Fast scrambling on sparse graphs

Abstract

Given a quantum many-body system with few-body interactions, how rapidly can quantum information be hidden during time evolution? The fast scrambling conjecture is that the time to thoroughly mix information among N degrees of freedom grows at least logarithmically in N . We derive this inequality for generic quantum systems at infinite temperature, bounding the scrambling time by a finite decay time of local quantum correlations at late times. Using Lieb-Robinson bounds, generalized Sachdev-Ye-Kitaev models, and random unitary circuits, we propose that a logarithmic scrambling time can be achieved in most quantum systems with sparse connectivity. These models also elucidate how quantum chaos is not universally related to scrambling: we construct random few-body circuits with infinite Lyapunov exponent but logarithmic scrambling time. We discuss analogies between quantum models on graphs and quantum black holes, and suggest methods to experimentally study scrambling with as many as 100 sparsely-connected quantum degrees of freedom. arXiv:1805.08215v3 [cond-mat.str-el]

Scrambling and complexity in phase space

“…< l a t e x i t s h a 1 _ b a s e 6 4 = " ( n u l l ) " > ( n u l l ) < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " ( n u l l ) " > ( n u l l ) < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " ( n u l l ) " > ( n u l l ) < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " ( n u l l ) " > ( n u l l ) < / l a t e x i t > 1/ p n th < l a t e x i t s h a 1 _ b a s e 6 4 = " ( n u l l ) " > ( n u l l ) < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " ( n u l l ) " > ( n u l l ) < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " ( n u l l ) " > ( n u l l ) < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " ( n u l l ) " > ( n u l l ) < / l a t e x i t > (a) This motivates us to turn our attention to OTOCs. In thermalizing many-body systems, the decay of OTOCs detects the spread of local operators in real space [6,8,9,[24][25][26][27][28][29][43][44][45][46][47]. In few-body CV systems, OTOCs have also found use due to their correspondence with diagnostics of classical chaos [37].…”

“…One of the defining features of such DV scrambling is the notion of operator growth, where the time evolution of an initially simple, local operator V, yields a more complex, late-time operator, V(t) = U † (t) VU (t) [4], whose decomposition is dominated by non-local operator strings. A particularly powerful quantitative diagnostic of operator growth is provided by the so-called out-of-time-order correlation (OTOC) function V † (t)W † (0)V(t)W(0) , which measures the spreading of V(t) via another local probe operator W [23][24][25][26][27][28][29]. In addition to its use on the theory front, OTOCs have also attracted a significant amount of experimental interest and attention [30][31][32][33][34].…”

“…they simply map one displacement operator to another), useful insights into genuine scrambling can be obtained by studying the stochastic evolution of quasi scramblers. Indeed, recent progress [9,[43][44][45][46][47] on the interplay between holography and quantum information theory has revealed that quantum circuits composed of short-range random Clifford unitaries can provide useful intuition for understanding entanglement and operator propagation in DV many-body systems [24][25][26][27][28][29].…”

Quantum epidemiology: operator growth, thermal effects, and SYK

“…where by size we mean the number of simple operators multiplied together in a typical piece of O (t). The intuition behind this growth is that the percentage of the q-body interactions utilized in [H, O (t)] is proportional to the size of O (t), and almost all the resultant operators obtained from [H, O (t)] are bigger than O (t) [5][6][7][8][9][10][11]. 1 Systems with both spatial locality and a large number of internal degrees of freedom-such as (chaotic) field theories in the large-N limit -display both linear spatial growth and exponential internal size growth [12][13][14][15][16].…”

Treelike Interactions and Fast Scrambling with Cold Atoms

“…Our model at s = 0 offers an alternative route to fast scrambling: despite its effective light cone vs. graph distance r ij ( Fig. 2a(iv)), the early-time growth of OTOCs is permitted to reach values C ∼ e −(rij −vt) ∼ 1/N α [42] due to the logarithmic graph diameter r max ≈ 1 2 log 2 (N ), where α is a constant of order unity and v ∝ J 0 log(N ) because each spin has log(N ) couplings. Subsequent Lyapunov growth C ∼ e λt /N α therefore allows OTOCs to reach saturation in a time t * ∼ α log(N )/λ.…”