by a distance r > r * , with r * to be chosen later. Assuming a conventional LR bound with velocity v, Eq. (95) can be bounded by two contributions: (97) See Ref. [8] for details, including the appearance of the Gaussian factor. Since the terms of Eq. (97) decay exponentially with r or faster, the sum over all i and j with |i − j| > r * is indeed O(N ) and scales as exp [−ar * ]. For summing over |i − j| < r * , simply use Eq. (96). Thus by setting r * = O(|ω|), both contributions decay exponentially with |ω|, and therefore The exponential decay with |ω| is the main result of Ref. [8].
Let us consider whether this conclusion is altered in non-translation-invariant systems by the use of our modified LR bound. As in the preceding subsections, we choose a set {β i } n i=0 and minimize Eq. (45) over β i . The integral over t in Eq. (95) splits into multiple terms (compare to Eq. (97)): δt 2 e −a0(r−v0t) (99) Although the latter terms are indeed much smaller than in the translation-invariant case, note that the first term is unaffected. Thus |σ ij (ω)| still scales as exp [−ar] at large r, we are still led to take r * = O(|ω|), and the final result in Eq. (98) is unchanged.
The lack of any significant reduction in the heating rate can be traced back to the fact that it is the tail of the LR bound which constrains |σ(ω)|, and the tail of the nontranslation-invariant LR bound is no tighter than that of the translation-invariant case. Of course, this is only a statement about the bounds -the physics involved in any specific system very well may imply a dramatically slower heating rate.
Strictly speaking, the above comments only apply within linear response theory. To go beyond linear response, one could perform a Magnus-like expansion along the lines of Refs. [19,62]. However, higher-order terms in the expansion involve higher-body interactions (i.e., terms supported on more than two sites), and so the results of this paper do not immediately apply. While we expect that our results could be generalized to higherbody interactions, we have not attempted to do so and leave this for future work.

D. Ground state correlations
One of the most well-known applications of LR bounds is for the proof that gapped ground states have exponentially decaying correlations. Considering a 1D nearestneighbor Hamiltonian H for concreteness (although this result holds much more generally), the statement is that if there is a non-vanishing gap ∆E between the ground state and first excited state energies, then for any local observables A 0 and B r , where expectation values are in the ground state of H. See Refs. [2,4] for full details of the proof. Here we only discuss the steps at which LR bounds enter.
The authors of Refs. [2,4] show that one can bound G(r) by (assuming A 0 = B r = 1 for simplicity) with q to be chosen later. Much as we described in the previous subsection, assuming a conventional LR bound with velocity v, split the integral into one over |t| < r/2v and one over |t| > r/2v. Using the LR bound in the former and the trivial bound [A t 0 , B r ] ≤ 2 in the latter, we find 14 Thus if ∆E > 0, setting q = O(r) gives Eq. (100). For non-translation-invariant systems, using our modified LR bound, we split the integral over t into additional terms as in the previous subsection (see Eq. (99)). Yet we again do not obtain any significant improvement over Eq. (102), since we still have a term scaling as exp [−ar/2].
In fact, it is unclear whether Eq. (102) itself applies to the systems considered here -Ref. [63] proves that ensembles of Hamiltonians are generically gapless if the norms of the interactions have continuous distributions extending to zero. However, the lack of a gap in this case is due to the existence with high probability of nearly-disconnected local regions hosting low-energy excitations [63]. Since those excitations are decoupled from the larger system, one does not expect them to give rise to long-range correlations on physical grounds. To our knowledge, it remains a significant open question whether (and under what conditions) the ground states of such disordered systems have rapidly decaying correlations.

E. Predicting properties of gapped ground states
An interesting recent application in which LR bounds enter is classical machine-learning algorithms for predicting properties of quantum many-body ground states. The authors of Ref. [64] consider a family of timeindependent Hamiltonians H(x) parametrized by a continuous (potentially multi-dimensional) variable x, with corresponding ground states ρ(x). They obtain rigorous results on the ability to predict Trρ(x)O for a certain x from knowledge of {Trρ(x i )O} n i=1 for other parameter 14 Note that [A 0 0 , Br] = 0 for any r = 0, and even more, one can show that [A t 0 , Br] = O(t) at small t. Thus the integrand of Eq. (101) does not diverge as |t| → 0 and the integral is finite. The LR bound as presented in this paper, however, does not share the property of vanishing as |t| → 0, but this can easily be remedied by combining the LR bound with the above observation that [A t 0 , Br] ≤ C|t|: use the former for |t| ≥ C −1 and the latter for |t| < C −1 (see Refs. [2,4]).
values {x i } n i=1 , where O is any observable that can be written as a sum of local terms.
A central ingredient is the result that if H(x) has a non-zero spectral gap uniformly in x, then one can bound the size of the gradient ∇ x Trρ(x)O. Specializing to 1D chains with H(x) = l H l (x) and O = i O i (although Ref. [64] treats more general systems), the authors show that whereû is an arbitrary unit vector in the parameter space, and W (t) is a filter function which decays faster than any polynomial as |t| → ∞.
Much as before, the conventional LR bound enters by dividing the terms on the right-hand side into two groups, one in which i and l are separated by a distance less than vt and the other in which they are separated by greater than vt. Use the trivial bound on the commutator for the former and the LR bound for the latter. Since there are O(|t|) terms in the former and the sum over the latter is O(1), Eq. (103) reduces to The remaining integral is finite (since W (t) decays sufficiently fast), and thus the gradient is bounded. This property is then used in Ref. [64] to establish the efficiency of algorithms capable of predicting Trρ(x)O. In a certain sense, our modified LR bound for nontranslation-invariant systems provides an improvement, since it allows us to replace the factor of |t| in Eq. (104) by |t| α . However, the important feature is merely that the resulting integral over t is finite. Thus while our modified bound does tighten the numerical value of the gradient, it does not seem to give any dramatic changes. Furthermore, since the analysis of Ref. [64] requires that the Hamiltonians H(x) be gapped, the caveats from our discussion of ground state correlations apply here as well. The more substantive question is whether the conclusions of Ref. [64] apply to disordered systems at all.
To begin, keep in mind that the fundamental objects in probability theory are subsets of the set of all possible outcomes, called "events", together with a function Pr that maps such subsets to the interval [0, 1]. Oftentimes not all subsets can be included in the domain of Pr, and Pr must obey certain natural properties, but we will not dwell on these here. The important thing to note is simply that we can perform all of the usual set-theoretic operations on events, such as the union or intersection, and many of the basic tools in probability theory involve relating the values of Pr with respect to those operations.
Our first tool is "countable subadditivity" (also known as the "union bound"): for any countable (potentially infinite) collection of events {E r }, We will primarily use Eq. (D1) after establishing that Pr[E r ] = 0 for all r -it then follows that Pr[∪ r E r ] = 0. In words, if each event E r separately has probability zero, then the probability that any of them occur (even if there are a countably infinite number) is also zero. The next tool is the "continuity" of probabilities. Suppose that {A r } is an "increasing" set of events, in that A 1 ⊆ A 2 ⊆ · · · , and similarly that {B r } is a "decreasing" set of events, in that B 1 ⊇ B 2 ⊇ · · · . We then have that For our purposes, we will have events either of the form A r = "the event that E r occurs for all r > r" or B r = "the event that E r occurs for some r > r". In terms of set-theoretic operations, these are given by Note that {A r } is an increasing set of events and {B r } is a decreasing set. We will specifically want to determine the probability that some A r occurs and the probability that all B r occur. The former event is denoted "E r a.a." with a.a. abbreviating "almost always", and the latter event is denoted "E r i.o." with i.o. abbreviating "infinitely often". These events are given by Note that E r a.a. and E r i.o. can be described in words as "there exists an r past which all E r occur" and "past every r there is some E r that occurs", i.e., "E r occurs almost always" and "E r occurs infinitely often" respectively, in exactly the same manner as we have used in discussing LR bounds. Continuity -Eq. (D2) -allows us to express the probabilities of these events as We lastly need the "first Borel-Cantelli lemma", which says that if a collection of events {E r } has probabilities such that r Pr[E r ] is finite, then the probability is 1 that only a finite number of the events occur: In particular, suppose we have a sequence of random variables {X r } and want to prove that it converges to zero with probability 1. The definition of {X r } not converging to zero is that there exist some > 0 such that for every R, |X r | > for some r > R. In other words, |X r | > i.o. for all > 0. It suffices to consider only rational > 0, and thus by countable subadditivity, we only need to prove that Pr[|X r | > i.o.] = 0 for each individual . By the Borel-Cantelli lemma, it further suffices (but need not be necessary) to prove that r Pr[|X r | > ] is finite. If each individual term Pr[|X r | > ] can be calculated or at least bounded directly, this is a very useful line of attack.