Supporting Information for Fast Scrambling on Sparse Graphs

Supporting Information for Fast scrambling on sparse graphs

A Bounding the Growth of Entanglement

The supporting information contains the technical details of the results announced in the first section. Henceforth, we will assume that the Hamiltonian is k-local on a hypergraph with L vertices, and that the local Hilbert space dimension is 2M . Here we are only interested in models where M is strictly finite: e.g. M = 1;1 thus log N log L. ⇡ In this section, we sketch a proof that 2 is finite under rather general circumstances: (i)thek-local Hamiltonian (2) is extensive, and (ii) the spectrum of the many-body Hamiltonian approaches a smooth distribution in the thermodynamic limit. The proof proceeds in two parts. Let ⇢v(t)bethereduced density matrix for vertex v at time t, and define

S(1)(t)= tr [⇢ (t) log ⇢ (t)] ,S(2)(t)= log tr ⇢ (t)2 . (S1) v v v v v (2) M ⇥ ⇤ First, we show that if M log 2 Sv = , and 2 1, ⌧ 2(M 1)/2 M log 2 S(1) 3/2 +O 2 . (S2) v 2 6 3 ⇣ ⌘ Secondly, we show that v(t) v0 (0) t |hO O i| > Ce v⇤ , (S3) kOvkkOv0 k where and are local operators on vertex v, and is a v-dependent decay rate which is independent Ov Ov0 v⇤ of N for almost all vertices. With these two results in hand, we now consider the von Neumann entanglement entropy of a region 1 A consisting of L0 = pL of the vertices in the graph, with 0

(1) 1 (2) ML0 log 2 S M log 2 S M log 2 S , (S4) A > v ⇡ 2 v v A v A X2 h i X2 h i and (2) 22vt M log 2 S e , (S5) v & if half of the vertices within A have decay rates 2v 6 2, then such vertices must (on average) have 4a in order for ML0 log 2 Sv a. In the thermodynamic limit (L with M ﬁnite), we will 6 L0 6 0 2t !1 show that there exists a positive constant C N for which & Ce ,whichproves(12). P / 1In SYK-like models, the local Hilbert spaces below should be interpreted as few fermion clusters on a given graph vertex, in contrast to our interpretation in (25), and in Appendix C.

1 A.1 Relating Renyi to von Neumann Entropy

First we show (S2). If 0 6 pi 6 1 denote the eigenvalues of the reduced density matrix ⇢v,then

(1) (2) 2 Sv = pi log pi,Sv = log pi . (S6) ! Xi Xi

Also observe that pi = 1. Let 1+✏ p = i , (S7) P i n (2) and let M log 2 Sv = .If2<1, then by deﬁnition 1 ✏2 =e 1 <+ 2 < 2. (S8) 2M i Xi Thus, for all i, ✏ p2M+1. (S9) | i| 6 Now, when 2M+1 1, ⌧ 1 1 1 S(1) = M log 2 ✏2 + ✏3 ✏4 + , (S10) v 2M+1 i 6 2M i 12 2M i ··· Xi · Xi · Xi and so 2 (1) 1 1 2 max( ✏i ) max( ✏i ) 1 2 M log 2 Sv ✏i 6 | | + | | + ✏i (S11) 2 2M 6 12 ··· 2M ⇣ ⌘ i ✓ ◆ i X X Taylor expanding the equality in (S8), and using (S9), we obtain (S2). A.2 Renyi Entropy and the Memory Function

(2) Next, we derive an exact, albeit non-local in time, equation of motion for Sv (t). To do so, let us consider the vector space of all Hermitian operators which act on the Hilbert space of vertex v. The density matrix a a ⇢v) is in this vector space. A basis set for this vector space is I), T ) ,whereT denote the Hermitian | {| | v } v generators of SU(2M ), acting on vertex v alone. An (orthogonal) inner product for this basis is

( ) tr ( ) . (S12) O1|O2 ⌘ O1O2 Our goal is now to find a linear equation for ⇢ (t)). This can be done, as the Schrödinger equation | v is linear. However, as we are integrating out degrees of freedom, the resulting equation will be nonlocal. This is the essence of the memory function formalism [58, 59, 60]. Unlike in the usual of the memory function equations (in real time), here we will also need to keep track of additional terms related to the initial conditions of the entire many-body state. We will ultimately argue that these additional terms cannot generally modify (S3). The key point is as follows. First, let us temporarily expand the vector space to include the set of all Hermitian matrices. Our goal is to project the dynamics back on to the set of reduced density matrices on vertex v, which is an exponentially small subset of all possible basis vectors. Let the initial many-body state be 0). Since tr⇢ = 1 for all times, we know that 0) and I) have some overlap, so we may write the | | | initial state as 0) = 0 )+ 0). (S13) | | v | e 2 The first term is the initial condition for ⇢v, tensored with the identity on all other sites; the second term makes up the remainder of terms. Let be the Liouvillian: L )= [H, ]), (S14) L|O | O with H the many-body Hamiltonian. The time evolution of the density matrix is given by e i t 0). L | Denoting p as the projection operator onto I), T a) , and q =1 p, we thus wish compute {| | v } i t ⇢ (t)) = pe L 0). (S15) | v | More specifically, since S(2)(t) e v =(⇢ (t) ⇢ (t)), (S16) v | v we wish to compute (2) dSv 2 d ⇢ (t)) = (⇢ (t) | v . (S17) dt (⇢ (t) ⇢ (t)) v | dt v | v Using the identity t i t i qt i q(t s) i s pe L = pe L ip dse L pe L , (S18) L Z0 we obtain t d ⇢v(t)) iq qt iq qs | = ip qe L 0) ip p ⇢ (t)) ds p qe L q p ⇢ (t s)). (S19) dt L | L | v L L | v Z0 e This is a linear equation (albeit nonlocal in time) with a source, given by the first term on the right hand side. We may write its solution as the sum of a particular solution and a homogeneous solution:

⇢ (t)) = ⇢0(t)) + ⇢ (t)) (S20) | v | v | v The first term above is a homogeneous solution of (S19), obeying p ⇢0)= ⇢0); the second term is a e | v | v particular solution of (S19) obeying ⇢ (0)) = 0. For generic initial conditions, we expect the decay of | v ⇢v(t)) to I) can be no faster than the decay of the homogeneous term. So we expect that it suces | ⇠| to bound the decay of ⇢0(t)). This is the first point at which we will sacrifice some rigor and simply | v e sketch out a proof of (S3). The fact that a generic linear equation decays no faster than the decay of the homogeneous solution is the formal version of the inequality we provided in (12). It is most natural to describe the dynamics of ⇢0(t)) via a Laplace transform. It is convenient to | v subtract out the identity component:

0 1 ⇢ (t)) ⇢ (t)) I). (S21) | v ⌘| v 2M | Since I) is a null vector of , it decouples from the dynamics and we will omit this basis vector in what | L follows. The assumption that entropy saturates to maximal then implies that ⇢ (t)) 0 as t . v ! !1 Letting 1 zt ⇢ (z)) dt e ⇢ (t)), (S22) | v ⌘ | v Z0 we ﬁnd that z ⇢ (z)) ⇢ (0)) = (ip p + (z)) ⇢ (z)), (S23) | v | v L K | v

3 where the memory function 1 (z) p q (z +iq q) q p. (S24) K ⌘ L L L The inverse Laplace transform then gives us the time-dependent reduced density matrix:

dz zt 1 ⇢ (t)) = e (z + (z)+ip p) ⇢ (0)) (S25) | v 2⇡i K L | v z0+iZ R

The real number z0 is chosen somewhere where the contour can be closed: i.e. for z0 6 Re(z ), where ⇤ z is the complex number with smallest real part such that z + (z)+ip p is a singular matrix. For ⇤ K L the moment, let us assume that z is ﬁnite. We will justify this in the next subsection, together with ⇤ Re(z ) < 0. Standard theorems then give that (for generic initial conditions) as t ⇤ !1 zt dz zt 1 dz e z t ⇢v(t)) p e (z + (z)+ip p) p ⇢v(0)) = C ⇢v(0)) = e ⇤ C ⇢v(0)) | ⇡ ⇤ 2⇡i K L ⇤| 2⇡i z z ⇤| ⇤| z0+iZ R z0+iZ R ⇤ (S26) where p projects onto all vectors with this most singular ‘eigen’vector, and C is a suitable matrix. ⇤ ⇤ Subleading corrections to this equation decay with a faster exponential rate. We then conclude that (S3) 2 holds, with v⇤ = 2Re(z ) and C =(⇢v(0) C ⇢v(0)). ⇤ | ⇤ | A.3 Finiteness of the Memory Function

It remains to justify why < Re(z ) < 0. First, let us suppose that E↵) denote the eigenfunctions of 1 ⇤ | q q in the image of q.Wewrite L p q = c )(E (S27) L ↵E↵ | ↵ ↵| ↵ X Let us further assume that the density of states of the many-body Hamiltonian is continuous, and is given by ⇢(E). We then ﬁnd that the density of states of both and q q is given by L L

⇢ (E)= dE0 ⇢(E0)⇢(E0 + E). (S28) L Z Because q q and are identical matrices up to an exponentially small number of entries, corrections to L L their relative continuous spectra are e L, and such e↵ects are comparable, in the thermodynamic limit, / to other “ﬁnite size” e↵ects that we neglect. We further assume that

2 ⇢ (E) (E) (E↵ E) c↵E↵ ↵)( ↵ (S29) L F ⌘ | | | | ↵ X is a smooth-valued positive definite matrix as a function of E, in the thermodynamic limit.2 If these two assumptions hold, we obtain ⇢ (E) (E) (z)= dE L F (S30) K z iE Z We first show that < Re(z ). We do this by bounding ,since 1 ⇤ F (↵ (z) ↵) < ⇡ max ((↵ ⇢ (E) (E) ↵).) . (S31) |K | E | L F | First, define = p. (S32) Lv L 2It is, by definition, positive semidefinite; we further assume that there are no null vectors.

4 For a k-local Hamiltonian, using the L operator norm (for simplicity): 1 1 2k H k k L0. (S33) L kLvk 6 L / v V X2 The last step uses extensivity. Using Markov’s inequality, this means that a ﬁnite fraction of vertices have M0 max ⇢ (E) > . (S35) ⇤ E L 1 To argue that Re(z ) < 0, observe that ⇤

Re( (0)) = ⇡ (0) dE⇢(E)2. (S36) K F Z Re(z ) < 0whenever (0) is positive definite. It is positive semidefinite by construction. We do expect ⇤ F that all eigenvalues are strictly positive in a typical chaotic model. One subtlety arises from the possiblity of Hamiltonians which have continuous many-body spectrum but are integrable. These include either the quantum version of “classical” Ising Hamiltonians with integer coecients, or many-body localized systems [61]. For these systems, we do expect that has null vectors or vectors that are so close to null F that the above proof fails. For example, in the many-body localized theories, one can generalize the above procedure to finite subsets of vertices. Localization of eigenstates will apply that some eigenvalues of F are extremely small, when the subset of vertices is finite in the thermodynamic limit, but large compared to the localization length. Subject to the caveats described above, this completes the argument for the finiteness of v⇤ for most vertices. Our assumptions seem quite plausible in chaotic many-body systems. As stated in the main text, it would be interesting if the “proof” sketched here can be made rigorous.

A.4 Possible Generalizations We now briefly comment on a few possible generalizations of our results. Firstly, we expect that the bounding of entanglement generation at late times by the decay of correlation functions also holds at finite temperature, though a generalization of our “proof” appears non-trivial. Nevertheless, we present some circumstantial evidence that hints towards a finite temperature generalization of (12). Combining results from [62] and [17], we find that scrambling at late times is set by the decay rate of two point functions at finite temperature in two dimensional holographic conformal field theories. Another example where the (partial) saturation of entanglement is related to the decay of two-point functions is the Sachdev- Ye-Kitaev model [37]. The presence of conservation laws, including energy conservation, may further slow down the growth of entanglement by preventing the rapid decay of mutual information between vertices within the subregion A [63, 64]. Such e↵ects further increase ts beyond log L/(22), consistent with the fast scrambling conjecture. The growth of entanglement is extremely rapid at early times, and slows down at late times.3 This is consistent with bounds on the rate of entanglement generation, which show that the generation of entanglement is proportional to the perimeter of A [67, 68, 69]. So we can formally bound the growth of

3Although saturation of entanglement entropy after a quench appears to be abrupt in translation invariant theories with a holographic dual [65, 66], such models do not saturate (1): the spatial spread of information is “slow”.

5 entanglement at early times. For simplicity, we will focus on a 2-local Hamiltonian with operator norms (18), though we will now let ↵, run from 1 to 22M 1: [68, 69] dSA c < 18M log 2 E c < 18cN log 2. (S37) dt ⇥ K ⇥| A,A | c In the above equation, E c counts the number of edges between A and A . On a generic, locally | A,A | tree-like graph, nearly every vertex is at the edge of A [24], so we do expect dS /dt N at early times. A ⇠ The fact that dS /dt N at early times was also noted in [7]; however, in [7], a N in (5), so that A / / t N 0 is possible. We view this N-independent time scale as a time scale for local thermalization, but s / not for scrambling.

B Lieb-Robinson Bounds

B.1 Derivation In this section we derive Lieb-Robinson bounds for arbitrary Hamiltonians of the form

H = JSHS, (S38) S 1,...,L ✓{X } e where the sum over S includes all possible non-empty subsets of vertices, JS > 0, and HS is an arbitrary unit norm Hermitian operator which acts non-trivially only on the Hilbert space . Observe that i S Hi the requirement that the Hamiltonian is k-local amounts to a requirement that J 2=0ife S >k.To NS | | derive our bound, we review the useful notion of a factor graph [70]. A factor graph G =(V,F,E) consists of a vertex set V = 1,...,L , a “factor vertex set” F ZV , which consists of the possible S in (S38), { } ⇢ 2 and an edge set E V F containing all edges obeying the rule e ⇢ ⇥ EvS e =(v, S) 2 2 . (S39) / Ev/S ⇢ 2 2 See Figure S1. As we will see below, it is convenient to take the factor vertex set F to only include the sets S for which JS > 0, together with all S of size 1 (corresponding to single vertices), but this is not strictly necessary. In this case, the factor vertices denote terms in H, and the edges from F to V simply denote which vertices each term “acts” upon. We can define a natural adjacency matrix 1(v, S) E = 2 , (S40) AvS 0(v, S) / E ⇢ 2 and we also define the matrix = J . (S41) JSS0 SS0 S Let us now return to the question of operator growth. Consider a pair of operators AS and BQ which act non-trivially only on the vertices in S F and Q F ,respectively.Define 2 2 [A (t),B ] C (t) sup k S Q k, (S42) SQ ⌘ A B AS ,BQ k Skk Qk with the supremum running over all possible Hermitian operators AS and BQ acting on the suitable Hilbert spaces. Without loss of generality below, we take t>0 and ✏>0. Our goal is to find an inequality governing the growth of CSQ(t). Defining

H J H . (S43) S ⌘ S0 S0 S :S S = 0 X\ 06 ; e 6 4

5 2

Figure S1: An example of a factor graph. Red circles denote vertices V , blue squares denote 2 factor vertices F , and black lines denote edges. A Hamiltonian which might be associated with 2 z x x y z 2 x y x y z z x y z z z x x x z this factor graph is H = 1 +3 +4 +5 +1x+2 3 +2 4 +35 +4 5 +123 +2 3 4 5. we observe that [A (t + ✏),B ] [A (t),B ] = [A (t),B ] i✏[[H, A (t)],B ] [A (t),B ] +O ✏2 k S Q kk S Q k k S Q S Q kk S Q k ✏ [[H, A ],B ( t)] +O ✏2 = ✏ [[H ,A ],B ( t)] +O ✏2 6 k S Q k k S S Q k 2✏ A [H (t),B ] +O ✏2 (S44) 6 k Skk S Q k We thus obtain [A (t),B ] [H (t),B ] C (t + ✏)= sup k S Q k +2✏k S Q k +O ✏2 SQ A B B AS ,BQ  k Skk Qk k Qk 2 6 CSQ(t)+2✏ JS0 CS0Q(t)+O ✏ S :S S = 0 X\ 06 ; 2 C (t)+2✏ S0 S J C (t)+O ✏ , (S45) 6 SQ | \ | S0 S0Q S X0 which, upon taking ✏ 0, leads to the di↵erential inequality ! dC (t) SQ 2 C . (S46) dt 6 ASvAvS00 JS00S0 S0Q where we have employed the Einstein summation convention above. In order to obtain (S46), we have used the fact that the matrix ( T ) = S S counts the number of vertices which the two sets S A A SS0 | \ 0| and S share in common. Observe that both C (0) and T are non-negative matrices – i.e., all 0 SQ A AJ components are non-negative. We can integrate (S46) to obtain

n n T 1 (2t) T CSQ(t) 6 exp 2t CS Q(0) = CS Q(0). (S47) A AJ SS 0 n! A AJ SS 0 0 n=0 0 h i X ⇣ ⌘ 7 By deﬁnition, we know that 2 S Q = C (0) \ 6 ; 2( T ) . (S48) SQ 6 0 S Q = 6 A A SQ ⇢ \ ; Suppose for simplicity that we are interested in the evolution of operators that act only on a single T vertex at time t = 0. In that case, the initial sets S = u and Q = v , and we may write Zu v v Q = { } { } ASu0 0 0 A 0 Zuv for any matrix Zuv. Using this identity, together with (S47), and denoting C u v (t)=Cuv(t), we obtain our most general Lieb-Robinson bound: { }{ }

n n 1 (2 t ) T T Cuv(t) 6 2 | | =2exp 2 t (S49) n! AJ A uv | |AJ A uv n=0 X ⇣ ⌘ h i ↵ Let us now focus on 2-local Hamiltonians with only huv = 0, as in the introductory section. Using 6 (18) we obtain

T 1 T = J u0v0 u (u0,v0) (u ,v ) v AJ A uv 2 { }A{ } A 0 0 { } ⇣ ⌘ uX0,v0 c c ( + )( + )= (D + A) . (S50) 6 2K uu0 uv0 vu0 vv0 K uv u v E 0X02 1 In the sum on the ﬁrst line, we put a factor of 2 to account for the double counting of u0v0 and v0u0. Combining (S49) and (S50)weﬁnd(19).

B.2 Fast Scrambling on “Regular Hypergraphs” In the main text, we described the constraints on fast scrambling in 2-local models. To generalize this to k-local models, we can now consider the following simple argument. Let us suppose that the number of J > 0 is given by . For simplicity, let us assume that S N LJ JS 6 ⇤ , (S51) N with J > 0, though it is straightforward to generalize the argument. Let 1v =(1,...,1). Since ⇤

T 1v J kJ 6 ⇤ S 6 ⇤ , e (S52) JA Sv L | | ⇣ ⌘ N N and thus e T 1v kJ kJ 6 ⇤ 1= ⇤ nu, (S53) AJ A uv L S:u S ⇣ ⌘ e N X2 N where nu denotes the number of terms in H that act non-trivially on u. This generalizes the degree ku of the vertex in the 2-local case. On a hypergraph, where n m /L for every vertex (note m 1), we u 6 N > thus arrive at T 1v 1u 6 kJ m , (S54) AJ A uv L ⇤ L and hence ⇣ ⌘ e e 2kJ mt C (t) e ⇤ uv . (S55) L2 6 L u,v X If both k and m are ﬁnite in the limit L , the exponent is ﬁnite and thus C only becomes O(1) !1 uv after a time t log L, consistent with the fast scrambling conjecture. ⇠

8 B.3 Towards the Traditional Bound We now derive (23), along with a generalization for arbitrary Hamiltonians. The observation is simply that we can tighten (S52)to

n T 1v 1u (kJ m) n > duv 6 ⇤ , (S56) AJ A uv L L ⇥ 0 n

n n n (2t) T duv (2et) n Cuv(t) 6 2 < 2e (kJ m) < 2 exp [2ekJ mt duv] . (S57) n! AJ A uv n! ⇤ ⇤ nX=duv ⇣ ⌘ nX=duv Using (S50), we obtain (23). More generally, we see that if m (as deﬁned in the previous subsection) and k are ﬁnite, fast scrambling is only possible on hypergraphs with diameter . log L.

C Sachdev-Ye-Kitaev Models on a Graph

v In this section, we describe the generalized Sachdev-Ye-Kitaev model [71, 72] on a random graph. Let i denote Majorana fermion i on vertex v V : v,u = uv, and i 1, 2,...,M , and consider the 2 { i j } ij 2{ } Hamiltonian

q u u u q uv u u v v H =i2 J ... +i2 J ... ... . (S58) i1,...,iq i1 iq i1,...,iq/2j1,...,jq/2 i1 iq/2 j1 jq/2 u i1<...

2 (q 1)! 2q 1 bk 2 (q/2)!2 2q 1 J u = 1 u J 2, J uv = bJ 2. (S59) E i1,...,iq q 1 E i1,...,iq/2j1,...,jq/2 q 1 M q 2 qM q ⇣ ⌘ ✓ ◆ ⇣ ⌘ The parameter 0 b 2 sets the relative coupling strength on the links; J sets the e↵ective coupling 6 6 kmax constant.

C.1 Large q Limit and Saddle Point At large M, one solves for the correlation functions of the fermions by computing an e↵ective action for the two-point Euclidean time Green’s function

M 1 G (⌧ ,⌧ )= u(⌧ )u(⌧ ) , (S60) u 1 2 M h i 1 i 2 i Xi=1 and an analogous self-energy ⌃u(⌧1,⌧2). The result is

1 J 2 1 S = log Pf (@ ⌃ ) + d⌧ d⌧ ⌃ G (Gq b⇤ Gq/2Gq/2) , (S61) e↵. ⌧ u uv 2 1 2 u u uv q u uv 2 uv u v u,v X  Z ✓ ◆

9 where the matrix ⇤ = D A is the graph Laplacian, which generalizes a discretized 2 to a general r graph. ⇤ encodes all spatial dynamics of the e↵ective action. Because ⇤ always has a null vector: 1 1v = (1, 1,...,1), (S62) pL the above action admits a simple saddle point, where Gv = G and ⌃v = ⌃ do not depend on vertex v. ⇤ ⇤ In the large q limit, we expand G and ⌃ to O(1/q) for this saddle point: [30] ⇤ ⇤ 2 1 g (⌧) g (⌧) G (⌧)= sgn(⌧) 1+ ⇤ ,⌃(⌧)=J e ⇤ , (S63a) ⇤ 2 q ⇤ q ✓ ◆ where the saddle point solution g (⌧) is given by ⇤ 2 ⇡⌘ g (⌧) cos 2 ⇡⌘ e ⇤ = ,J= ⇡⌘ (S64) 1 t 2cos ⇡⌘ | | 3 cos 2 2 4 ⇣ ⌘5 where ⌘ is a parameter determined by the coupling constant: in the high temperature limit J 1, ⌧ ⌘ J ; in the low temperature limit J 1, ⌘ 1 2 . ⇡ ⇡ ⇡ J In the discussion that follows, we will assume that q is large, but not the largest parameter in the problem. Namely, we will take M L q. We assume that the latter inequality may be safely taken, but have not explicitly checked this assumption.

C.2 Out-Of-Time-Ordered Correlators By expanding around this saddle point, we can compute the connected piece of the averaged, regularized OTOC M u v u v tr yi (0)yj (t1)yi (0)yj (t2) conn. (t ,t ), (S65) ⇣ M 2 ⌘ ⌘Fuv 1 2 i,jX=1 1/4 where y = ⇢ ,with⇢ the thermal density matrix at inverse temperature . Because of the combined the large M limit and large q limit, we can describe the exponentially growing behavior of (t ,t ) at Fuv 1 2 all temperatures in this model. Indeed, following [28, 30], we ﬁnd that obeys the following linear Fuv equation in the exponential growth regime:

(t ,t )= dt dt KR (t ,t ; t ,t ) (t ,t ), (S66) Fuv 1 2 3 4 uw 1 2 3 4 Fwv 3 4 Z R where Kuv is a retarded kernel with ‘spatial’ dynamics: 2 2 R 2⇡ v ⇥(t13)⇥(t24) Kuv(t1,t2; t3,t4)= Suv, (S67) 2 2 ⇡⌘ cosh t34 ⇣ ⌘ and4 b S = ⇤ . (S68) uv uv q 1 uv The exponentially growing ansatz has the following form:

t1+t2 (t ,t )=e L 2 f (t ). (S69) Fuv 1 2 uv 12 4See [71, 72] for similar discussions and diagrams for spatially dependent kernels in generalized SYK models on lattices.

10 R Because the vertex dependence in Kuv comes entirely through Suv, we can solve for the spatial dynamics by studying (S66) for each individual eigenvector of S with eigenvalue s. Suitable combinations of the solution to (S66) for these eigenvectors can be used to construct uv for any initial conditions. Using the F u explicit form of the retarded kernel, applying a pair of derivatives, @1@2,to(S66), and denoting t12 = ⇡v , we find: 2 2 L 2 2s fs(u)= @ + fs(u) (S70) 4⇡2⌘2 · u cosh2 u ✓ ◆ This equation is the Schrödinger equation in a cosh-potential in one spatial dimension. The following exact bound state solution is known, together with a corresponding eigenvalue parametrized by a: 1 2 2 f (u) L = a2, 2s = a(a + 1). (S71) s / cosha u ) 4⇡2⌘2 Importantly, we have now fixed the Lyapunov exponent . Focusing for simplicity on the OTOC (t)= L Fuv uv(t, t), we conclude that F 1 (t)= C eL(s)t, (S72) Fuv M u v X where C are undetermined constants, and v denote eigenvectors of ⇤, and therefore S: b S = s = 1 . (S73) uv v u q 1 u v X ✓ ◆ In the large q limit, the eigenvalues of Suv, s, are close to 1. Therefore 2⇡ 2⇡ 2b = ⌘a ⌘ 1 . (S74) L ⇡ 3(q 1) ✓ ◆ It remains to fix C to solve the problem. Unfortunately, this is highly non-trivial in general, and depends on complicated details of the early time physics in the SYK model, e.g. on the dynamics at early time 0

C.3 High Temperatures

J 2b First, we describe the high temperature limit J 1. In this regime, ⌘ ⇡ and L =2J 1 3(q 1) . ⌧ ⇡ It is reasonable to expect that uv(t) is approximately local at early times 0

C . (S75) u v / uv X Since the set of u form an orthonormal basis, we conclude that C is a constant, independent of .This implies that 1 2b (t) exp 2Jt 1 ⇤ . (S76) Fuv / M 3(q 1)  ✓ ◆uv We can interpret this result in a simple way: (t) is proportional to a concentration of “infected random Fuv walking individuals” located on vertex u at time t, given that the only infected individuals were located on vertex v at time t = 0. The assumptions we make are that infected individuals grow at a constant rate of 2J, and that infected individuals perform a random walk on G: traversing any given edge at a 4bJ constant rate of 3(q 1) . This connection between the growth of chaos and the spread of infections has been observed for some time, and is also visible in the RUC.

11 C.4 Graphs with Finite Spectral Gap Next, we relax the high temperature assumption, but require that the graph Laplacian ⇤ has a ﬁnite spectral gap >0. The gap of ⇤ implies a gap of Lyapunov spectrum, i.e.

2⇡ 2b 1 2⇡ ⌘t ⌘(1 )t (t)= C e 1 1 + C e 3(q 1) + ... (S77) Fuv M 0 u v 1 u v ⇣ ⌘ In the long time limit, e.g. t q, the subleading terms are negligible comparing to the ﬁrst term, related to the null vector 1 = 1 (1, 1,...,1).5 Therefore we have: u pL

2⇡ ⌘t 2⇡ ⌘t e e (t) C 1 1 = C (S78) Fuv ⇡ 0 u v M 0 N

In the last step we used N = ML. The constant C0 is set by the overlap of the initial condition with the spatially uniform component of , before the exponential growing regime, and we therefore expect C Fuv 0 to be an O(1) number. Thus, for the graphs with finite spectral gap, we obtain t = log N (S79) ⇤ 2⇡⌘ at leading order in N, independently of any details of the graph structure. As we are more interested in the spatial dynamics on the graph than in the dynamics of a single-site SYK model, we wish to take the limits M,L in such a way that log M 0. In this case, it is !1 L ! necessary for the spectral gap to remain finite even in the thermodynamic limit L , in order to !1 obtain (S79). Rather remarkably, it is a famous result in graph theory that is strictly finite on any graph where the ‘perimeter’ of any subset of vertices scales proportionally to the number of vertices: see Appendix E. Therefore, on a typical sparse and locally treelike graph, the SYK model described above is just as chaotic as it would be on a fully connected graph. This serves as an explicit example of a chaotic quantum system where some amount of sparsity to the connectivity graph does not a↵ect the time scales of quantum information loss and operator growth.

C.5 Low Temperatures Another commonly discussed solvable limit is the low temperature limit M J 1. In this limit, we do not need to assume that q is large. In this limit, the saddle point equation can be approximately R solved by a conformal ansatz, where Kuv takes the following form (for simplicity, we set q = 4 for this subsection): R 3⇡⇥(t13)⇥(t24) Kuv(t1,t2; t3,t4)= 1 Suv, (S80) 2 ⇡ ⇡ ⇡ 2 cosh t34 sinh t13 sinh t24 R ⇣ ⌘⇣ ⇣ ⌘ ⇣ ⌘⌘ Similarly to the large q discussion, Kuv can be diagonalized by the eigenvectors of Suv, and eigenfunctions of the temporal part [30, 28], which determines the Lyapunov exponent through the following equation 3 b 1 =1. (S81) 3 1+ ⇡ L ✓ ◆ We hence obtain b 2⇡ = 1 . (S82) L 2 ✓ ◆ 5Technically, we also require ML = N eq for this long time limit to be sensible.

12 Therefore the gap in ⇤ also indicates a gap in the Lyapunov spectrum for the q = 4 model at low temperature (a similar result also applies to arbitrary q > 4). Following the same argument as before, 2⇡ we can ignore the subleading terms and only focus on the null vector 1u and leading exponent L = , which saturates the chaos bound [42]: 2⇡ t e (t) . (S83) Fuv / ML independently of the graph structure. We can gain further intuition about the spatial dynamics by using an alternative treatment of the leading exponent in the low temperature limit. Following [71, 72], we describe the contribution to the leading exponent by the dynamics of reparametrization modes on every vertex. The generalization of these works is straightforward but as the computation is rather technical we do not write it explicitly. We ﬁnd that the OTOC becomes

2⇡ 1 t 1 ↵ b 2⇡ t e 1 (t) + ⇤ e Z . (S84) Fuv / M J 3 ⌘ M uv ✓ ◆uv where ↵ is a numerical constant. The identity matrix is implicitly assumed to be multiplying constants in the expression above. Note that this calculation assumes that 2⇡ for all modes; we see from (S82) L ⇡ that this is analogous to approximating b 0. ! 1 J At leading order in the small parameter 1/J, we reproduce the previous result Zuv ↵L independent 1 ⇡ of indices u and v. Regarding the higher order e↵ects, we can interpret Zuv in terms of a simple statistical problem: at vertex u, we release one random walker per unit time onto the graph. At any given instant in time, a random walker may walk along an edge of the graph to a neighboring vertex – this occurs with b ↵ rate 3 . Finally, the random walkers die with rate constant J . Let nv be the expected number of random walkers on vertex v. In steady state, the rate of incoming walkers equals the rate of outgoing/dying walkers: b ↵ b + n = + k n . (S85) uv 3 w J 3 v v wv E ✓ ◆ X2 1 We conclude that nv = Zuv > 0. Interestingly, this does not have the interpretation of a di↵using random walker. The meaning of the di↵erent mechanisms for the spread of chaos at low vs. high temperature is not clear to us, although its origins can be straightforwardly understood: the mechanisms responsible for chaos at high [8] and low [30] temperatures are very di↵erent. See also the recent discussion in [73], which argues that the high temperature behavior is more generic at ﬁnite M.

D Random Unitary Circuit Model

D.1 Mapping OTOCs to a Classical Stochastic Process In this subsection, we review the stochastic process which the computation of OTOCs in the m-local RUC maps on to. These results were all found previously in [33]. First, consider a growing operator (t)=U U ,with a local operator which acts non-trivially Ov Ov † Ov only on vertex v. Recall that t =1/L in order to recover extensive quantum dynamics. For simplicity, let us assume that U will act on a subset A of m vertices: namely, U is a random 2Mm 2Mm unitary ⇥ matrix. U U = if v/A.Ifv A, after averaging over all U with uniform measure, U U will be Ov † Ov 2 2 Ov † an equal superposition of all non-trivial 22Mm 1 Hermitian operators acting on the subset A.Observe

13 that the fraction of these operators which act non-trivially on ` 6 m vertices is given by (22M 1)` m p = . (S86) ` 22Mm 1 ` ✓ ◆ If we ﬁx vertices u and v, and average over all Us acting on the subset A, and also average over local operators and ,weﬁnd Ou Ov m ` 2 p` u, v A E u,U vU † 8 m { }⇢ . (S87) O O / X`=1  <> 0 otherwise The proportionality constant is related to the normalization:> of the v and is not needed for our purposes. O The right hand side should be interpreted as the weight in the operator U U which act non-trivially Ov † on u. At later time steps, the above procedure generalizes straightforwardly. For an operator = A Oi which consists of a large number of complicated terms, U U can be evaluated term-by-term. In each A † PN term ,ifthereexistsavertexv A for which is not the identity, then we replace with Ov 2 Ov u A Ou a random sum of all possible 22Mm 1 Hermitian operators with coecients whose squares sum2 to one. N N This leads to the following observation: after averaging over all possible circuits, we can compute the proportion of the growing operator U(t) U(t) which acts non-trivially on the subset S V by mapping Ov † ⇢ onto the following stochastic process. Let n 0, 1 denote whether a vertex is “infected” or not; at u 2{ } time t = 0, n = 1 and all other n = 0 for u = v. At each time step, pick a random allowed subset v u 6 A V of m vertices. If n = 0, then do nothing; otherwise – regardless of the microscopic state ⇢ u A u – with probability p ,set` vertices2 in A, chosen uniformly at random, to be infected, and the remaining ` P m ` vertices to be uninfected. In the limit M , p = 1. The infection only grows. This is why, when m = 2, the RUC maps !1 m onto a discrete time analogue of the SI epidemic model. This is the example which we focused on in the main text.

D.2 2-Local Operator Dynamics: Mean Field Methods and their Breakdown As we saw at infinite M in the main text, the RUC wih 2-local dynamics maps on to the SI epidemic model which has super-exponential infection growth on heterogeneous networks. In this section, we will show that this e↵ect is an artifact of the M limit for 2-local RUCs. The following section will give !1 an example of super-exponential operator growth with a 3-local RUC. For now, we follow the literature on epidemics on complex networks [35] and use a mean field approximation to solve for the dynamics of the stochastic process of the previous section with m = 2. The key approximation is a closed set of equations for Pv P[nv = 1]: ⌘ dP 2 v = [(1 p )(1 P )P p P ] . (S88) dt K 1 v u 1 v u v X⇠ The first term corresponds to the rate at which the vertex v becomes infected: this occurs with probability 1 p (as we do not care whether its neighbor gets uninfected in the process!), and requires the neighbor 1 to be infected while v should be uninfected. The latter term corresponds to the rate at which a vertex uninfects itself, which occurs at rate 2p1/K per edge, regardless of the state of the neighbor. At early times, we may write the growth equation as dP 2 k [(1 p )A p D ] P . (S89) dt ⇡ K 1 vu 1 vu u

14 The growth rate of the epidemic is thus related to the spectrum of a particular linear combination of adjacency and degree matrices. In order to make further progress, we resort to a further “degree-based” mean-field description [35]. Let Pk denote the probability that nv = 1 for a vertex v with degree k, and let ⇢k be the probability that a randomly chosen vertex in the graph has degree k.Wedefine k⇢ ✓ = k P , (S90) K k Xk which is the probability that a randomly chosen edge points to a node which is infected. The mean-field approximation is that each node e↵ectively sees each neighbor infected with probability ✓.Withthis approximation, Pk obeys the closed di↵erential equation dP 2k 2k k = (1 p )(1 P )✓ p P . (S91) dt K 1 k K 1 k The overall factor of 2k/K arises because this is the rate at which a random edge which connects to a vertex of degree k is chosen. The first term in (S91) counts the rate at which the vertex is uninfected, and an edge is chosen between the given vertex and one of its neighbors: the final factor of 1 p counts the probability that the infection spreads to the vertex. The second term counts the probability that a random edge is chosen, and this causes the infection to decay from the central vertex. At early times, we expect that P eLt: namely, the largest eigenvalue of the linearized equation of k ⇠ motion dominates growth and sets a scrambling time, as defined by OTOCs. This eigenvalue equation gives that (K +2kp ) P =2k(1 p )✓ (S92) L 1 k 1 when P 0. Combining (S90) and (S92), we find that k ! k⇢ 2k(1 p ) 1= k 1 . (S93) K KL +2kp1 Xk Let us begin with some formal bounds on L. A lower bound on L can be found from (S92)by applying Jensen’s inequality on the convex function x2/(x + 1) (for x>0):

2K(1 p ) 1 > 1 . (S94) KL +2p1K Hence > 2(1 2p ). (S95) L 1 A lower bound can be found by applying Jensen’s inequality to the concave function x/(x + 1), assuming the probability distribution on degrees k⇢k/K, instead of ⇢k: 1 < 2(1 2p ) ⇢ k2. (S96) L 1 K2 k Xk This latter probability upper bound is proportional to the growth rate of an SI epidemic on a heterogeneous 1 2 graph [36]: K2 ⇢kk . Unsurprisingly, infections spread at a reduced rate at finite M. This accordingly 1 increases the time for the operator to grow on a regular graph by the factor (1 2p1) . As noted in P [33], this is analogous to the emergence of a butterfly velocity [74], which plays the role of an e↵ective Lieb-Robinson velocity, accessible in correlation functions [75]. In general, we expect that the upper bound above is better for heterogeneous graphs with a finite variance of the degree distribution, as it is known that epidemics do spread faster on heterogeneous

15 2 networks, and our model reduces to the SI model as p1 0. However, if k ⇢kk diverges, as it can on ⌫ 1 ⌫ ! scale free graphs with ⇢k K k ,with2<⌫6 3, then this upper bound is lousy. We can estimate / P the value of L on these scale free graphs by observing that

KL/2p1 2 1 2 ⌫ dk ⌫ 1 2k (1 p1) dk ⌫ 1 k(1 p1) 1 p1 L 1 K + K , (S97) / k⌫ K2 k⌫ Kp / p 2p L 1 1 ✓ 1 ◆ Z1 KLZ/2p1 which gives us that as p1 0 ! 3 ⌫ ⌫2 p . (S98) L / 1 This suggests that for any ﬁnite p1, the “epidemic spreading” dynamics of the RUC is fundamentally di↵erent from the SI model: in particular, L remains ﬁnite. Indeed, for 2-local dynamics, we are not able to construct any explicit examples where we can prove that = . One way to try is to consider the “star graph” (the interaction graph of (26)). Here (S93) L 1 becomes 1 p L(1 p ) 1 1 + 1 . (S99) ⇡ 2L +2p1 2L +2Lp1 This equation is approximately solved by

L(1 2p ); (S100) L ⇡ 1 namely, the mean-field description implies that diverges in the thermodynamic limit L . However, L !1 on the star graph, we may also give a more explicit construction of the dynamics. For simplicity, let us consider dynamics where the infection starts on the central node, 1. At time steps 1/L, we act on the / central node (and another) with a random unitary; if the central node is infected, at each time step there is a finite probability p1 that this node is “uninfected”. With high probability, at t 1/(p1L), the central 1 / vertex becomes uninfected. If Q p of the other vertices were infected in this time frame, the time it / 1 takes for the central vertex to be reinfected is Q 1 p . So we can estimate that the time it takes to ⇡ / 1 infect a finite fraction of vertices as p p t p + 1 + 1 + p log L. (S101) ⇡ 1 2 3 ···⇠ 1 This gives us a finite Lyapunov rate: 1 L . (S102) ⇡ p1 We have not mathematically shown that any 2-local RUC on any graph has a larger Lyapunov exponent than this. It is possible that the mean field treatment of the dynamics on scale free graphs is also inaccurate.

D.3 A 3-Local Model with = L 1 One may ask – is the observation that < when M is ﬁnite a general feature of the RUC, or is it an L 1 artifact of the 2-local dynamics above. We now show that it is the latter. Consider a 3-local RUC on a generalized star hypergraph consisting of R “inner” nodes and N “outer” nodes: we take R/N 0inthe ! thermodynamic limit. Allowed unitaries in the RUC act on a single outer vertex and two inner vertices: all such pairs are allowed. See Figure S2. As on the star graph under 2-local dynamics, we can understand 3-local dynamics on the generalized star by focusing on the fraction s of inner vertices which are infected at any given time. Denoting with q

16 Figure S2: A 3-local RUC on a generalized star graph consisting of R inner nodes (solid circles) and N outer nodes (hollow circles). The black triangle denotes a sample triplet of vertices on which a unitary can act. the fraction of outer edges which are infected, the probability that s transitions at each time step is

2 1 P s s + =(p3 + p2)q(1 s) 1 s , (S103a) ! R R ✓ ◆ ✓ ◆ 1 1 P s s + = 2(p3 + p2)s (1 s)+2N(p1 + p2)q(1 s) 1 s , (S103b) ! R R ✓ ◆ ✓ ◆ 1 1 P s s =2p1s(1 s)+2N(p1 + p2)s s , (S103c) ! R R ✓ ◆ ✓ ◆ 2 1 P s s = p1s s . (S103d) ! R R ✓ ◆ ✓ ◆ At early times s R 1 and q N 1. Note that the rates of transition are the above probabilities / / multiplied by N (the total number of vertices, in the thermodynamic limit). The dominant infection- spreading at these early times are associated with unitaries that act on one infected inner node, and two uninfected nodes: one inner and one outer. This means that Rs undergoes a biased random walk from 0 to R. Let us now ask – given that at some time the random walk is at Rs = m, what is the probability p that the random walk reaches m = 0 before m R? This probability obeys the recursive equations m ⇠ p0 = 1 and [76] p3 + p2 p1 pm = pm+1 + pm 1. (S104) p3 + p2 + p1 p3 + p2 + p1 In the R limit, this set of equations is solved by !1 p m p 1 . (S105) m ⇡ p + p ✓ 2 3 ◆ What this means is that once a single vertex in the inner core is infected, there is a rather high probability that the inner core never becomes uninfected. Generalizing this logic, we conclude that for any finite m, there is a finite probability that Rs > m for all future times. What this means is that we may approximate ds N (p + p p )s, (S106) dt ⇡ R 3 2 1 and so after an initial time R log R t , (S107) init ⇡ N(p + p p ) 3 2 1 a finite fraction of the nodes inner core are infected. So long as (R log R)/N 0 in the thermodynamic ! limit, t 0. init !

17 Once a ﬁnite fraction of the inner core is infected, then we may approximate dq (p3 +2p2 + p1)s (2 s )(1 q) (2p1 + p2)q, (S108) dt ⇡ ⇤ ⇤ where p + p p s = 3 2 1 (S109) ⇤ p3 +2p2 + p1 is the fraction of inner vertices that are infected once t t . The time it takes for q to be O(1) is ﬁnite init in thus given by a constant in the thermodynamic limit. By construction, we have shown that there exist 3-local RUCs with small local Hilbert space dimensions, for which = . L 1 D.4 2-Local Entanglement Dynamics Here we provide a more serious bound on the growth of entanglement for the random unitary circuit than the one found in the main text. For simplicity, we focus on the 2-local case. The discussion below follows [32], which studied the growth of entanglement on regular lattices. As in the main text, we assume that our initial state is a product state. To bound the growth of entanglement we will need the following two inequalities for the von Neumann entropy [77]:

S S S S + S . (S110) | A B| 6 AB 6 A B Here A and B are disjoint subsets of the vertex set V . This implies that

SA(t) 6 SA+v(t)+M log 2,SA(t) 6 SA v(t)+M log 2. (S111) Here A + v = A v and A v = A v c. The second useful fact that we will need is that S [ ]= [{ } \{ } A | i S [U ]ifeither i, j A or i, j A = : this is easy to show by writing out the explicit formula A ij| i { }⇢ { }\ ; for SA in terms of a partial trace. (S111) may be used to get sharper bounds on the growth of entanglement. In particular, the result can be phrased as optimizing the following “cutting” problem on a graph: choose a (possibly) time-dependent subset B(t) V of vertices for which at every discrete time t, the unitary U (t) acts either entirely ✓ itjt within B(t) or entirely outside B(t). At time steps where the subset B(t) changes, we either remove or add one vertex to B(t). Thus we conclude that

S (t 1 ) B(t)=B(t 1 ) S (t) B L L . (S112) B 6 S (t 1 )+M log 2 B(t) = B(t 1 ) ⇢ B L 6 L

For simplicity, we simply write SB(t) instead of SB(t)(t). We now look for ﬂuctuating subsets B(s)with the boundary condition B(t)=A, and arbitrary initial subset B(0). As the initial state is a tensor product, SB(0) = 0, and so we conclude from (S112) that

S (t)=S (t) M log 2 n [B(t)] (S113) A B 6 ⇥ cut where ncut denotes the number of time steps at which the subset B changes. See Figure S3 for an illustration of the algorithm described above. The entropy SA(t) is clearly bounded by the choice of ﬂuctuating subset B(t) with the minimal number of cuts. In the limit of large M, SA(t) was observed to be given exactly by the minimal number of cuts (up to a factor of M log 2) [32].

18 t

Figure S3: Bounding the entanglement generation in a 2-local RUC with L = 8 vertices. After 5 time steps, we see that the set B(t) has changed during 2 time steps, and so we conclude that SA(t) 6 2M log 2.

E The Cheeger Inequality

In this appendix we show that ⇤ has a ﬁnite spectral gap on any graph with sucient “nonlocality”. The result goes by the name of the Cheeger inequality: the derivation below follows [78] and we review it here for completeness. Let v be any eigenvector of ⇤, and let >0 be the associated non-zero eigenvalue:

= k . (S114) v v v u u:uv E X2 The spectral gap is given by the smallest possible choice of , and so we will simply look for bounds on . Let V V be the vertices where > 0. Note that V = V and V = as ⇤ is a symmetric matrix + ⇢ v + 6 + 6 ; with orthogonal eigenvectors, and ⇤ has a null vector (1,...,1). We may write

k v v v u v V+ u:uv E ! = X2 X2 . (S115) b 2 v v V+ X2 Deﬁning v v V+ v 2 (S116) ⌘ 0 otherwise ⇢ we obtain b

k k ( )2 v v v u v v v u u v v u:uv E ! v u:uv E ! uv E = X X2 > X X2 = X2 b b 2 b b 2 b b 2 b v v v v v v X X X ( )2b ( + )2 b b u v u v uv E uv E > X2 X2 (S117) b 2 b b b2 v 2 kvv v v X X b b 19 2 2 Now, using the identity (for real numbers) a + b > 2ab, and thus

(a b)2(c + d)2 +(a + b)2(c d)2 2 a2 b2 c2 d2 , (S118) > together with the Cauchy-Schwarz inequality on the vectors v and pDv, we obtain

2 2 2 b b u v 1 uv E > 0 X2 1 . (S119) 2 2 pb kvb B v v C B C B P C @ b A Without loss of generality, label the vertices such that .Then 1 > 2 > ···> L L v 1 2 2 = A 2 2 (S120) u v vu ` `+1 uv E u=1 v>u `=u X2 X X X ⇣ ⌘ b b b b Denote with the number of edges between 1,...,i and i +1,...,L .Then Ei { } { } L 1 2 2 = 2 2 (S121) u v E` ` `+1 uv E `=1 X2 X ⇣ ⌘ b b b b In the mathematics literature, it is more helpful to express in terms of a constant which we define as Ei h Ei . (S122) i ⌘ i L min kv, kv v=1 ! X v=Xi+1 h denotes the fraction of edges that go between 1,...,i and i +1,...,L . In fact, the Cheeger i { } { } constant, defined as EA,Ac hG =min | | (S123) A V min( E , E c ) ⇢ | A| | A | c where E c E consists of the edges between A and A ,iswellknowntobefinite in the thermodynamic A,A ⇢ limit for many families of random graphs [78]. This is a quantitative measure of how “locally treelike” the graph G is. Clearly, hG 6 hi.So

2 2 h 2 2 k = h k 2 (S124) u v > G ` `+1 j G ` ` uv E ` j ` ` X2 X ⇣ ⌘ X6 X b b b b b and hence 2 2 k`` 2 2 2 h ` h k > G 0 X 1 > G min . (S125) 2 b 2 2 k kvv max B C B v C @X p A On any graph where hG > 0 and kmin/kmax is a ﬁniteb positive constant in the thermodynamic limit, this proves that the spectral gap of ⇤ is strictly positive.

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