PHYSICAL REVIEW X 7, 021021 (2017)

Quantum Entanglement in Neural Network States

Dong-Ling Deng,1,* Xiaopeng Li,2,3,1 and S. Das Sarma1 1Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA 2State Key Laboratory of Surface Physics, Institute of Nanoelectronics and Quantum Computing, and Department of Physics, Fudan University, Shanghai 200433, China 3Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China (Received 25 January 2017; revised manuscript received 18 March 2017; published 11 May 2017) Machine learning, one of today’s most rapidly growing interdisciplinary fields, promises an unprecedented perspective for solving intricate quantum many-body problems. Understanding the physical aspects of the representative artificial neural-network states has recently become highly desirable in the applications of machine-learning techniques to quantum many-body physics. In this paper, we explore the data structures that encode the physical features in the network states by studying the quantum entanglement properties, with a focus on the restricted-Boltzmann-machine (RBM) architecture. We prove that the entanglement entropy of all short-range RBM states satisfies an area law for arbitrary dimensions and bipartition geometry. For long-range RBM states, we show by using an exact construction that such states could exhibit volume-law entanglement, implying a notable capability of RBM in representing quantum states with massive entanglement. Strikingly, the neural-network representation for these states is remarkably efficient, in the sense that the number of nonzero parameters scales only linearly with the system size. We further examine the entanglement properties of generic RBM states by randomly sampling the weight parameters of the RBM. We find that their averaged entanglement entropy obeys volume-law scaling, and the meantime strongly deviates from the Page entropy of the completely random pure states. We show that their entanglement spectrum has no universal part associated with random matrix theory and bears a Poisson-type level statistics. Using reinforcement learning, we demonstrate that RBM is capable of finding the ground state (with power-law entanglement) of a model Hamiltonian with a long- range interaction. In addition, we show, through a concrete example of the one-dimensional symmetry- protected topological cluster states, that the RBM representation may also be used as a tool to analytically compute the entanglement spectrum. Our results uncover the unparalleled power of artificial neural networks in representing quantum many-body states regardless of how much entanglement they possess, which paves a novel way to bridge computer-science-based machine-learning techniques to outstanding quantum condensed-matter physics problems.

DOI: 10.1103/PhysRevX.7.021021 Subject Areas: Computational Physics, Condensed Matter Physics, Quantum Physics

I. INTRODUCTION even numerically. Fortunately, physical states usually only access a tiny corner of the entire Hilbert space and can often Understanding the behavior of quantum many-body be characterized with much less classical resources. systems beyond the standard mean-field paradigm is a Constructing efficient representations of such states is thus central (and daunting) task in condensed-matter physics. of crucial importance in tackling quantum many-body One challenge lies in the exponential scaling of the Hilbert problems. Notable examples include quantum states with space dimension [1–3]. In principle, a complete description area-law entanglement [4], such as ground states of local of a generic many-body state requires an exponential gapped Hamiltonians [5] or the eigenstates of many-body amount of information, rendering the problem unattainable, localized systems [6], which can efficiently be represented in terms of matrix product states (MPS) [7–9] or tensor- *Corresponding author. network states, in general [10–12]. These compact repre- [email protected] sentations of quantum states play a vital role and are indispensable for tackling a variety of many-body problems Published by the American Physical Society under the terms of ranging from the classification of topological phases the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to [13,14] to the construction of the AdS/CFT correspondence the author(s) and the published article’s title, journal citation, [15,16]. In addition, they are also the backbones of a and DOI. number of efficient classical algorithms for solving intricate

2160-3308=17=7(2)=021021(17) 021021-1 Published by the American Physical Society DENG, LI, and DAS SARMA PHYS. REV. X 7, 021021 (2017) many-body problems, e.g., density-matrix renormalization the artificial neural networks, whose connections to the group (DMRG) [9,17,18], time-evolving block decimation entanglement features of the corresponding quantum states (TEBD) [19], projected entangled pair states (PEPS) are particularly desirable to address. In this paper, we study [10,12], and multiscale entanglement renormalization the entanglement properties, such as the entanglement ansatz (MERA) methods [20,21]. Recently, a novel entropy and spectrum, of the neural-network states. We neural-network representation of quantum many-body focus on the quantum states represented by the restricted states has been introduced [22] in solving many-body Boltzmann machine (RBM), which is a stochastic artificial problems with machine-learning techniques. However, neural network with widespread applications [40,48–50]. the entanglement properties (which are crucial for the We first prove the general result that all short-range RBM renowned MPS or tensor-network representations) of these states obey an entanglement area law, independent of neural-network states remain unknown. In this paper, we dimensionality and bipartition geometry. Since the one- fill this crucial gap by studying the entanglement properties dimensional (1D) symmetry-protected topological (SPT) of these many-body neural-network quantum states both cluster states and toric code states (in both 2D and 3D) have analytically and numerically. Our work provides an impor- an exact short-range RBM representation [51], it follows tant connection between the physical properties of many- immediately that they all have area-law entanglement. For body quantum entanglement and the computer-science long-range RBM states, calculating their entanglement properties of neural-network-based machine leaning. entropy and spectrum analytically is very challenging (if Machine learning is the core of artificial intelligence and not impossible), and we thus resort to numerical simula- data science [23]. It powers many aspects of modern tions. We randomly sample the weight parameters of the society, and its applications have become ubiquitous RBM states and compute their entanglement entropy and throughout science, technology, and commerce [24,25]. spectrum. We find that their entanglement entropy exhibits In fact, perhaps because of the dominant presence of big a volume-law scaling, in general. However, surprisingly, data in our modern world, the terms artificial intelligence, their entropy is noticeably less than the Page entropy for machine learning, neural networks, deep learning, etc. have random pure states, and their entanglement spectrum has no generically entered the lexicon of the cultural world, well universal part associated with random matrix theory and outside the technical world of computer science where they bears a Poisson-type level statistics. This indicates that the originated, often appearing in everyday press and popular — RBM states with random weight parameters live in a very articles or stories for example, the software technology restricted subspace of the entire Hilbert space (in spite of underlying automated self-driving cars crucially depends manifesting a volume-law entanglement entropy) and are on artificial intelligence and machine learning. Within not irreversible—namely, there exists an efficient algorithm physics, applications of machine-learning techniques have to completely disentangle these states [52]. recently been invoked in various contexts such as gravi- In addition, we analytically construct a family of RBM tational wave analysis [26,27], black hole detection [28], states with maximal volume-law entanglement. These material design [29], and classification of the classical states cannot be described in terms of matrix product liquid-gas transitions [30]. Very recently, these techniques states or tensor-network states with a computationally have been introduced to many-body quantum condensed- tractable bond dimension. In sharp contrast, their RBM matter physics, raising considerable interest across different representation is remarkably efficient, requiring only a – communities [22,31 44]. Exciting progress has been made small number of parameters that scales linearly with the in identifying quantum phases and transitions among them system size. This shows, in an exact fashion and in the most (either conventional symmetry-broken [33,35,37,38] or explicit way, the unparalleled power of neural networks in topological phases [32]), modeling thermodynamic observ- describing many-body quantum states with large entangle- ables [36], constructing decoders for topological codes ment. Unlike MPS or tensor-network states, entanglement [39], accelerating Monte Carlo simulations [41,42], and is not the limiting factor for the efficiency of the neural- establishing connections to renormalization group tech- network representation. As an important consequence, we niques [45,46], etc. In addition, machine-learning ideas are able to calculate (through a reinforcement-learning have also been explored in measuring quantum entangle- scheme [22,53]) the ground state, whose entanglement ment and wave-function tomography through the analyses has a power-law scaling with system size, of a spin of data extracted from quantum gas microscopes in cold- Hamiltonian with long-range interaction. Finally, we show atom experiments [47]. The fledgling field of machine- that the RBM representation could also be used as a tool to learning applications in physics appears to be in its rapidly analytically compute the entanglement entropy and spectrum growing early phase with many future breakthroughs for certain quantum states with short-range RBM descrip- expected as it matures. tions. We demonstrate this by using a concrete example of From the numerical perspective, the applications of the 1D SPT cluster states. Our results not only demonstrate machine-learning techniques to many-body problems explicitly the exceptional power of artificial neural networks would rely vitally on the underlying data structures of in representing quantum many-body states, but they also

021021-2 QUANTUM ENTANGLEMENT IN NEURAL NETWORK STATES PHYS. REV. X 7, 021021 (2017) reveal some crucial aspects of their data structures, which provide a valuable guide for the emerging new field of machine learning and many-body quantum physics.

II. NEURAL-NETWORK REPRESENTATION AND QUANTUM ENTANGLEMENT: CONCEPTS AND NOTATIONS An artificial-neural-network representation of quantum many-body states has recently been introduced by Carleo and Troyer in Ref. [22], where they demonstrated the remarkable power of a reinforcement-learning approach in calculating the ground state or simulating the unitary time evolution of complex quantum systems with strong inter- actions. We show elsewhere that this representation can be used to describe topological states, even for those with long-range entanglement [51]. To start with, let us first briefly introduce this representation in the RBM architec- ture. We consider a quantum system with N spins living on FIG. 1. A 2D pictorial illustration of artificial-neural-network a d-dimensional cubic lattice Ξ σr ; σr ; …; σr . ¼ð 1 2 N Þ quantum states in the restricted-Boltzmann-machine architecture. Correspondingly, we introduce ΞY for spins in a subsystem The yellow balls (green cubes) denote the neurons on the visible Y as ΞY ¼fσr∶r ∈ Yg. The geometric details of the lattice (hidden) layer, corresponding to the physical (auxiliary) spins. do not matter. Here, we choose cubic lattices and focus on The brown lines show the connections between visible and 1 W spin-2 (qubits) systems for simplicity. A RBM neural hidden neurons, with the weight parameter denoted by rr0 network contains two layers, one visible layer with N (only a small portion of the connections are shown for the best nodes (visible neurons) corresponding to the physical visualization). Here, we also show a typical bipartition of the system into two subsystems A and B in order to study the spins, the other a hidden layer with M auxiliary nodes h ;h ; …;h entanglement properties of the neural-network states. ð r1 r2 rM Þ (hidden neurons). The neurons in the hidden layer are connected to those in the visible layer, but there is no connection among neurons in the same layer theorems [57–59]. Nevertheless, the approximation may (see Fig. 1 for a 2D illustration). The RBM neural-network require a huge number (exponential in system size) of representation of a quantum state is obtained by tracing out neurons and parameters, thus rendering the representation the hidden neurons [22]: impractical, especially in numerical simulations. A ques- P P P X z z tion of both theoretical and practical interest is as follows: arσrþ b 0 h 0 þ W 0 h 0 σr ΦMðΞ; ΩÞ¼ e r r0 r r rr0 r r r ; ð1Þ What kind of many-body quantum states can be efficiently fhrg described by RBMs with a numerically feasible number of neurons and parameters? It is now established that entan- M where fhrg¼f−1; 1g denotes the possible configura- glement plays a crucial role in determining whether a tions of the hidden spin variables and the weights Ω ¼ quantum state can be efficiently represented by a MPS or ðar;br0 ;Wrr0 Þ are parameters that need to be trained to best tensor network or not. Quantum states with volume-law represent the many-body quantum state. It is worthwhile to entanglement cannot be described efficiently by a MPS or mention that the RBM state defined in Eq. (1) is a tensor network and thus cannot be simulated efficiently by variational state, with its amplitude and phase specified DMRG, PEPS, or MERA. In sharp contrast, as we will by ΦMðΞ; ΩÞ. The actual quantum state should be under- show in the following sections, RBMs are indeed capable stood asP (up to an irrelevant normalization constant) of efficiently describing certain specific quantum states jΨðΩÞi ≡ ΞΦMðΞ; ΩÞjΞi, similar to the Laughlin-like with volume-law entanglement, giving rise to the great description of the resonating-valence-bond ground state of potential of numerically simulating these states with new the exactly solvable Haldane-Shastry model [54,55]. machine-learning algorithms based on RBMs. We remark that RBMs can be trained in either supervised In this paper, we study the quantum entanglement or unsupervised ways, and in the machine-learning com- properties of the RBM states. In particular, we investigate munity, RBMs have had successful applications in the entanglement entropy, the Rényi entropy [60], and the classification [50], dimensionality reduction [48], feature entanglement spectrum [61], which are three of the most learning [56], and collaborative filtering [49], etc. broadly used quantities for characterizing many-body Mathematically, the ability of the RBM to approximate entanglement of a pure quantum state. These quantities any many-body state is assured by representability can be defined as follows: Considering a pure many-body

021021-3 DENG, LI, and DAS SARMA PHYS. REV. X 7, 021021 (2017) quantum state jψi, we divide the system into two sub- outside of the black hole [79–81]. Although it is apparent regions, A and B (a typical bipartition of a 2D system is that entanglement in “natural” quantum systems should shown in Fig. 1). We then construct the reduced density roughly live on the boundary and many numerical simu- matrix of subsystem A by tracing out the degree of freedom lations indeed support this intuition, rigorously proving the in B: ρAðjψiÞ ¼ TrBðjψihψjÞ. The αth order Rényi entropy area law for a given family of quantum states is notoriously is defined as challenging and often involves sophisticated mathematical techniques [4]. A breakthrough was first made by Hastings 1 A α in Ref. [5], where he proved an entanglement area law for Sα ≡ log½TrðρAÞ: 1 − α the ground states of 1D gapped local Hamiltonians by using the Lieb-Robinson bound [82]. More recently, this proof The zeroth-order (α ¼ 0) Rényi entropy is related to the has been simplified and generalized to ground states with a rank, namely, the number of nonzero singular values of ρA. When α → 1, the first-order Rényi entropy reduces to the finite number of degeneracy by a combinatorial approach von Neumann entropy, based on Chebyshev polynomials [83,84]. Unfortunately, both the Lieb-Robinson bound approach and the combi- A natorial approach seem unlikely to carry over to the case of S1 ≡ −TrðρA log ρAÞ: higher dimensions. Establishing the area-law entanglement In the literature, the entanglement entropy usually means for ground states of gapped Hamiltonians in more than the von Neumann entropy. However, throughout this paper, one dimension remains a major open problem (and argu- we do not differentiate between the entanglement entropy ably the most important one) in the field of Hamiltonian and the Rényi entropy since most of the results are valid for complexity [3]. the Rényi entropy to all orders. To define the entanglement Here, we prove that short-range RBM states obey the spectrum, we first define the entanglement Hamiltonian by area law of entanglement in any dimension for arbitrary taking the log of ρA: bipartition geometry. To be precise, we call a RBM state an R-range RBM state if each hidden neuron is only con- H ≡ − ρ ; ent log A nected to these visible neurons within an R neighborhood; 0 i.e., Wrr0 ¼ 0 if jr − r j > R. For instance, in Ref. [51],we and then the entanglement spectrum is defined as the have demonstrated that both the 1D SPT cluster states and H spectrum of ent. We mention that entanglement is nowa- toric code states (both 2D and 3D) can be represented days a central concept in many branches of quantum exactly by RBMs, with hidden neurons being connected physics. In condensed-matter physics, the entanglement only to nearest visible neurons. These states are 1-range entropy and spectrum have proven to be powerful tools RBM states. For general R-range RBM states, we have the characterizing topological phases [4,61–64], quantum following theorem: phase transitions [65–67], and many-body localization Theorem 1.—For an R-range RBM state, the Rényi [68–70], etc. A number of theoretical proposals have been entropy for all orders satisfies introduced to measure the entanglement entropy [71–74] and spectrum [75] in many-body systems. Notably, exper- A Sα ≤ 2SðAÞR log 2; ∀ α; ð2Þ imental measurements of the second-order Rényi entropy have been achieved in recent cold-atom experiments in where SðAÞ denotes the surface area of subsystem A. This optical lattices [76,77]. We expect that our study of area law is valid in any dimension and for arbitrary entanglement properties of neural-network states would bipartition geometry. also provide novel inspiration in this context. Proof.—For RBMs, since there is no intralayer connec- tions between neurons, we can explicitly factor out the III. AREA-LAW ENTANGLEMENT FOR hidden variables and rewrite ΦMðΞ; ΩÞ in a product form: Q z Q SHORT-RANGE NEURAL-NETWORK STATES arσr ΦMðΞ; ΩÞ¼ re r0 Γr0 ðΞr0 Þ, where we have intro- We start with short-range RBM states and prove that they r ≡ r∶ r − r

021021-4 QUANTUM ENTANGLEMENT IN NEURAL NETWORK STATES PHYS. REV. X 7, 021021 (2017)

In order to utilize the locality feature of Γr0 factors, we orthogonal component jφAijφBi are independent of spin can further divide the subregion A (B) into three parts: A1, configurations outside Y. Points (i) and (ii) are crucial for A2, and A3 (B1, B2, and B3), as illustrated in Fig. 2. the validity of the proof, and they are made evident by the Explicitly, the subregion with r directly coupled (via one deliberate partitions of both subregions A and B further into hidden neuron) to the lattice sites in B is defined to be A3, three smaller parts. the one directly coupled to A3 within A is defined to be A2, We emphasize that in the above proof, we did not specify and the rest of A is A1. The subregions B1;2;3 are introduced the dimensionality or the geometry of the bipartition. The correspondingly. One can regard the subregions A2, A3, B2, proof works for any dimension and any bipartition of the and B3 as hypersurfaces with thickness R in high dimen- system. Thus, it might shed new light on the important sions. Let Y ¼ A2 ∪ A3 ∪ B2 ∪ B3; then, we can rewrite challenging problem of proving the entanglement area law the R-range RBM state as for local gapped Hamiltonians in higher dimensions [5,83– X Y 85], given the possibility that all ground states of these Hamiltonians are perhaps representable by short-range jΨðΩÞi ¼ Γr0 ðΞr0 ÞjφAijφBi; ð3Þ 0 RBMs, although a rigorous proof of this still remains ΞY r ∈A3∪B3 unclear. Intuitively, one can increase the number of hidden P Q neurons to increase the number of local weight parameters. where jφAi¼ Ξ r0∈A ∪A Γr0 ðΞr0 ÞjΞAi and jφBi¼ P Q A1 1 2 When there are enough free parameters, the corresponding Ξ r0∈B ∪B Γr0 ðΞr0 ÞjΞBi. From Eq. (3), we only have, B1 1 2 RBMs should be able to represent the ground states of at most, 2jYj terms, with jYj denoting the number of spins in general local gapped Hamiltonians. This would work region Y, in the summation, and each term is a tensor because of a crucial aspect of our proof—the numbers product of orthogonal states of A and B. This gives the of hidden neurons and weight parameters are unlimited as upper bound in Eq. (2) after tracing out the degrees of long as the connections are finite-ranged. freedom in region B. We stress two crucial aspects of As shown in our previous work [51], the 1D SPT cluster Eq. (3): (i) jφAi is independent of spin configurations in states, the toric code states in both 2D and 3D, and the low- region B, and jφBi is independentQ of spin configurations in energy excited states with Abelian anyons of the toric code A 0 Γ 0 Ξ 0 region ; (ii) the coefficients r ∈A3∪B3 r ð r Þ for each Hamiltonians can all be represented exactly by short-range RBMs with R ¼ 1. An immediate corollary of the above theorem is that the entanglement of all these states fulfill an area law. In fact, based on the RBM representation, one can even analytically compute the entanglement spectrum of the 1D SPT cluster state, as we will show in Sec. V.Itis important to clarify that although the RBMs are short- ranged, their represented quantum states can capture long- range entanglement. The RBM representations of the toric code states (both in 2D and 3D), which have intrinsic topological orders (long-range entangled), are such exam- ples. The area law of short-range RBM states does not imply short-range entanglement. This distinction between short-ranged in the RBM sense and short-ranged in the entanglement sense is an important point. In 1D, the area-law bound in Eq. (2) gives rise to an interesting relation between the RBM and MPS represen- FIG. 2. A sketch for the proof of area-law entanglement for tations of quantum many-body states. It has been proved short-range restricted-Boltzmann-machine states. The system is that a bounded Rényi entropy of all orders in 1D neces- divided into two subsystems, A and B, with the red line showing sarily guarantees an efficient MPS representation [86] (note A the interface boundary. In order to show that the Rényi entropy Sα that a counterexample does exist if only the von Neumann α A B obeys an area law for any , the subsystem ( ) is further entropy is bounded [87]). As a result, our area-law results divided into three parts: A1, A2, and A3 (B1, B2, and B3). Since the Γ r ∈ A ∪ A imply that all 1D short-range RBM states can be efficiently neural network is short range, the r factors with 1 2 described in terms of MPS. However, the validity of the (r ∈ B1 ∪ B2) are independent of spin configurations in region B inverse statement is unknown. It would be interesting to (A); thus, we can group the spins in regions A1 and B1 with their Γr factors. The entropy of the reduced density matrix ρA then only find out whether all MPS descriptions with small bond depends on the degree of freedom in region A2 ∪ A3, which is dimensions have efficient RBM representations or not, and proportional to the surface area of region A. This gives us a clear if so, what the general procedure is for recasting MPS into A geometric picture of why Sα is upper bounded by the surface area RBM states. It would also be interesting to investigate the of A, up to an unimportant scaling constant as given in Eq. (2). relations between higher-dimensional RBM states and

021021-5 DENG, LI, and DAS SARMA PHYS. REV. X 7, 021021 (2017) tensor-network states, PEPS, or MERA. Nonetheless, it is presents serious practical limitations in solving problems worth emphasizing here that the entanglement scaling of involving volume-law entanglement states. As introduced RBM states is sharply distinctive from MPS—the maximal in the previous section, the construction philosophy of entanglement entropy of an R-range RBM state scales neural-network states is very different from that of MPS or linearly with R, whereas a bond-dimension (χ) MPS has an tensor-network states. This gives rise to the possibility for entanglement entropy scaling as log χ. This implies that neural networks to efficiently represent quantum states and even though a RBM state can be generically converted to a solve problems with volume-law entanglement. We also MPS, the parametrization in RBM states is much more stress that our exact results here provide an important efficient for representing highly entangled quantum states. anchor point for future theoretical and numerical studies We remark that our rigorous proof of entanglement area and should have far-reaching implications in the applica- law for short-range RBMs provides a valuableguide for some tions of machine-learning techniques in solving currently practical numerical calculations. For instance, in some intractable many-body problems. In Sec. IV C, we indeed circumstances, we know that the problem may only involve use RBMs to solve the ground state (with massive power- a small amount (an area-law) entanglement; then, we may law entanglement) of a modified Haldane-Shastry model use short-range, rather than long-range, RBMs to reduce with long-range interactions by using the reinforcement the number of parameters and consequently speed up the learning. calculations (we have tested this in a numerical experiment of We first give a 1D example. Let us consider a 1D system finding the ground state of the transverse-field Ising model of N qubits. The goal is to construct a RBM state with via reinforcement learning, and a considerable speedup has maximal volume-law entanglement entropy. To this end, indeed been obtained). On the other hand, if the problem to be we introduce a RBM with N visible and M ¼ ⌊ð3NÞ=2⌋ − 1 solved involves large entanglement (such as some quantum hidden neurons. Here, the floor function ⌊x⌋ denotes the criticality or quantum dynamic problems), then short-range largest integer less than or equal to x. The weight RBMs will not necessarily work, and we should choose a parameters of ΦMðΞ; ΩÞ, which characterize the RBM as long-range RBM to begin with. defined in Eq. (1), are chosen to be a 0; ∀ k ∈ 1;N ; IV. VOLUME-LAW ENTANGLEMENT IN k ¼ ½ ð4Þ LONG-RANGE NEURAL-NETWORK STATES 8 iπ < − 4 k ∈ ½1;N− 1 In the last section, we proved that all short-range RBM b h i k ¼ : iπ 3N ð5Þ states satisfy an area-law entanglement. What about RBM 2 k ∈ N;⌊ 2 ⌋ − 1 ; states with long-range neural connections? From the linear- R R in- entanglement-entropy scaling of -range RBM states iπ 0 4 ðk ;kÞ ∈ S derived in the last section, we would anticipate that long- Wk0k ¼ ð6Þ range RBM states could exhibit volume-law entanglement. 0 otherwise; In this section, we explicitly show that this is indeed true by S S ≡ a rigorous exact construction and a numeric benchmark. where is a set of paired integers defined by fði;jÞ∶i ∈ ½1;N−1 and j ¼ i;iþ1 or i ∈ ½N;⌊ð3NÞ=2⌋−1 and j ¼ iþ1−N;iþ1−⌈N=2⌉g. The ceiling function ⌈x⌉ A. Exact construction of maximal volume-law denotes the smallest integer greater than or equal to x.A entangled neural-network states pictorial illustration of this RBM is shown in Fig. 3.Now, Here, we analytically construct families of neural- we show that the quantum states described by the above network states with volume-law entanglement. These states RBM have volume-law entanglement entropy for any are exact and have unified closed-form RBM representa- contiguous region no larger than half of the system size. tions. More strikingly, the RBM representation of these To be more precise, we have the following theorem. states is surprisingly efficient—the number of nonzero Theorem 2.—For a 1D RBM state with weight param- parameters scales only linearly with the system size. We eters specified by Eqs. (4)–(6), if we divide the system into stress that efficient representations of quantum states play a two parts A and B, with A consisting of the first l vital role in solving many-body problems, especially when (1 ≤ l ≤ ⌊N=2⌋) qubits and B the rest, then the corre- numerical approaches are employed. A prominent example sponding Rényi entropy of ρA is is the advantageous usage of the MPS representation in A DMRG [9] (for the ground states), TEBD [19] (for time Sα ¼ l log 2; ∀ α: ð7Þ evolution), and DMRG-X [88] (for highly excited eigen- states of local Hamiltonians deep in the many-body Proof.—As mentioned in Sec. III, since there localization region) algorithms. However, the MPS or is no intralayer connection between neurons for tensor-network representation is efficient only in describing a RBM, we can explicitly factor out the hidden quantum states with area-law entanglement and thus variables and rewrite ΦMðΞ; ΩÞ in a product form:

021021-6 QUANTUM ENTANGLEMENT IN NEURAL NETWORK STATES PHYS. REV. X 7, 021021 (2017) P jΨðΩÞi ≡ Ξ χΞ jΞAijΞB ∶ΞB ¼ ΞAijΞB i, where A∪B2 A∪B2 1 1 2 χΞ ¼C is a coefficient depending on ΞA∪B, with C A∪B2 a positive constant. By tracing out the degrees of freedom in region B and putting back the normalization constant, we l obtain the reduced density matrix ρA ¼ I=2 , with I the identity matrix of dimension 2l × 2l. This completes the proof. It is worthwhile to mention that the subregion A does not necessarily have to be at the left end. In fact, A can be any contiguous region of length l, and Eq. (7) still remains valid, although the details of the proof would change slightly in this situation. We choose A to be at the left end just for convenience. In the limit N → ∞, for any contiguous region, its entanglement entropy scales linearly FIG. 3. An illustration of the constructed 1D neural-network with the size of the region—a volume-law entanglement. state with a maximal volume-law entanglement entropy. This For the 2D case, we can construct volume-law-entangled N ⌊ 3N =2⌋ − 1 restricted Boltzmann machine has visible and ð Þ RBM states in a similar manner. We consider a system of N hidden neurons. For 1 ≤ k ≤ N − 1 (N ≤ k ≤ ⌊ð3NÞ=2⌋ − 1), the L L Λ k k qubits living on an x × y square lattice denoted as .We th hidden neuron is connected to two visible neurons at sites L L and k þ 1 (k þ 1 − N and k þ 1 − ⌈N=2⌉) with connection assume x and y are even integer numbers, for simplicity weight parameters equal to ðiπÞ=4. The on-site potentials for (one can use the floor and ceiling functions to deal with the the visible neurons are chosen to be zero, ak ¼ 0 (∀k). For the case of odd numbers, but the notations will be more hidden neurons, bk is chosen as bk ¼ −½ðiπÞ=4 (1 ≤ k ≤ N − 1) cumbersome). We label each vertex of the lattice by a and bk ¼½ðπiÞ=2 (N ≤ k ≤ ⌊ð3NÞ=2⌋ − 1). The scissors show a pair of indices (kx, ky)(1 ≤ kx ≤ Lx and 1 ≤ ky ≤ Ly) and cut of the system into two subsystems (A and B) with equal sizes, N L L A attach a qubit ¼ x × y to it. We construct a RBM with and for this bipartition, the Rényi entropy is Sα ¼ ⌊N=2⌋ log 2, 5 N visible and LxLy − Lx − Ly hidden neurons. The proportional to the system size. This is also the maximal amount 2 of entropy one can have for a system with N qubits. hidden neurons are divided into three groups. The first (second) group, denoted by X (Y), has ðLx − 1Þ × Ly Q Q [Lx × Ly − 1 ] neurons that connect nearest visible neu- N a σz M ð Þ Φ Ξ; Ω e k k Γ Ξ Γ Ξ Mð Þ¼Pk¼1 k0¼1 k0 ð Þ, with k0 ð Þ¼ rons along the x (y) direction. The third group, denoted by 2 b W σz a 0 1 coshð k0 þ k k0k kÞ. FromQ Eq. (4), k ¼ for all Z, contains Lx × Ly hidden neurons that connect visible N a σz 2 k ∈ 1;N e k k ½ ; thus, the first term k¼1 simply equals neurons nonlocally. One can draw an analogy with the 1D 1 and can be omitted from ΦMðΞ; ΩÞ. Consequently, example: The neurons in groups X and Y connect nearest the variational wave function ΦMðΞ; ΩÞ only depends visible neurons, and they correspond to the first N − 1 on the Γk0 factors, which correspond to the hidden neurons. hidden neurons in the 1D case; similarly, those in group Z 0 As shown in Fig. 3, for k ∈ ½1;N− 1, Γk0 ¼ correspond to the remaining ones. The hidden neurons in 2 − iπ=4 iπ=4 σz σz X Y Z x x ;x y y ;y cosh½ ð Þþð Þð k0 þ k0þ1Þ connects its corre- , , and are labeled by ¼ð 1 2Þ, ¼ð 1 2Þ, and 0 sponding hidden neuron at site k to two nearest-neighbor z ¼ðz1;z2Þ, respectively. Following the 1D example, the k0 k0 1 Γ visible neurons at sites and ffiffiffiþ . Note that k0 has only weight parameters can be chosen as p z z two possible values: Γk0 ¼ − 2 if σ 0 ¼ σ 0 ¼ −1 and pffiffiffi k k þ1 a 0; ∀ k ;k ∈ Λ; 0 kxky ¼ ð x yÞ Γk0 ¼ 2 otherwise. For k ∈ ½N;⌊ð3NÞ=2⌋ − 1, Γ 2 iπ=2 iπ=4 σz σz iπ iπ k0 ¼ cosh½ð Þþð Þð k0 1−N þ k0 1−⌈N=2⌉Þ con- ðXÞ ðYÞ ðZÞ þ þ bx ¼ by ¼ − ;bz ¼ ; k0 4 2 nects its corresponding hidden neuron at site to two far-away, separated, visible neurons at sites iπ ðX=Y=ZÞ ðX=Y=ZÞ 4 ðx=y=z; kx;kyÞ ∈ S2D k0 1 − N k0 1−⌈N=2⌉ Γ W ¼ þ and þ . Here, k0 vanishes if x=y=z;kxky 0 ; σz −σz Γ 2 Γ −2 otherwise k0þ1−N ¼ k0þ1−⌈N=2⌉ and k0 ¼ ( k0 ¼ )if z z z z σ 0 ¼ σ 0 ¼ −1 (σ 0 ¼ σ 0 ¼ 1). ðXÞ ðYÞ ðZÞ k þ1−N k þ1−⌈N=2⌉ k þ1−N k þ1−⌈N=2⌉ where S , S , and S are the three sets that specify the Γ 2D 2D 2D These features of the k0 factors are crucial in the following connections between the visible neurons and the proof of Eq. (7). three groups of hidden neurons. They are defined as For convenience, we define two sets of integers: B1 ¼ ðXÞ ðYÞ S2D ≡ fðx; kx;kyÞ∶kx ¼ x1;x1 þ 1; ky ¼ x2g, S2D ≡ fj∶N þ 1 − ⌈N=2⌉ ≤ j ≤ N þ l þ 1 − ⌈N=2⌉g and B2 ¼ ðZÞ fj∶lþ1

021021-7 DENG, LI, and DAS SARMA PHYS. REV. X 7, 021021 (2017) if ðLx=2Þ

(a) (b) (c)

6 N = 14 6 = 1 = 1 6 = 2 N = 16 = 3

=2 3 N = 18 = 4

A

N=20 1/2 6 = 3 4 4 1 A S

= 4 S Page entropy 2 2

) 4 2 10 15 20

1 A A 2 1 A N

S 024 S 4 ( S =1

= 1 = 2 = 3 = 2 1 = 4 = 3 2 2 = 4

0 10 15 20 246 10 15 20 A N S 1 N

FIG. 4. Entanglement scaling and distributions of neural-network states with random weight parameters. (a) The averaged von A A Neumann entropy S1. For small γ (γ ¼ M=N denoting the ratio between the number of hidden and visible neurons), S1 scales linearly A with the system size, indicating a volume-law entanglement. However, S1 tends to saturate when γ becomes large (γ ¼ 4). Moreover, for A all γ values studied, S1 deviates significantly from the Page value for the entanglement entropy averaged over random pure states. A (b) The entanglement distribution of S1 for different γ. The peak shifts to the left as γ increases, which is consistent with the observation A 4 that the averaged S1 decreases as we increase γ. Here, the system size is chosen to be L ¼ 20, and we have used 10 random samples. A A The inset shows S1 as a function of γ for different system sizes. Note that S1 reaches its maximal value at γ ≈ 0.7. However, even this A maximal value of S1 is still noticeably smaller than the Page value for random states. (c) The scaling of the Rényi entropy with the A A A system size. The second-order Rényi entropy S2 behaves similarly to the von Neumann entropy S1. The inset shows the results for S1=2. 0 For simplicity, we have chosen the on-site potentials ak (k ∈ ½1;N) and bk0 (k ∈ ½1;MÞ to be zero. The connection weight parameters Wkk0 are complex numbers randomly drawn from uniform distributions with ReðWkk0 Þ ∈ ½−3=N; 3=N and ImðWkk0 Þ ∈ ½−π; π.

021021-8 QUANTUM ENTANGLEMENT IN NEURAL NETWORK STATES PHYS. REV. X 7, 021021 (2017) less dependent on the spin configuration Ξ as γ increases. In and dB denote the Hilbert space dimensions of subsystems other words, in the represented many-body quantum wave A and B, respectively [89]. An interesting observation in function, the difference between the coefficients of each Fig. 4(a) and the inset of Fig. 4(b) is that the entanglement component becomes smaller and smaller. Thus, the state entropy is always smaller than the Page entropy for all γ. becomes closer and closer to a product state, and therefore, This implies that the pattern of entanglement for the RBM the entanglement decreases. This result is further justified states with random parameters is distinct from that of A in the inset of Fig. 4(b), where the von Neumann entropy S1 random pure states, which is consistent with the fact that is shown as a function of γ for different system sizes. From the RBM states live in a very small restricted subspace of A the entire Hilbert space. This also indicates that a random this figure, we see that S1 reaches its maximal value at a critical γ ≈ 0.7, independent of system size. When γ > γ, state in the Hilbert space is probably not well described by a A RBM efficiently. In Fig. 4(b), we plot the entanglement S decreases as we increase γ. It is also worthwhile to 1 distribution for different γ. We find that, as γ increases, the mention that when we fix M as a finite number (γ → 0 in N → ∞ SA distribution becomes broader, and the density peak shifts the thermodynamic limit ), then 1 is upper towards smaller values. This result is in agreement with the M 2 bounded by log , regardless of the system size. This observation in Fig. 4(a) that the entanglement decreases as result can be understood heuristically by imagining all the γ increases. Figure 4(c) shows the results for the Rényi B hidden neurons being grouped into subsystem ; then, entropy of orders α ¼ 2 and α ¼ 1=2. As expected, S2 A 2M subsystem can only have, at most, degrees of freedom behaves very similarly to S1.ForS1=2, we find a similar B SA that are entangled with . Consequently, 1 is bounded by volume-law scaling of entanglement, but the bending M log 2. This explains the numerical observation in the feature does not show up at γ ¼ 4 because of finite-size A inset of Fig. 4(b), that for M ¼ 1 (smallest γ), S1 ≈ log 2 effects. independent of the system size. The entanglement entropy studied above provides a In order to compare the RBM states with random wealth of information about the data structure of the parameters with generic random pure states, we also RBM states. However, as has been realized in a number calculate the so-called Page entropy [89], which is the of different physical contexts, the entanglement entropy averaged entanglement entropy over pure states drawn cannot capture the full entanglement structure of the system randomly from the entire Hilbert space of the system. [96–101]. Much greater information can be extracted from The Page entropy provides an estimate for entanglement the entanglement spectrum. In order to obtain a more in extended thermal states [90] and has been widely comprehensive understanding of the data structure of the used in the context of quantum chaos [91], black hole RBM states with random weight parameters, we have information [92,93], and many-body localization [94,95]. therefore also calculated their entanglement spectra and From the random matrix theory,P it can be computed the entanglement Hamiltonian level statistics. In Fig. 5(a), S − d − 1 =2d dAdB 1=k d γ as Page ¼ ½ð A Þ Bþ j¼dBþ1ð Þ, where A we plot the averaged entanglement spectrum for different .

(a) 0 (b) 0.2 (c) 1

= 1

= 1 =2 = 3 = 2 -10 Poisson = 3 GOE

) GUE k ) ) r

e GSE ( ( 0.1 0.5 -20 log( = 1

= 2

= 3 -30 Marchenko-Pastur 0 0 0 200 400 0 1020300246 k e r

FIG. 5. Entanglement spectrum and level statistics for neural-network states with random weight parameters. Here, the lattice size is chosen to be L ¼ 20, and we have used 104 random samples. The random parameters are drawn from the same distribution as specified in Fig. 4. (a) Averaged entanglement spectrum with different γ. Here, ξk denotes the kth eigenvalue of ρA (the eigenvalues are arranged in descending order). The red line denotes the spectrum of a completely random state (derived from a Marchenko-Pastur distribution). It is evident that the entanglement spectra of the restricted-Boltzmann-machine states with random parameters are completely distinct from the Marchenko-Pastur distribution, indicating that their corresponding entanglement Hamiltonians are very different from Wishart matrices. (b) Density of states for the entanglement Hamiltonians. Here, e denotes the eigenspectrum of Hent. When increasing γ, the distribution of the eigenvalues broadens, and the peak shifts rightwards. (c) Distributions of the ratios of consecutive spacings for the entanglement spectrum. These distributions follow a Poisson law and differ significantly from the predictions of GOE, GUE, or GSE.

021021-9 DENG, LI, and DAS SARMA PHYS. REV. X 7, 021021 (2017)

We find that the entanglement spectrum for the RBM states random matrix theory and is governed by the same random is completely different from the Marchenko-Pastur distri- matrix ensemble as the energy spectrum. However, in the bution derived from random matrix theory [102]. More many-body localized phase, the entanglement spectrum specifically, the Marchenko-Pastur distribution describes shows a semi-Poisson distribution, in contrast with the the asymptotic average density of eigenvalues of a Wishart energy spectrum following a Poisson law. For the RBM matrix (a matrix of the form Y ¼ XX†, with X a random states with random weight parameters studied in this rectangular matrix). It was shown recently in Ref. [97] that section, we find that their entanglement spectra follow a the entanglement spectrum of highly excited eigenstates in Poisson distribution, as shown in Fig. 5(c). The averaged 4 the delocalized phase bears a two-component structure: value of rn (over 10 random samples) with N ¼ 20 and (i) a universal part that is associated with random matrix γ ¼ 3 equals 0.378, which is in good agreement with the theory, i.e., a universal tail that follows the Marchenko- Poisson predicted value 2 ln 2 − 1 ≈ 0.386 [107]. The small Pastur distribution and thus is model independent, and (ii) a deviation could be attributed to finite-size effects. Thus, model-dependent nonuniversal part that dominates the these RBM states are distinct from the eigenstates of weights in the spectrum. In the localized phase, the Hamiltonians in either the delocalized or localized phases universal part of the spectrum disappears in the thermo- on average. In addition, we also remark that the Poisson dynamic limit, leaving only the nonuniversal part that leads behavior and the lack of a universal part in the entangle- to an area-law scaling of the entanglement entropy. In our ment spectra of these RBM states imply that they are not case for the RBM states with random weight parameters, irreversible—namely, there exists an efficient algorithm to the universal part disappears completely even for a system completely disentangle these states [52]. Finding the size as small as N ¼ 20. In this sense, these RBM states are disentangling algorithm would provide some insight into less random than the highly excited eigenstates in both the the nature of neural-network quantum states and is an delocalized and localized phases. This further shows that, interesting topic for future investigation. although these RBM states obey a volume law of entan- glement entropy on average, they are living in a very C. Reinforcement learning of ground restricted subspace of the entire Hilbert space. In Fig. 5(b), states with power-law entanglement we plot the density of states for the entanglement The above discussion shows that, unlike MPS or tensor- Hamiltonian of these RBM states. We find that a broader γ network states, entanglement is not the limiting factor for distribution shows up as we increase , and the peak moves the efficiency of the RBM representation. As an important to the right, which is consistent with the surprising results γ consequence, RBM might be capable of solving some from Fig. 4 (i.e., entanglement decreases as increases). quantum many-body problems where massive entangle- Another quantity that is also useful in understanding the r ment is involved. To demonstrate this unprecedented data structure of the RBM states is the adjacent gap ratio power, in this section, we consider the problem of finding defined as rn ¼½minðδn; δn−1Þ= maxðδn; δn−1Þ, with δn the n n − 1 the ground state (with power-law entanglement) of a spin- level spacing between the th and the ( )th eigenstates 1=2 Hamiltonian with long-range interaction, through a of the entanglement Hamiltonian. We note that the impor- N r reinforcement-learning scheme [22,53]. We consider tance of the distribution of has been broadly appreciated spin-1=2 particles living on a ring (see Fig. 6) with a in various contexts. In quantum chaos [103], it is argued modified Haldane-Shastry [54,55] Hamiltonian given by that whereas the level statistics for integrable quantum Hamiltonians obeys a Poisson law [104], the case for XN 1 Hamiltonians with chaotic dynamics must follow one of the H −σˆ xσˆ x − σˆ yσˆ y σˆ zσˆ z ; MHS ¼ d2 ð i j i j þ i jÞ ð8Þ three classical ensembles from random matrix theory [105], j

021021-10 QUANTUM ENTANGLEMENT IN NEURAL NETWORK STATES PHYS. REV. X 7, 021021 (2017)

(a) (b) 0.8 RBM, z z Numerical data 1 1+j RBM, x x Power-law fit 1 1+j ED, z z 1 1+j 1.2 ED, x x 0.4 1 1+j 1 A S

1 0 Spin correlations

0.8 -0.4 5101520 246810 N j (c) (d) (1) W 0 2 W (2) (3) W -1.6 (4) 1 W j )

-1.8 f z 1+ ( j N / 1 z 0 W 0 -0.2 E 1 -2 FIG. 6. The modified Haldane-Shastry model. The N spin-2 particles form an equally spaced lattice on a ring. Each spin is -1 -2.2 interacting with all the other spins. The interaction between spin j 0 200 400 600 -2 Iteration number and k is of Heisenberg XXZ type, and its strength is inversely -0.4 0 5 10 15 20 0 1020304050 proportional to the square of the chord distance djk [see Eq. (8)]. j j The ground state of this model has power-law entanglement and can be calculated with a restricted Boltzmann machine through FIG. 7. Reinforcement learning of the ground state of the reinforcement learning. modified Haldane-Shastry model with power-law entanglement. (a) The von Neumann entropy calculated by exact diagonaliza- tion (ED) of the Hamiltonian with different system sizes. It has an reinforcement learning, despite the fact that the ground A 0.106 excellent power-law fit: S1 ∼ 2.473N − 2.033. Here, the state has a large amount of entanglement entropy. Since the entropy is calculated from an equal bipartition of the system. Hamiltonian has a lattice translation symmetry, we can use (b) Spin correlations for small system sizes calculated by ED and this symmetry to reduce the number of variational param- a restricted Boltzmann machine, respectively. (c) The learned eters, and for integer hidden-variable density (γ ¼ 1; 2; …), feature maps for representing the ground state with a restricted WðfÞ Boltzmann machine. In panels (b) and (c), the hidden-unit density the weight matrix takes the form of feature filters j with γ 4 N 20 f ∈ 1; γ ¼ , and the system size is fixed to ¼ . (d) Reinforcement ½ , as described in Ref. [22]. In Fig. 7(b), we plot the learning of the spin correlations and ground-state energy density different spin correlations obtained via reinforcement for larger system size N ¼ 100. The inset shows the variational learning, for small system sizes. We compare the RBM energy density as a function of the iteration number of the result with that from exact diagonalization. As shown in learning process. As the iteration number increases, the this figure, the RBM result matches the ED result very well. energy converges smoothly to an asymptotic value around The accuracy of the RBM result can be improved by E0=N ≈−2.12. increasing γ and the number of iterations in the training process. In Fig. 7(c), we show the feature maps after a typical reinforcement learning process with γ ¼ 4 and long-range interactions are involved. Moreover, as pointed N ¼ 20. The accuracy of the trained RBM can be quanti- out in Ref. [22], the RBM approach also works in higher ϵ dimensions and for dynamical problems. We also mention fied by the relative error on the ground-state energy rel ¼ H that MHS might be realized with trapped ions in a ring ðRBMÞ ED ED jðE0 − E0 Þ=E0 j [22]. For the parameters shown in geometry [109,110]. Thus, the numerically calculated ϵ ∼ 10−5 Fig. 7(c), we find rel . We then calculate the correlations could be experimentally verified in the future. correlation functions and ground-state energy density for larger system sizes, which are far beyond the capability of V. AN ANALYTICAL RBM RECIPE FOR the ED technique. We plot some of the results for N ¼ 100 σˆ z σˆ z CALCULATING ENTANGLEMENT in Fig. 7(d). We find that the correlation h 1 1þji has a sharp jump at j ¼ 2, which is also obtained in our ED In Sec. III, we proved that all short-range RBM states calculations for smaller system sizes, as shown in Fig. 7(b). obey an area-law entanglement. Can we calculate the We remark that the DMRG/MPS-based simulations are entanglement entropy and spectrum analytically? For a particularly challenging for the above problem and would general many-body state, this is an outstanding challenge, presumably require a substantially larger number of varia- especially for a system with a finite size. In fact, most of the tional parameters than the RBM approach [9,108]. In this past works focus on the thermodynamic limit and compute regard, the reinforcement-learning-based RBM technique entanglement entropy asymptotically. The methods used has apparent advantages when large entanglement and often involve complicated mathematics [4]. For instance,

021021-11 DENG, LI, and DAS SARMA PHYS. REV. X 7, 021021 (2017) X the Fisher-Hartwig formula has been used to evaluate the 1 jΨi ¼ Γα Γα Γβ Γβ jΨAijΨBi; ð11Þ cluster 4 2 4 2 4 asymptotic behavior of the entanglement entropy for the Ξ critical XX model and other isotropic models [111–114]. A2∪B2 Another notable approach is the use of conformal field P Q theory, where some universal properties of entanglement with jΨAi¼jΨAðΞA Þi ¼ 2 Ξ Γα Γα k0∈A Γk0 ðΞAÞjΞAi 2 P A1 1 3Q 1 entropy have been established for critical (1 þ 1)- and jΨBi¼jΨBðΞB Þi ¼ 2 Ξ Γβ Γβ k0∈B Γk0 ðΞBÞjΞBi. dimensional systems [115–117]. For calculating the exact 2 B1 1 3 1 Equation (11) is crucial in calculating the entanglement entanglement entropy of a finite system, the quotient entropy and spectrum. Compared with Eq. (10), it contains group method has been used to calculate the entropy of 28 2N an arbitrary bipartition of the 2D toric code states only , rather than , terms in the summation. Noting that jΨAi (jΨBi) only depends on the spin configurations within [118,119]. Here, however, we introduce an alternative 0 subregion A2 (B2) and hΨAðΞA ÞjΨAðΞA Þi ¼ δΞ ;Ξ 0 , one approach and show that the RBM representation would 2 2 A2 A2 also be helpful in the analytical calculation of the entan- can perform a unitary transformation UA (UB) within glement entropy and spectrum. As an example, we consider subregion A (B) to rotate the basis of the Hilbert space Ψ H H A B the 1D SPT cluster state j icluster, which is the ground A ( B)of ( ). Note that this rotation will not affect the H state of the cluster Hamiltonian cluster defined on a 1D entanglement entropy and spectrum. In the new basis, H Ψ Ξ Ψ Ξ H H latticeP with a periodic boundary condition: cluster ¼ j Að A2 Þi [j Bð B2 Þi] is just a basis vector of A ( B). By − N σˆ z σˆ xσˆ z B k¼1 k−1 k kþ1. This state is a topological state pro- tracing out the degrees of freedom in subregion and tected by Z2 × Z2 symmetry [120]. It serves as a simple toy plugging in the parameter values in Eq. (9), we find a very model for studying SPT phases and has important appli- simple expression for the reduced density matrix ρA in the cations in measurement-based quantum computation [121– new basis, 123]. An exact and efficient RBM representation of the 1D cluster state has been found in Ref. [51]. This representa- ρA ¼ M1 ⊕ M1 ⊕ M2 ⊕ M2 ⊕ 0; ð12Þ tion has N hidden neurons, with each one connecting only locally to the visible neurons within a distance of 1. The weight parameters are specified as iπ iωμ if jk − jj¼1 ak ¼ 0;bk ¼ ;Wkj ¼ ð9Þ 4 0 otherwise; where ωμs(μ ¼ 1,0,−1) are positive real numbers given by ðω1; ω0; ω−1Þ¼ðπ=4Þð2; 3; 1Þ. In the product form, the normalized 1D cluster state reads

X YN Ψ Γ Ξ Ξ ; j icluster ¼ k0 ð Þj i ð10Þ Ξ k0¼1

where the Γk0 factor only depends on the configurations of Γ Ξ Γ σz ;σz ;σz three nearest visible neurons k0 ð Þ¼ k0 ð k0−1 k0 k0þ1Þ¼ π=4 1 2σz 3σz σz cos½ð Þð þ k0−1 þ k0 þ k0þ1Þ (note that we returned Γ the normalization constant and rescaled all the k0 factor FIG. 8. A sketch for analytically computing the entanglement by 1=2). entropy and spectrum of the 1D symmetry-protected-topological In order to study its entanglement properties, we con- cluster state, through the corresponding exact short-range re- sider an arbitrary bipartition of the system into two parts, A stricted-Boltzmann-machine representation. The scissors show a and B, and we aim to calculate the entanglement entropy cut of the system into two subsystems A and B. We calculate the and spectrum of subsystem A analytically. For convenience, entanglement entropy and spectrum from the reduced density ρ we further divide the subregion A (B) into two parts, A1 and matrix A. In order to conveniently explore the short-range feature of the restricted-Boltzmann machine, we further divide A2 (B1 and B2), with A2 (B2) containing only four sites, α1, A (B) into A1 and A2 (B1 and B2). It is important that the Γk0 α2, α3, and α4 (β1, β2, β3, and β4), as shown in Fig. 8. factors in subregion A1 (B1) are independent of spin configura- Using the fact that the RBM is short-ranged and pffiffiffi tions in B (A), such that the summations of spin configurations Γ Ξ 2=2 Ψ k0 ð Þ¼ð Þ, we can rewrite j icluster in the follow- in A1 and B1 are interchangeable and can be factorized out ing form, where the subregions A and B appear explicitly, explicitly, as shown in Eq. (11).

021021-12 QUANTUM ENTANGLEMENT IN NEURAL NETWORK STATES PHYS. REV. X 7, 021021 (2017)

1 1 where M1 ¼ 4 jψ 1ihψ 1j and M2 ¼ 4 jψ 2ihψ 2j are four-by- For these states, the RBM representation is remarkably 1 efficient, requiring only a small number of parameters four matrices with jψ 1i¼2 ðj0iþj1iÞ ⊗ ðj0i − j1iÞ and 1 scaling linearly with the system size. These results explic- jψ 2i¼2 ðj0i − j1iÞ ⊗ ðj0i − j1iÞ, respectively; 0 is a zero NA NA itly show, in an exact fashion, the remarkable power of matrix of dimension 2 1 × 2 1 , with NA denoting the 1 artificial neural networks in describing quantum states with A number of spins in subregion 1. From Eq. (12), the massive entanglement. Unlike MPS or tensor-network ρ ρ eigenvalues of A can readily be obtained. Note that A has states, entanglement is not the limiting factor for the only four nonzero eigenvalues that are degenerate and equal 1 efficiency of the neural-network representation of quantum to 4. As a result, the Rényi entropy is given by many-body states. Through reinforcement learning of a

A modified Haldane-Shastry model, we have shown that Sα ¼ 2 log 2; ∀ α: RBM is capable of calculating the ground state, which has power-law entanglement. The corresponding ground- H For the entanglement spectrum, ent has a fourfold state energy and correlations can also be efficiently degeneracy, with the four smallest eigenvalues equal to obtained. Finally, we also demonstrated, through a concrete 2 2 log and the rest infinite. The fourfold degeneracy is a example, that the RBM representation could be used as a – signature of SPT phases [61,124 126]. tool to analytically compute the entanglement entropy and We expect that this RBM approach would carry over to spectrum for finite systems. Our results reveal some crucial calculating the entanglement entropy and spectrum for the aspects of the data structures of neural-network quantum 2D and 3D toric code states, whose RBM representation has states and provide a useful guide for the practical appli- already been given in Ref. [51], although the calculation will cations of machine-learning techniques in solving quantum be more technically involved. Undoubtedly, like all analyti- many-body problems. cal methods in calculating entanglement, our RBM approach Many open questions remain. First, what are the limiting has obvious limitations and cannot be applied, in general, to factors for RBMs in efficiently representing quantum an arbitrary short-range RBM state. First, given a specific states? In the future, it would be interesting and important quantum many-body system, there is so far no systematic to find the necessary and sufficient conditions under which way to write down its wave function in terms of RBM. a many-body state can be represented efficiently by neural Second, we need certain symmetries (such as the transla- networks, as well as to discover how to convert a general tional symmetry) to substantially simplify the equations. quantum state satisfying these conditions into a RBM. Thus, this approach works only in certain specific circum- These findings would help develop new machine-learning stances. However, we emphasize that our method does not algorithms for solving many-body problems and advance contain sophisticated mathematics and is a completely new the understanding, from a physical perspective [127], of the approach that has never been considered in the literature. power of machine learning itself. It would also be interest- ing to study entanglement properties in other types of VI. CONCLUSION AND OUTLOOK artificial-neural-network states [33,34]. Another interesting In summary, we have studied the entanglement proper- direction worth more investigation is the relation between ties of neural-network quantum states in the RBM archi- the MPS or tensor-network representation and the neural- tecture. In particular, we have proved that all short-range network representation. In this context, we note a recent RBM states satisfy an area law of entanglement for work on supervised machine learning with quantum- arbitrary dimensions and bipartition geometry. This result inspired tensor networks, where the tensor network repre- not only immediately implies an area law for the entangle- sentation has been shown useful in training neural networks ment of the 1D SPT cluster states and the 2D/3D toric code in a supervised way [128]. From Sec. III, we now know that states (with or without anyonic excitations), but it also all short-range RBM states in 1D can be represented in sheds light on the open problem of proving the entangle- terms of MPS. What about higher dimensions and the ment area law for the ground states of local gapped inverse statement? Can we rewrite all states bearing a MPS Hamiltonians in higher dimensions. For generic long-range or tensor-network representation with small bond dimen- RBM states with random parameters, we numerically sions in terms of short-range RBMs? These questions are studied their entanglement entropy and spectrum. We found worth exploring in the future. that (i) the averaged entanglement entropy follows a volume law but is significantly smaller than the Page ACKNOWLEDGMENTS entropy for random pure states and that (ii) their entangle- ment spectrum has no universal part associated with We thank Frank Verstraete, Roger Melko, Lei Wang, random matrix theory and manifests a Poisson-type level Giuseppe Carleo, Zlatko Papic, Ignacio Cirac, Mikhail D. statistics. In addition, we analytically constructed families Lukin, Subir Sachdev, and Matthias Troyer for helpful of RBM states (in both 1D and 2D) with maximal volume- discussions. D. L. D. thanks Zhengyu Zhang for his help in law entanglement, which cannot be represented efficiently developing the code for the reinforcement learning calcu- in terms of matrix product states or tensor-network states. lations. This work is supported by JQI-NSF-PFC and

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