A Separation of Out-of-time-ordered Correlation and Entanglement

TL;DR

This paper demonstrates that out-of-time-ordered correlations and entanglement entropy can saturate at different times in quantum systems, challenging the assumption they always scramble information simultaneously, especially in graph-structured circuits.

Contribution

It provides a rigorous proof of the fundamental difference in scrambling times between OTOC and entanglement in quantum circuits on graphs with bottlenecks.

Findings

OTOC and entanglement saturate at different times in certain graph-based circuits.

The results generalize previous OTOC studies from lattices to complex graph geometries.

Supports the idea that black holes are slow scramblers of quantum information.

Abstract

The out-of-time-ordered correlation (OTOC) and entanglement are two physically motivated and widely used probes of the "scrambling" of quantum information, a phenomenon that has drawn great interest recently in quantum gravity and many-body physics. We argue that the corresponding notions of scrambling can be fundamentally different, by proving an asymptotic separation between the time scales of the saturation of OTOC and that of entanglement entropy in a random quantum circuit model defined on graphs with a tight bottleneck, such as tree graphs. Our result counters the intuition that a random quantum circuit mixes in time proportional to the diameter of the underlying graph of interactions. It also provides a more rigorous justification for an argument in our previous work arXiv:1807.04363, that black holes may be slow information scramblers, which in turn relates to the black hole information problem. The bounds we obtained for OTOC are interesting in their own right in that they generalize previous studies of OTOC on lattices to the geometries on graphs in a rigorous and general fashion.

Figures

Tables

Table 1. Table 1: Comparison of the time scales of OTOC and entanglement saturation in various graphs. n 𝑛 n is the total number of vertices in the graph.

Models/graphs	OTOC	Entanglement
Euclidean lattices in $D$ dimensions	$n^{1/D}$	$n^{1/D}$
Hyperbolic space with $D=3$	$\log n$	$\sqrt{n}$ 111See Section 4 of Shor18 .
Binary tree	$\log n$	$n$
Tree with degree $z\gg d^2$	$n^{1-\frac{\log d^2}{\log z}}$	$n/z$
Dumbbell graph	$\log n/n$	$n$

Equations

$$C(t)=\frac{1}{2}⟨[O_1(x,0),O_2(y,t){]}^{†}[O_1(x,0),O_2(y,t)]⟩,$$

$$U^{†}{\sigma }_{\vec{p}}U=\sum _{\vec{q}}{\alpha }_{\vec{q}}{\sigma }_{\vec{q}},{\alpha }_{\vec{q}}\equiv \frac{1}{d^n}\mathrm{Tr}[U^{†}{\sigma }_{\vec{p}}U{\sigma }_{\vec{q}}^{†}].$$

$${\mathbb{E}}_U{\alpha }_{\vec{q}}{\alpha }_{\overrightarrow{q^{\prime }}}^{\ast }=\frac{1}{d^{2n}}{\mathbb{E}}_U\mathrm{Tr}[U^{†\otimes 2}({\sigma }_{\vec{p}}\otimes {\sigma }_{\vec{p}}^{†})U^{\otimes 2}({\sigma }_{\vec{q}}^{†}\otimes {\sigma }_{\overrightarrow{q^{\prime }}})].$$

$$\frac{1}{d^2-1}\sum _{i=1}^{d^2-1}\frac{1}{2d}\mathrm{Tr}([{\sigma }_i,P{]}^{†}[{\sigma }_i,P])=\frac{d^2}{d^2-1},$$

$$E((U_{AB}\otimes {\mathrm{id}}_{A^{\prime }{B}^{\prime }})|\psi {⟩}_{A{A}^{\prime }B{B}^{\prime }})-E(|\psi {⟩}_{A{A}^{\prime }B{B}^{\prime }})\le 2\log d.$$

$$E(|\psi {⟩}_{A{A}^{\prime }B{B}^{\prime }})+2E(|{\Phi }_d⟩)\ge E((U_{AB}\otimes {\mathrm{id}}_{A^{\prime }{B}^{\prime }})|\psi {⟩}_{A{A}^{\prime }B{B}^{\prime }}),$$

Full text

††preprint: MIT-CTP/5131

I Introduction and overview

The “scrambling” of quantum information is a phenomenon of fundamental importance, deeply connected to many important research topics in physics, such as black holes HP07 ; SS08 ; MSS16 ; SS14 and many-body chaos NH15 ; PH10 . In recent years, a great amount of research effort has been devoted to the detection and characterization of scrambling. The so-called out-of-time-ordered correlation (OTOC) LO69 is a commonly used measure of quantum chaos and scrambling. A variant based on commutators (also known as the OTO commutator) is given by

[TABLE]

where $O_{1}(x,0)$ is an operator acting on site $x$, and $O_{2}(y,t)$ is a Heisenberg operator at time $t$ that only acts on $y$ at time 0, i.e. $O_{2}(y,t)=U^{\dagger}(t)O_{2}(y,0)U(t)$ where $U(t)$ is the unitary for the evolution from time 0 to $t$. The average is taken w.r.t. the thermal state at some temperature, which we take to be infinite in this work. Intuitively speaking, it characterizes parameters like sensitivity to initial conditions via the spread of local operators. Also notice that the scrambling phenomena exhibit a truly quantum nature—the state of the entire system remains pure during the unitary evolution (although it is effectively randomized), thus no information is really lost; the generation of global entanglement leads to the scrambling of initially localized quantum information, spreading and hiding it from observers that only have access to part of the system. This observation leads to another fundamental probe of a stronger form of scrambling, namely the entanglement between parts of the system SS08 ; LSH+13 ; HQRY16 ; LLZZ18 (which measures the equivalent effect as the tripartite information HQRY16 in the case of unitary dynamics; see LLZZ18 for more detailed discussions).

To understand and characterize the dynamical behaviors of scrambling systems, several explicit models have been proposed and investigated, such as the Sachdev-Ye-Kitaev (SYK) model PhysRevLett.70.3339 ; k . Another leading approach is the random quantum circuit model HP07 ; HQRY16 ; NVH18 ; NVH17 ; PhysRevX.8.031057 , capturing the key kinematic feature of chaos that the evolution appears to be random, and the locality of physical interactions. In these well-studied physical scrambling models, the saturation of OTOC and that of entanglement are expected to occur at a similar time scale NVH18 ; NVH17 ; HQRY16 . More generally, one could consider the dynamics of many small quantum systems (say qubits) connected according to some graph HK06 ; NS06 , with random unitary gates being applied to each edge. Suppose that we apply gates in a random order such that on average each edge has one gate applied to it per unit time. A natural conjecture here, which would be compatible with all previous results, is that the scrambling time for a constant degree graph is proportional to its diameter, i.e. the maximum distance between any two vertices. This would correspond to information traveling through the graph at a linear velocity and is assumed implicitly in previous works. However, no proof exists, outside of the special case of Euclidean lattices in a fixed number of dimensions. Even for Euclidean lattices in more than one dimension, this result was only recently proven HM18 .

Our main results are the following. The first one (Theorem 1 and Theorem 2) shows that for arbitrary graphs with sufficiently low degree, the OTOC saturation time scales linearly in the graph diameter. Here by low degree, we mean $d^{2}\geq z$ where $d$ is the dimension of the quantum system and $z$ is the degree of the graph. On the other hand, we use bounds on entanglement growth to show that the time needed to establish substantial entanglement between parts of the system scale at least as the number of vertices and thus could be longer than the OTOC saturation time, for graphs with bottlenecks (see Corollary 1). Such graphs include e.g. binary trees, which we explicitly analyze in this paper, and discretizations of hyperbolic space around black holes, originally proposed by Shor18 , which are expected to exhibit similar behaviors (as argued below). In other words, we have established an asymptotic separation between the time scales of OTOC and entanglement saturation. Refs. BGL18 ; Lu19 studied scrambling on certain peculiar graphs via a Hamiltonian model, but the relations between OTOC and entanglement were not fully understood and the physical correspondences were not clear; here we rigorously proved the separation in a general setting and studied the implications to physics.

Our results have the following major implications:

i) Scrambling in non-Euclidean geometries. Existing work studied scrambling mostly on Euclidean lattices NVH18 ; NVH17 ; JHN18 . The general assumption is that after time $t$, a localized perturbation will affect everything within some ball of radius $v_{\text{butterfly}}t$, where $v_{\text{butterfly}}$ is known as the “butterfly velocity” which characterizes the speed of information spreading. However, this has not been proved and previous works only gave heuristic arguments for it that included uncontrolled approximations. For the random circuit models defined on general graphs, we find that if the local dimension is large relative to the graph degree then indeed there is a linear butterfly velocity. We also find apparent counter-examples which suggest that linear butterfly velocity no longer holds in high-degree graphs. Some of these examples are not rigorously analyzed but we present a heuristic argument suggesting that the scrambling time for some families of graphs should grow more rapidly or more slowly than the diameter of the graphs.

ii) Black hole information scrambling. Our results can be regarded as a more rigorous argument that fleshes out the idea of a recent paper by one of the authors Shor18 , which concerns whether it is possible for the fast scrambling conjecture of black holes SS08 to hold if one assumes that the causality structure of general relativity holds around a black hole, and if the medium by which the information is scrambled is Hawking radiation. In the model of Shor18 , the space around the black hole is divided into cells, each of which contains a constant number of bits of Hawking radiation. It then gives arguments for why the Hawking radiation is not adequate for fast scrambling if the entanglement definition of scrambling is used. The cell structure around the black hole looks like a patch of a cellulation of hyperbolic geometry, where the cells on the event horizon are the boundary of this patch. The tree graph we shall consider captures a key feature of this geometry: the leaves lie on the event horizon, and the density of nodes decreases as one moves outwards radially. As the assumptions essentially suggest that information is processed via local interactions of the Hawking radiation, we may consider a random circuit defined on the underlying graph to be a toy model that captures key features of the black hole scrambling process. Our mathematical results then indicate that the scrambling time scales given by entanglement and OTOC are fundamentally different. Another way to interpret our model is that the information “wavefront” could reach the farthest side rather quickly since there exists short paths, but it takes a longer time, which scales with the number of degrees of freedom, to establish truly global entanglement. This is consistent with recent holographic calculations (see e.g. LiuSuh14:tsunami ; CENSX19 ), which suggest that the entanglement entropy grows roughly linearly after a quench in chaotic systems.

We would also like to remark upon the task of recovering quantum information falling into the black hole from Hawking radiation (commonly known as Hayden-Preskill decoding HP07 ), which plays central roles in recent studies of the black hole information problem. Yoshida and Kitaev recently proposed an explicit protocol 2017arXiv171003363Y whose decoding fidelity is at least the order of $1/{d_{A}^{2}(1-C(t))}$, where $d_{A}$ is the Hilbert space dimension of the input message. Here $C(t)$ takes the form of Eq. (1) and considers $O_{1}$ and $O_{2}$ averaged over all Pauli operators on the infalling system and Hawking radiation respectively; see Sections 2–4 of 2017arXiv171003363Y for details. By simple calculations one can see that our results imply a possible time window in which the decoding could be achieved with high fidelity without substantial entanglement when the infalling quantum state is sufficiently small compared to the black hole. However, it appears that adding a small number of qubits to a Schwarzschild black hole can only be done by photons whose wavelength is comparable to the size of the black hole. It does not seem surprising that the information carried by such photons can be extracted by a black hole quickly; when the information is absorbed by the black hole, it is already spread out over the entire black hole, and so does not need to migrate from a localized region to a state where it is delocalized on the black hole.

iii) Inequivalence of convergence to 2-designs in different measures. The speed of convergence of a random circuit to a 2-design (distributions that approximately agree with the Haar measure up to the first two moments, which have found many important applications as an efficient approximation to Haar randomness L10 ) has been the subject of a large amount of research. In particular, HL09 ; HM18 ; BHH16 ; BF12 ; BF15 show that the speed of convergence depends on the graph of interactions, and suggest that it should be proportional to the diameter. Note that 2-designs are very powerful measures of convergence, in the sense that a distribution being close to a 2-design implies that the distribution has mixed with respect to not only OTOC but also von Neumann and Rényi-2 entanglement entropies HQRY16 ; PhysRevLett.120.130502 , and other important signatures of information scrambling such as decoupling 1367-2630-15-5-053022 . Our work provides several examples where a random circuit approximates the OTOC but not the entanglement properties of a 2-design, and therefore implies that a strong approximation of 2-designs (in terms of e.g. the frame operator LLZZ18 ) may not be achieved in time proportional to the diameter.

II Models and Notation

Let $G$ be a graph with $V$ vertices and $E$ edges. The model we study consists of a graph with a $d$-dimensional Hilbert space associated with each vertex of $G$. Each edge has Haar-random unitary gates applied to qudits on its endpoints according to a Poisson process with rate 1, meaning a Poisson distribution such that $k$ unitaries are applied in time $t$ with probability $t^{k}e^{-t}/k!$). The mixing times for OTOC and entanglement, $\tau_{\rm{OTOC}}^{(x,y)}$ and $\tau_{\rm{ent}}^{(A)}$, are defined as follows.

Definition 1**.**

$\tau_{\rm{OTOC}}^{(x,y)}$* (resp. $\tau_{\rm{ent}}^{(A)}$) is defined to be the minimum amount of time needed for OTOC between vertices $x$ and $y$ (resp. the entanglement entropy between $A$ and the complement of $A$) to become at least a constant fraction of its equilibrium value. In this work we take the constant to be $1/(d^{2}+1)$.*

Here we expect that qualitatively similar behavior will hold with $1/(d^{2}+1)$ replaced by any constant strictly between 0 and 1. We will study how $\tau_{\rm{OTOC}}$ and $\tau_{\rm{ent}}$ scale with parameters such as local dimension, degree, and number of vertices.

We study the pair of $(x,y)$ that has largest $\tau_{\rm{OTOC}}^{(x,y)}$, and the set $A$ that has largest $\tau_{\rm{ent}}^{(A)}$, as they could best characterize OTOC and entanglement properties for $G$.

Instead of studying this model directly we can consider the process in which a random edge is picked every $1/E$ time units. This is because in our Poisson process model, each edge is equally likely to be picked. The number of unitaries applied within time $t$ is of order $Et$ (see Appendix A), so the two models above are equivalent up to a constant factor.

III OTOC

To analyze the saturation time of OTOC, we describe the process of operator spreading as a Markov chain. Consider an arbitrary Pauli operator $\sigma_{\vec{p}}$ acting on $n$ $d$-dimensional qudits, $\vec{p}\in\{0,\ldots,d^{2}-1\}^{n}$, and apply some unitary $U$ to it. We expand the resulting operator on Pauli basis and have

[TABLE]

The expected value of the cross term for $\alpha_{\vec{q}}$ averaged over the distribution of $U$ would be

[TABLE]

According to the construction of random circuit, this is zero for $\vec{q}\not=\vec{q^{\prime}}$ for $U$ being the unitary in a single step. Therefore in each step the values of $\overline{\alpha_{\vec{q}}\alpha_{\vec{q}}^{*}}$ undergo linear transformation, which we can interpret as a distribution because they are positive and sum to 1.

If we start from a Pauli operator located at a single vertex $x$, on each vertex all non-identity Pauli operators will have the same probability as long as $x$ has been touched at least once in the process. So we only care if the operator on a vertex is identity (I) or non-identity (N). And the norm of the time-evolved operator with a Pauli operator $P$ on some vertex $y$ would be proportional to the probability of having nonzero Pauli operator on that site, and the factor of proportionality would be

[TABLE]

which is just the commutator averaged over all non-identity Pauli operators.

In summary, the object we will study is the OTOC between Pauli operator on vertex $y$ and time-evolved Pauli operator on vertex $x$ after $T$ steps of random circuits on graph $G$, which equals to $\frac{d^{2}}{d^{2}-1}$ times the probability of having “N” on vertex $y$ after $T$ steps in the Markov chain $M_{0}$ defined below. The state space of $M_{0}$ is the set of all the configurations in which each vertex of $G$ is assigned a label “N” or “I”. The initial state of $M_{0}$ has “N” assigned to vertex $x$ and “I” assigned to all other vertices. The update rule is that in each step a uniformly random edge is picked and the labels on the two corresponding vertices are updated. “II” remains “II”, and otherwise they has a probability of $\frac{d^{2}-1}{d^{4}-1}=\frac{1}{d^{2}+1}$ for becoming “IN” or “NI” each, and $\frac{d^{2}-1}{d^{2}+1}$ for becoming “NN” HL09 .

Now we prove an upper bound for the OTOC saturation time. For illustration we present an outline of the proof here; full proof could be found in Appendix C. Note that $O(\alpha)$ that appears here and in the following is represents a quantity that scales asymptotically as $\alpha$, i.e. $\geq c_{1}\alpha$ and $\leq c_{2}\alpha$ for constants $0<c_{1}<c_{2}$.

Theorem 1** (OTOC upper bound).**

Let $G$ be a graph with $V$ vertices and $E$ edges, and suppose the degree for each vertex at most $d^{2}$, where $d$ is the Hilbert space dimension for each vertex. Then for any pair of vertices $x$ and $y$, $\tau_{\rm{OTOC}}^{(x,y)}=O(D(x,y))$ with high probability, where $D(x,y)$ is the distance between $x$ and $y$. The probability of failure is exponentially small in $D(x,y)$. As a consequence the perfect binary tree has $\tau_{\rm{OTOC}}^{(x,y)}=O(\ln V)$, where $x$ and $y$ are the farthest pair of vertices.

Proof.

As explained earlier, the OTOC saturation time corresponds to the number of steps needed for $M_{0}$ to have constant probability of having a label “N” on $y$. We will first prove Lemma 1, which states that with probability $1-e^{-O(D(x,y))}$ the vertex $y$ gets hit by a label “N” within order of $E\cdot D(x,y)$ steps. As shown in Appendix A, this needs order of $D(x,y)$ time units with high probability. Then we will show in Lemma 2 that after this happens, the probability for having an “N” on $y$ remains constant. ∎

Lemma 1**.**

Suppose that $G$ is a graph with the degree for each vertex being at most $d^{2}$. For any pair of vertices $x$ and $y$ with distance $D(x,y)$, the expected number of steps for $y$ to be labeled “N” is of order $E\cdot D(x,y)$ in $M_{0}$ starting from $x$. Besides, with high probability the vertex $y$ gets labeled “N” in time of order $E\cdot D(x,y)$. The probability of failure is exponentially small in $D(x,y)$.

Proof.

(Sketch) We will first construct a Markov chain $M$ which has the same initial state as $M_{0}$, and in each step the update rule of $M$ is applied, followed by changing all “N” into “I” except the one closest to vertex $y$. By a simple coupling argument the number of steps needed for $y$ to get an “N” in $M_{0}$ is lower bounded by that in $M$. The distance between the vertex with label “N” and vertex $y$ in Markov chain $M_{0}$ can be described by a biased random walk, from which we can obtain the desired bound. More details could be found in Appendix C. ∎

Lemma 2**.**

After a label “N” reaches the target vertex $y$, the probability for having an “N” on $y$ will remain order one.

Proof.

(Sketch) We again consider the modified chain which only keeps one label “N” after each step. We will show that vertex $y$ has constant probability of having label “N” in the equilibrium distribution. This probability is monotonically non-increasing as a function of the number of steps, so the probability is order one in any step. More details could be found in Appendix D. ∎

Theorem 1 states that the number of steps needed for OTOC saturation in a low-degree graph is at most of order $E\cdot D(x,y)$. However, we expect that in a graph with high degree, the number could be much larger. Some intuitions are given in Appendix E.

Besides this upper bound we also derive a lower bound for OTOC saturation.

Theorem 2** (OTOC lower bound).**

Let $G$ be a graph with $V$ vertices and $E$ edges, and suppose the degree for each vertex is $O(1)$. Then for any pair of vertices $x$ and $y$, $\tau_{\rm{OTOC}}^{(x,y)}$ is at least of order $D(x,y)$ with high probability, where $D(x,y)$ is the distance between $x$ and $y$. The probability of failure is exponentially small in $D(x,y)$.

The proof is presented in Appendix F.

IV Entanglement

Here we only need to consider the case where the evolution is unitary and the system is pure.

Entanglement entropy of pure state $|\psi\rangle_{AB}$ is given by $E(|\psi\rangle):=S(\rho_{A})$ where $\rho_{A}=\operatorname{Tr}_{B}[|\psi\rangle\langle\psi|]$ and $S$ is the von Neumann entropy. Notice the following simple, general fact:

Lemma 3**.**

Let $U_{AB}$ be a unitary operator acting on two $d$-dimensional systems $AB$. Then for any $|\psi\rangle_{AA^{\prime}BB^{\prime}}$ with ancilla systems $A^{\prime}$, $B^{\prime}$,

[TABLE]

Proof.

Adapted from the proof of Lemma 1 of BHLS03 . Suppose Alice holds $AA^{\prime}$ and Bob holds $BB^{\prime}$. In addition, they share two copies of the maximally entangled state $|\Phi_{d}\rangle=\frac{1}{\sqrt{d}}\sum_{i=1}^{d}|i\rangle|i\rangle$, $E(|\Phi_{d}\rangle)=\log d$. Consider the following double teleportation protocol. Alice consumes a $|\Phi_{d}\rangle$ and classical communication to teleport A to Bob, who performs $U$ locally and then consumes a $|\Phi_{d}\rangle$ and classical communication to teleport system $A$ back to Alice. The protocol is LOCC, under which the entanglement entropy between Alice and Bob is monotonically nonincreasing. Therefore, by the additivity of $S$ (and thus $E$) on tensor products,

[TABLE]

and so the claimed bound follows. ∎

Note that the proof also applies to e.g. the Rényi-2 entropy, which is a variant of the entanglement entropy that can be more easily measured in experiments Islam2015 ; 2018arXiv180605747B .

By Lemma 3, the entanglement entropy between the two trees increases by at most $2\log d$ when the random unitary is acted across the middle edge. This edge only has a probability of $1/E\sim 1/V$ to be selected in each step. So in order to reach the maximum entropy of order $V\log d$, we need at least an order of $V^{2}$ steps or equivalently order $V$ time. This is much larger than the OTOC time of order $\log V$.

From Lemma 3 we can get the following result for a general graph.

Corollary 1**.**

For a general graph $G$ with vertices partitioned into sets $A$ and $B$, the expected entanglement saturation time is at least of order $\frac{\min\{|A|,|B|\}}{C(A,B)}$, where ${C(A,B)}$ is the number of edges with one endpoint in $A$ and one in $B$.

This also lower-bounds the time it takes for the random circuit to converge to 2-designs HQRY16 ; PhysRevLett.120.130502 .

V OTOC vs. entanglement in different models

According to our results, a simple graph that can give a separation of OTOC and entanglement saturation times is a perfect binary tree, as depicted in Fig. 1. Here we consider the OTOC of operators located on the pair of farthest vertices in the graph (a leaf vertex of left subtree and a leaf vertex of the right subtree), and the entanglement entropy between left and right subtrees. The cell structure roughly equivalent to the hyperbolic geometry in 3 dimensions, or indeed any constant number of dimensions, exhibits such a separation as well. These graphs are regarded as toy models of quantum information scrambling around black holes Shor18 , as motivated in the introduction; see also Appendix B.

The behaviors of OTOC and entanglement on some other graphs are also studied. These include: i) The “dumbbell graph” consisting of two complete graphs connected by a bottleneck edge. A careful analysis could still show a separation between OTOC and entanglement; ii) High degree graphs. We demonstrate two examples in which the OTOC saturation time can be much longer or shorter than the bound we have on low degree graphs. See Appendices E, G for more details. Also note that there is no separation on Euclidean lattices. These results are summarized in Table 1.

Other than the Poisson process on each edge, different orders of choosing the edges have also been studied in Appendix G.

VI Conclusion

Random quantum circuits have widespread applications in quantum information, and are also very important models of scrambling and chaotic quantum systems in theoretical physics. There are several ways to characterize scrambling and randomness in quantum processes, among which the OTOC and entanglement are two important types of measures. This work aims to understand whether they are equivalent to each other as the signature of scrambling. To this end, we carefully analyze local random quantum circuits defined on, e.g., a binary tree, which exhibit the property that OTOC mixes rather fast since the light cone can quickly reach the far end (time of order $\ln V$), while it takes a much longer time for entanglement between the left and right subtrees to grow (time at least of order $V$). We furthermore generalize the result to any bounded-degree graph with a tight bottleneck. That is, the generation of entanglement is slow, even if the graph has small diameter. Our result indicates that unitary $t$-designs can be much more expensive than we thought: They require a random quantum circuit to have depth much larger than the diameter of the underlying graph. This result provides a more rigorous evidence for arguments made in Shor18 : if we consider the model discussed in Shor18 ; SS08 , then the scrambling of quantum information as seen by strong measures such as entanglement or decoupling can be much slower than we thought before. It would be interesting to explore further implications of this phenomenon, and more generally, quantum information processing, to the black hole information problem, many-body physics, and beyond.

Acknowledgements.

AWH was funded by NSF grants CCF-1452616, CCF-1729369, and PHY-1818914; ARO contract W911NF-17-1-0433; and the MIT-IBM Watson AI Lab under the project Machine Learning in Hilbert space. ZWL is supported by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. LK was funded by NSF grant CCF-1452616. SM was funded by NSF grant CCF-1729369. PWS is supported by the National Science Foundation under grants No. CCF-1525130 and CCF-1729369 and through the NSF Science and Technology Center for Science of Information under Grant No. CCF-0939370.