Entangled Quantum Cellular Automata, Physical Complexity, and Goldilocks Rules

Entangled quantum cellular automata, physical complexity, and Goldilocks rules

Logan E. Hillberry,1, 2 Matthew T. Jones,1 David L. Vargas,1 Patrick Rall,3, 4 Nicole Yunger Halpern,4, 5, 6, 7 Ning Bao,4, 8 Simone Notarnicola,9, 10 Simone Montangero,10, 11, 12 and Lincoln D. Carr1 1Department of Physics, Colorado School of Mines, Golden, CO 80401, USA 2Center for Nonlinear Dynamics, The University of Texas at Austin, Austin, TX 78712, USA 3Quantum Information Center, The University of Texas at Austin, Austin, TX 78712, USA 4Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA 5Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA 6Department of Physics, Harvard University, Cambridge, MA 02138, USA 7Research Laboratory of Electronics, Massachusetts Institute of Technology,Cambridge, MA 02139, USA 8Berkeley Center for Theoretical Physics, University of California, Berkeley, CA 94720, USA 9Dipartimento di Fisica e Astronomia, Universitàdegli Studi di Padova, I-35131 Italy 10Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Padova, I-35131 Italy 11Dipartimento di Fisica e Astronomia, Universitàdegli Studi di Padova, I-35131 Italy 12Theoretische Physik, Universitätdes Saarlandes, D-66123 Saarbrücken,Germany (Dated: May 18, 2020) Cellular automata are interacting classical bits that display diverse behaviors, from fractals to random-number generators to Turing-complete computation. We introduce entangled quantum cellular automata subject to Goldilocks rules, tradeoffs of the kind underpinning biological, social, and economic complexity. Tweaking digital and analog quantum-computing protocols generates persistent entropy fluctuations; robust dynamical features, including an entangled breather; and network structure and dynamics consistent with complexity. Present-day quantum platforms— Rydberg arrays, trapped ions, and superconducting qubits—can implement Goldilocks protocols, which generate quantum many-body states with rich entanglement and structure. Moreover, the complexity studies reported here underscore an emerging idea in many-body quantum physics: some systems fall outside the integrable/chaotic dichotomy.

I. INTRODUCTION: PHYSICAL measures consistent with classical complexity, and (iv) COMPLEXITY IN QUANTUM SYSTEMS nontrivial network and entanglement dynamics. Such systems, we show, can be created on present-day analog quantum-computers, or quantum simulators [6]: Ryd- Classical cellular automata evolve bit strings (se- berg atoms in optical lattices [7, 13] and trapped ions [14, quences of 1s and 0s) via simple rules that generate di- 15]. Our QCA can also be implemented on digital- verse emergent behaviors. Quantum cellular automata quantum-computing platforms, including superconduct- (QCA) evolve quantum bits (qubits), supporting super- ing qubits, used to demonstrate quantum supremacy [16]. positions and entanglement (See 1). In computer sci- Quantum simulators [6, 17, 18] have energized the de- ence, complexity characterizes the number of steps in an veloping ﬁeld of nonequilibrium quantum dynamics [19]. algorithm. The algorithmic-complexity perspective on Our work provides a direction for such exploration, a cellular automata has received much attention [1]. In physical question: the origins of features discovered in contrast, physical complexity characterizes emergent be- multiscale complexity science, which, we show, occur in haviors, such as materials’ rigidity, spontaneous symme- simple, abiological quantum systems. try breaking [2], and self-organized criticality. Biological, social, and economic complexity features complex net- We emphasize physical complexity in our work because works, natural selection, diversity, power-law statistics,

arXiv:2005.01763v2 [quant-ph] 15 May 2020 many early QCA studies spotlighted single-particle or and robustness-fragility tradeoffs [3, 4]. Does the com- semiclassical approximations [1, 20–22] or weak correla- plexity arise in the quantum regime, or only at classical tions at the level of density functional theory [23, 24]. An- and biological scales? other line of work has highlighted Clifford operators [25– We study QCA’s physical complexity [9–11], iden- 28], which can efficiently be simulated classically [29]. tifying Goldilocks QCA rules that enforce tradeoffs. Our QCA generate states whose quantum entropies scale These rules generically generate complexity, quantified in a manner between area and volume laws and remain with the quantum generalizations of measures used exceptionally structured and dynamic. Therefore, semi- in electroencephalogram/functional-magnetic-resonance- classical QCA have little relevance to our study of phys- imaging (EEG/fMRI) measurements of the brain [12]: ical complexity. Other entangled QCA appear in the lit- mutual-information complex networks. Non-Goldilocks erature, such as in [9], which concerns the entanglement QCA tend toward equilibration, whereas Goldilocks generated from specific initial conditions by a single QCA QCA generate (i) robust dynamical features, (ii) persis- model within our paradigm, in [30], which concerns in- tent entropy fluctuations, (iii) average complex-network formation transport and entanglement distribution, and 2

C T strates our central claim: Goldilocks QCA generate phys- 201 1 ical complexity despite their simple underlying structure, , 1 54 (a) (b) (c) ↓ based in unitary quantum mechanics. Sections II-III suf- ﬁce for grasping the power of Goldilocks to QCA to produce complexity. For those who wish to reproduce our

t 36 work completely and understand every aspect, our QCA z σj , sj models’ mathematical details are described in Section IV; h i and our quantiﬁers, in Section V. In Section VI, we pro- Time, 18 pose a physical implementation of our models in Ryd- berg systems, an experimentally realized quantum simulator [6]. In section VII, we conclude with this work’s 0 ↑, 0 signiﬁcance. 0 9 18 Site, j II. OVERVIEW OF DISCRETE AND CONTINUOUS QCA

The best-known classical cellular automata are the elementary cellular automata (ECA), one-dimensional length-L bit strings. Each bit updates in discrete time steps, according to a function defined on the bit’s neighborhood. The neighborhood is defined as the bit and its nearest neighbors. The 256 possible transition functions are encoded in the rule number, CR = 0, 1,..., 255. A time step t ends when all bits have been updated. Rule C110 is Turing-complete, or capable of simulating any FIG. 1. (a) The classical-elementary-cellular-automaton [5] computer program. rule C201 flips a classical bit (black=1, white=0) if and only Our entangled QCA are one-dimensional. They extend if its nearest neighbors are 0s. This L=19-site chain is ini- ECA’s neighborhood-dependent update rule while re- tialized with a 1 centered in 0s. A periodic pattern, called maining spatially discrete. Five new features distinguish a blinker, propagates upward in discrete time steps. (b) We QCA: First, the classical bit is traded for a qubit pa- extend C201, for QCA, to rule T1. The classical bit becomes a rameterized by two angles: cos(θ/2) 0 + eiφ sin(θ/2) 1 , z quantum bit in an L=19-site chain. The average spin σˆj has wherein 0 and 1 denote the 1 eigenstates| i of the Pauli| i richer discrete-time dynamics, oscillating between the classi- | i z | i ± z-operator,σ ˆ , and θ, φ R. Second, time updates en- cal extremes of 0 (black) and 1 (white) and filling the lattice. ∈ A truly quantum analog of the classical blinker—a quantum tangle qubits. The entanglement generates quantum en- entangled breather—appears in Figs. 2(g)-(h). (c) The quan- tropy and long-range nonlocal features absent from ECA. tum lattice evolves into a high-entropy entangled state. sj Third, our QCA can run in discrete or continuous time. denotes the site-j second-order Rényi entropy, routinely mea- Hence QCA can be implemented on digital and analog sured in quantum simulators [6]. (d) Such an analog quantum quantum computers. Fourth, updates can depend on computer has implemented rule T1, recently named the PXP more neighbors, 4. Fifth, in digital QCA, all sites model. Optical tweezers were used to trap a Rydberg-atom cannot update simultaneously,≤ due to the no-cloning the- chain [7, 8]. (e) A quantum circuit can realize digital T1 dy- orem. Site updates within a time step must be ordered. namics. One QCA time step requires two layers of controlled- Similarly, the update of a site must not depend on the controlled-Hadamard gates. The first layer evolves even-index site itself. Our digital three-site QCA therefore parallel qubits; and the second, odd-index. The dashed line represents the 16 locally invertible ECA. the boundary qubits, which remain fixed in |0i’s. Section IV will detail our QCA mathematically. Briefly, each QCA is defined by a rule number that encodes under what conditions on a qubit’s neighbors the in [31–34], which studies entanglement growth in Clif- qubit evolves nontrivially. Site j has a set Ωj of neigh- ford and Clifford-like QCA. Some experimental work has bors: the sites, excluding j, within some radius r of j. touched upon QCA-like systems including nuclear mag- The boundary conditions are fixed: For sites within < r netic resonance studies on three-spin molecules [35] and sites of a boundary, we imagine neighbors extending to the PXP model realized in Rydberg atom chains [7]. additional sites fixed to 0 ’s. Ωj corresponds to a Hilbert However, QCA’s physical complexity has not been ad- space on which is defined| i a vector space spanned by pro- dressed. jectors ˆ(i). The i enumerates the neighbors’ possible This paper is organized as follows. Section II overviews PΩj ˆ(i) 1ˆ the time-evolution protocols in our QCA models. With configurations, and Ω = . i P j this high-level understanding in place, Section III demon- To define digital time evolution, we convert an ECA P 3 rule number into a neighborhood-sized unitary gate. The PXP model is a QCA, albeit not a Goldilocks rule. In gate evolves different neighborhoods in different circuit this analog QCA, the activation operator, a spin flip, acts layers, in a time-ordering scheme illustrated in Fig. 1(e). if and only if no neighbor occupies 1 . Adding a longi- The single-neighborhood unitary has the form tudinal field to a common quantum-simulator| i Hamilto- nian, the transverse-field Ising model suffices to realize ci (i) Uˆj = Vˆ ˆ . (1) more-general QCA. We propose an implementation of j PΩj i the five-site F -rules in Rydberg-atom chains in Fig. 9 X of Section VI. Vˆj denotes a unitary activation operator on site j. The rule number’s binary expansion determines the ci 0, 1 . The matrix power implies that Vˆ ci 1ˆ, Vˆ . We∈ { } ∈ { } III. DEMONSTRATION: GOLDILOCKS QCA illustrate with radius-1 digital QCA whose activation is GENERATE COMPLEXITY the Hadamard gate (the Bloch-sphere rotation that in- terchanges the x- and z-axes) times a phase gate (which rotates about the z-axis). Even-j sites are updated first; Figure 2 shows a sampling of dynamical QCA out- and odd-j sites, second [Fig. 1(e)]. We denote these comes. Although rule T1 has been explored experimentally and rule T14 has been explored theoretically [38], QCA by TR. T evokes three-site neighborhoods, and R = 0, 1,..., 15 denotes the rule number. only the Goldilocks rule T6 displays nonequilibrating To define analog time evolution, we convert the rule complexity at late times, visible in robust dynamical pat- ˆ terns. Complexity is produced by such Goldilocks rules number into a many-body Hamiltonian j Hj. The single-neighborhood Hamiltonian for diverse initial states, activation unitaries, and time- P evolution schemes. Conversely, rule T13 activates site j if ˆ (i) (i) 1 appears as a left neighbor or (ii) zero or two neigh- Hˆj = hj ci ˆ . (2) PΩj | i i bors occupy 1 ’s. T13 quickly equilibrates in almost all X cases. | i ˆ hj denotes a Hermitian activation operator. The rule Figure 2(g)-(j) concern analog quantum computation number’s binary expansion determines the ci 0, 1 . and five-site blocks. Rule F4 is a Goldilocks rule. We explore analog, radius-2, totalistic QCA with∈ { acti-} It activates a site whose neighbors contain two 1 ’s ˆ x but not zero, one, three, or four. The initial condi-| i vation hj =σ ˆj , the Pauli x-operator. Totalistic rules update a neighborhood’s central site conditionally on tion ... 000 101 000 ... generates a quantum entangled | i the total number of neighbors in 1 . We denote these breather, which parallels the breather of discrete nonlin- | i simulations by FR. F refers to five-site neighborhoods; ear wave theory [39]. In contrast, the non-Goldilocks rule R = 0, 1,..., 31 denotes the totalistic rule number. F26 rapidly destroys the state’s structure. An example QCA is the Goldilocks rule T6. The lo- The Goldilocks rules T6 and F4 exhibit late-time dy- (0) (1) cal update unitary has the form Uˆ = Pˆ UˆH Pˆ + namical features. These features appear in the expecta- (1) (0) (m) ⊗ ⊗ z Pˆ UˆH Pˆ . The Pˆ = m m for m = 0, 1 are the tion value σˆj (top row of Fig. 2) and the second-order ⊗ ⊗ | ih | projectors onto the single qubitσ ˆz eigenspace. The acti- Rényi entropy (bottom row). The non-Goldilocks rules z x vation operator, the Hadamard gate UH = (ˆσ +σ ˆ )/√2, exhibit no such persistent features. How can we quantify acts if exactly one neighbor is in 1 —not zero (too few) this lack of persistence? What is complexity? or two (too many); hence the name| i “Goldilocks.” Such Figure 3 quantifies complexity with the quantum tradeoffs generate and maintain complexity [36]. We ex- mutual information, which generalizes a measure used plored diverse digital and analog activation operators, throughout classical complexity theory. Complex net- site-update orderings, boundary conditions, and neigh- works can be imposed on quantum systems, such as net- borhoods. Complexity outcomes, quantified with entropy worked NISQ (noisy intermediate-scale quantum) com- fluctuations and network structure and dynamics, were puters and in a quantum-Internet [40]. Yet complex net- always similar. This Report summarizes our findings works can also emerge from a quantum state generated by with a few representative cases. Only Goldilocks rules noncomplex systems or models. This emergence has been produce complexity, defined below, independently of all demonstrated with the Ising and Bose-Hubbard models other variables. near quantum critical points [41, 42]. Figure 1(a) features the ECA rule C201, well-known to Figure 3(a) shows mutual-information networks for produce a periodic pattern called a blinker. The equiva- various states. We compare states produced by QCA lent digital QCA rule, T1, generates Figs. 1(b)-(c). T1 with well-known examples: random ( R ) [16], GHZ produces richer dynamics, including the generation of ( G ), W ( W ), and cluster ( C ) states.| i The cluster | i | i | i quantum entropy. The analog version of T1 is the PXP state is defined in terms of a 1 19 lattice and a phase model [7], realized with Rydberg-excited neutral atoms angle υ = π/4 (see Section V× for the definition). The [Fig. 1(d)]. Such models display diverse quantum dy- Goldilocks rule T6 has a visibly unusual network. Fig- namical features, including many-body scars [37]. The ure 3(b)-(c) show the time dependence of the network- important point is that a QCA has been created exper- averaged clustering ( ) and disparity ( ). These mea- imentally. However, it is not widely realized that the sures (also defined in SectionC V) quantifyY complexity near 4

T6 T14 T13 F4 F26 (a) (c) (e) (g) (i) ↓ 950

t 36 σz h j i Time, 18

0 ↑ 1 (b) (d) (f) (h) (j) 950

t 36

sj Time, 18

0 0 0 9 18 Site, j

FIG. 2. Entangled QCA create diverse dynamical behaviors. For (a)-(f), an L=19-site chain was initialized with one |1i centered in |0i’s, then evolved in discrete time. The rule-T6 (a) spin and (b) entropy dynamics show patterns that resist equilibration for long times, as shown in each panel’s upper block. (d) Under rule T14, the spin dynamics rapidly equilibrate. (d) The entropy exhibits persistent patterns on a lower-entropy background. (e)-(f) In contrast, the rule-T13 dynamics rapidly equilibrate to a high-entropy state. For (g)-(j), we extend QCA to 5-site rules, initialize to |101i, and evolve in continuous time. (g)-(h) The Goldilocks rule F4 generates a quantum generalization of the blinker in Fig. 1(a), a quantum entangled breather. (i)-(j) In contrast, rule F26 promotes rapid equilibration to a high-entropy state.

bond quantum critical points. Solid curves show an activation central bond entropy, sL/2 , for various initial conditions. phase angle of υ = 0. Dashed curves show υ = π/2, for T6 generates complexity similarly to, but higher bond T14; other rules’ features do not depend qualitatively on entropy than, the cluster state. The random state has υ. We studied many other complexity measures, includ- a large bond entropy, low average clustering, and low ing degree and path length [3]. Two measures suffice to average disparity, like the most non-Goldilocks rule, T13. illustrate that only rule T6 consistently maintains more Figure 3(g) shows how various initial conditions lead to complexity: First, a large clustering often signals com- states whose average disparity contrasts with well-known plexity [3]. The clustering remains large despite network quantum states’. The GHZ and W states have high clus- growth only under the Goldilocks rule T6 [Fig. 3(d)]. In terings, low disparities, and low bond entropies. A high fact, the network-and-time-averaged T6 clustering grows bond entropy can signal volume-law entanglement and with the network. Second, disparity fluctuations, while chaos; and a low bond entropy, area-law entanglement studied less commonly, signal a lack of equilibration. In and integrability. Goldilocks rules create new kinds of Fig. 3(e), the disparity fluctuates the most under T6. states, which are highly entangled (with bond entropies Figure 3(f) shows the time-averaged clustering vs. the larger than the W and GHZ states’) and highly complex 5

(a)

T6 T1 T14 T13 R W G C | i | i | i | i 1 1 10− 10− 3 3 C 10− C 10− 5 (b) 5 10− 10− (d) (f)

0.4 (c) (e) (g) 0.4 Y 0.2 Y 0.2 0.0 0.0 0 250 500 750 1000 6 10 14 18 1 3 5 7 9 bond Time, t Size, L sL/2

FIG. 3. (a) Four QCA were initialized with a central |1i centered in |0i’s, then evolved for 500 time steps. The QCA complex networks (left-hand side) contrast with four well-known entangled states’ networks (right-hand side). The networks’ labels provide a legend for the panels below. (b)-(c) Clustering and disparity, well-known complex-network measures [3], averaged over the networks, vs. time. Solid lines result from an activation phase-gate angle υ = 0. Dashed lines result from υ = π/2 for T14; the other rules’ behaviors depend on υ negligibly. Under the Goldilocks rule T6, the clustering remains high. The disparity oscillates rapidly, evidencing a lack of equilibration. In contrast, non-Goldilocks rules’ clusterings decrease, and the disparities oscillate less. (d)-(e) Clustering and disparity, averaged over the network and time, vs. system size. Solid lines show exponential ﬁts. Clustering tends to decrease exponentially with L, except for rule T6, whose clustering increases. Disparity, too, tends to decrease, but at size-dependent rates (the rule-T1 slope even changes sign). Each colored band depicts one standard deviation. Only T6 generates ﬂuctuations larger than the data points. Black lines in (b)-(c), and black data points in (d)-(e), follow from initializing the system in |Ri and evolving with one of the four QCA, then averaging over the QCA. (f)-(g) Time-averaged clustering and disparity vs. central-bond entropy for various initial conditions (one or a few |1i’s). QCA generate complexity only when initialized in low-enough-entropy states.

(exhibiting high disparity fluctuations and clustering). The Page curve distinguishes area-law and volume-law Complexity is signaled by the convergence of several entanglement. Let ` = 0, 1,...,L 2 denote the bipar- − lines of evidence [3]. We emphasize high disparity fluc- titionings of the lattice. The bond entropies for all the bond tuations and clustering, together with entropy fluctua- bipartitionings, s` , form the Page curve. Figure 4(c) tions (discussed below). Overall, Fig. 3 shows that the shows the Page curves for QCA-generated states. The T6 Goldilocks rule T6 stands out from other rules and from curve, being lowest, suggests an area law the most. If the well-known quantum states. For small-enough systems, initial state contains exactly one 1 , the Page curve fits | i T presents a borderline case. the scrambled-state ansatz in [47]. Many initial 1 ’s lead 1 | i QCA defy the chaos/integrability classification scheme to T6 Page curves (not shown) that disobey a volume law based on entanglement spectra, shown in Fig. 4(a) and and deviate significantly from the scrambled-state predic- detailed in Section V. Figure 4(b) shows the bond en- tion. Similarly, many-body scars—nonequilibrating fea- tropy’s time derivative, ∆sbond. The derivative remains tures discovered under T1 evolution (the PXP model)— L/2 are theorized to arise from a few area-law energy eigen- an order of magnitude higher for rule T6 than for the other rules. This largeness signals complexity alterna- states states amidst a volume-law sea. Yet observ- tively to network properties [45]. Dark lines show a ing many-body scars requires special initial conditions. moving average over L time steps. Faint lines, showing Fine-tuning is not necessary for resistance to equilibra- unaveraged data sampled every L time steps, illustrate tion under Goldilocks rules: Complexity emerges for di- fluctuations. The random background (gray shading) ex- verse initializations, activation operators, time-evolution tends between the mean the standard deviation of data schemes, and update orderings. from initializing with R±, then evolving under different This complexity survives perturbations and noise. Fig- rules in different trials.| i ure 4(d) demonstrates robustness of the quantum en- 6

dence of complexity’s emergence under Goldilocks rules. T6 T1 T14 T13 (a) 1.0 IV. MODEL DETAILS

η 0.5 0.0 In this section, we present explicit formulae for the T -type unitaries and the F -type Hamiltonians, as func- 0 tions of rule number. We then interrelate the two time- 10 (b) evolution schemes by considering a three-site analog pro- 2 /

bond 2 L 10− tocol. Our rule numbering is inspired by elementary-

∆s cellular-automata rule numbering, so we ﬁrst discuss 4 10− those classical models. 0 250 500 750 1000 Time, t A. Classical Elementary Cellular Automata 8 (c) (d) τ A classical-ECA neighborhood consists of a central bit 500 slope: -1.3 and its two nearest neighbors. Classical ECA rules are

bond ` 4 150 denoted by CR. The rule number R encodes the local s 50 transition function, which prescribes a bit’s next state Lifetime, for any possible neighborhood configuration. A bit has 0 two possible values (0 or 1), and ECA neighborhoods 1 8 16 0.5 1.5 5 consist of three sites. Hence a neighborhood can occupy 3 Cut, ` Admixture, ε (%) 23 = 8 configurations, which lead to 22 = 256 rules. The system is evolved through one time step as fol- FIG. 4. Dynamical many-body features distinguish lows. First, a copy of the bit string is produced. Each Goldilocks QCA. (a) The Brody parameter η interpolates site’s neighborhood is read off of the copy and is inputted between integrable systems’ Poisson statistics (η = 0) and into the local transition function. The function’s output chaotic systems’ Wigner-Dyson statistics (η = 1). We cal- dictates a site’s next state, which is written to the origi- culate η from entanglement spectra [43]. η grows to chaotic nal bit string. The copying allows all sites to be updated values under rules T and T , remaining Poissonian under 13 14 independently, simultaneously. T1. The T6 η defies the paradigm and so is not plotted: Many eigenvalues ≈ 0. Bars represent the error prescribed by the The rule encoding works as follows. Given CR, ex- η-extraction procedure. (b) Entropy fluctuations distinguish pand R into 8 binary digits, including leading zeroes: 7 n integrable from chaotic systems alternatively to the Brody R = n=0 cn2 . The index n enumerates the neigh- parameter [44] and suggest complexity [45]. The fluctuations borhood configurations when expressed as a sequence of P 2 k are 1-3 orders of magnitude larger for T6 than for T1, T14, and three binary digits: n = m=0 km2 . The binary ex- the random background (gray shading). (c) The Page curve pansion of n is the equivalence of the base-10 number n distinguishes between area and volume laws, characteristic P and the base-2 number k2k1k0. The binary expansion of integrable and chaotic many-body models [46, 47]. The n k2k1k0 encodes the neighborhood’s state. In the bi- T13 curve, bending far upward, suggests a volume law; T6 nary≡ expansion of R, the coefficients c 0, 1 dictate and T come closer to achieving area laws. The T curve n 1 14 the next state of the neighborhood’s central∈ { site.} Fig- deviates from the scrambled-state ansatz. Dots represent time-averaged Page curves; solid curves, the best fits to the ure 5 shows the update table for rules R = 30, 60, and scrambled-state ansatz; and shaded regions, time fluctuations. 110. Also depicted are the time evolutions of a 1 centered (A-C) report on L = 19 sites initialized with one centered |1i. in 0s. (d) The continuous-time rule F4 produces robust quantum entangled breathers [Fig. 2(g)-(h)]. F4 was perturbed with a non-Goldilocks rule, F26, at strength ε. The breather’s life- B. Entangled quantum cellular automata time decays as a power law: τ ∝ ε−1.3. Entangled QCA can be realized with 1D chains of L qubits, or sites. Dynamics are governed by a QCA Hamil- tangled breather generated with the Goldilocks rule F4 tonian (equivalently, by analog quantum computation, [Fig. 2(g)-(h)]. We perturbed F4 with the non-Goldilocks or a quantum simulator) or by a quantum circuit (by a rule F26, with strength ε. The breather’s lifetime, τ, digital quantum computer). Each time-evolution scheme varies with ε as a slow power law. Quantum entangled specifies how a site j updates conditionally on the state breathers survive also imperfect initializations and limi- of the site’s neighbors. The neighbors within a radius r tations on entanglement (Schmidt truncation). Emergent of site j form a set denoted by Ωj. (Ωj does not include power laws, with features such as high clustering, dispar- j.) Some of our QCA have three-site neighborhoods, of ity fluctuations, and entropy fluctuations, provide evi- a central site j and its nearest neighbors. Some of our 7

Under a totalistic rule, whether the activation opera- C30 C90 C110 t 32 tor transforms a neighborhood’s central site depends only on the total number of neighboring 1 ’s. If the neighbor- 16 (a) (b) (c) hood consists of three sites, zero, one,| i or two neighbors Time, 0 can be in the state 1 . Hence 23 = 8 totalistic three-site | i 2 0 32 64 0, 1 rules exist. They form a subset of the general 22 = 16 5 Site, j TR rules. Similarly, 2 = 32 FR rules exist. They form 4 a subset of the 22 = 65536 general five-site rules. The number of rules grows combinatorically with the neighborhood’s size. Under a Goldilocks rule, whether a site updates depends on a neighborhood’s total value, satisfy- ing the “not too many and not too few” tradeoff common in complexity theory [36]. If we did not invoke totalistic labeling of five-site-neighborhood rules, rule F4 would have rule number R = 5736. FIG. 5. (a-c) Classical-elementary-cellular-automata (ECA) To specify a QCA model, one must specify, in addi- evolutions of a 1 centered in 0s. (d) Rule tables for C30, tion to a rule number R, a time-evolution scheme (dig- which generates pseudorandom numbers; C60, which gener- ital or analog) and an activation operator. Equations ates a fractal structure; and C110, which is Turing-complete (1) and (2) of Section II formalize these requirements. (capable of simulating any computer program). We narrow down the options to two illustrative choices: The T -type simulations discussed involve digital evolution with a Hadamard-and-phase-gate activation. The F - QCA have five-site neighborhoods, of site j and its near- type simulations discussed involve analog evolution and est neighbors and next-nearest neighbors. The QCA rule aσ ˆx activation. (Below, we present the T -rule unitaries’ number encodes the coupling’s form, i.e., conditions on and F -rule Hamiltonians’ forms.) Yet analog and digi- the neighbors which activate site j. tal evolutions can be performed with any neighborhood Neither of our update schemes (discrete-time or size. We discuss analog TR rules (i) when interrelating continuous-time) decomposes into a component governed the digital and analog evolution schemes and (ii) when by left-hand neighbors and a component governed by proposing physical implementations. Digital evolution of right-hand neighbors. Our schemes’ bidirectionality con- rules with larger neighborhoods can be generalized as fol- trasts with the structures of most many-body Hamilto- lows: Recall that r denotes the radius of neighbors that nians in the literature and solved on quantum comput- determine whether a central site updates. Instead of up- ers. Examples include the Hubbard, Ising, and Heisen- dating even sites, then odd sites, we run r + 1 layers of berg models; the PXP model constitutes an exception [7]. a quantum circuit. Layer k 0, 1, . . . , r updates site j Nevertheless, our scheme can be realized experimentally, if j mod (r + 1) = k. ∈ { } as we show later. Our QCA’s definitions differ from ECA’s definitions in four ways: (i) Sites are evolved in place, rather than C. T -type simulations via copying. How the lattice evolves, therefore, depends on the order in which sites update. Complexity out- Rules TR denote digital simulations of the 3-site- comes, we find, do not depend on how site updates are neighborhood rules. To expose such a rule’s dynamics, ordered. For instance, site 0 can update, followed by site, convert R from base-10 into four base-2 bits (including 1, and so on: ( 0 , 1 ,... ). Alternatively, the even sites leading zeroes). The leftmost bit should be the most { } { } can update simultaneously, followed by the odd sites: significant; and the rightmost bit, the least significant: ( 0, 2,... , 1, 3,... , 0, 2,... ,...). Other options ex- R = c 23 + c 22 + c 21 + c 20 = 1 c 22m+n. { } { } { } 11 10 01 00 m,n=0 mn ist, e.g., ( 0, 3, 6,... , 1, 4, 7,... , 2, 5, 8,... ,...). We The m specifies the left-hand neighbor’s value; and n { } { } { } P therefore present about only the second scheme, even- specifies the right-hand neighbor’s value. cmn 0, 1 odd ordering. (ii) Rules dictate how a site evolves, not specifies whether the central bit evolves under a∈ single- { } the state to which the site evolves. (iii) In QCA, how qubit unitary Vˆ (cmn = 1) or under the identity 1ˆ site j evolves depends only on the site’s neighbors, not (cmn = 0), given the neighbors’ values. More concretely, on the site itself. (iv) An ECA neighborhood consists of the three-site update unitary has the form three sites. We present QCA that have three-site neighborhoods and QCA that have five-site neighborhoods. 1 ˆ ˆ ˆ(m) ˆ cmn ˆ(n) Our QCA models’ rule numbering is defined in terms Uj(V ) = Pj−1 Vj Pj+1 . (3) m,n=0 of the neighborhood size, the rule number R, and X whether the rule numbering is totalistic. The three- ˆ site-neighborhood rules are labeled as TR; the five-site- The superscript on Vj denotes a matrix power. We up- neighborhood rules are labeled as FR. date the lattice by applying Uˆj(V ) to each 3-site neigh- 8 borhood centered on site j, for every j. Even-j neigh- E. Relation between analog and digital borhoods update first; and odd-j neighborhoods, second. time-evolution schemes Each boundary qubit is fixed to 0 . We explored alter- | i native boundary conditions, including periodic boundary We will prove that evolution under analog three-site conditions and boundaries fixed to 1 ’s. The complexity QCA is equivalent to evolution under the digital three- | i outcomes are similar. site QCA, TR. [48] Consider the radius-1 analog QCA The local update unitary Vˆ is the product of the 2m+n j TR, with rule number R = m,n cmn2 . According Hadamard and phase gates familiar from quantum com- to Eq. (2), the three-site Hamiltonian has the form puting: P 1 iυ 1 1 e (m) (n) Vˆ Vˆ (υ) = Uˆ Uˆ (υ) = . (4) Hˆ = c Pˆ σˆx Pˆ . (7) j j H P 1 eiυ j mn j−1 j j+1 ≡ √2 m,n=0 − X The Hadamard gate rotates the Bloch sphere’s north pole to the x-axis. A phase gate of υ = π/2 rotates the x-axis The state vector is evolved for a time δt by the propagator ˆ ˆ into the y-axis. These gates can rotate a qubit state to U = exp( iδtH). For all j and k such that k j = ˆ ˆ − | − | 6 any point on the Bloch sphere. In most of the simulations 1, [Hj, Hk] = 0. Hence the propagator factorizes into 2 ˆ ˆ investigated here, υ = 0; the dynamics is restricted to the even and odd parts at order δt : U Πj∈{1,3,5,...}Uj xz-plane. Behaviors are mostly robust with respect to ≈ · Π Uˆ . The local propagator has been defined changes in υ. T14 forms an exception: Nonzero υ values j∈{0,2,4,...} j strongly suppress entropy fluctuations. as Uˆj = exp( iδtHˆj). This Uˆ is equivalent to the circuit − that runs the digital TR simulations, up to the unitary activation operator Vˆj. D. F -type simulations Now, we relate the discrete-time scheme’s Vˆj to the continuous-time scheme’s hˆ =σ ˆx. We expand the local F denotes analog simulations of five-site totalistic j R ˆ 1ˆ ˆ 2 ˆ 2 3 ˆ 3 ˆ x propagator as Uj = iδtHj δt Hj /2! + iδt Hj /3! + rules with a Hermitian activation operator hj =σ ˆj . The − − ˆ(m) ˆ(n) ˆ(m) value of R dictates the form of the many-body Hamilto- ... The projectors are orthogonal, Pj Pj = δmnPj , ˆ ˆ x 2 nian H = j Hj. We expand R into 5 digits of binary: and the Pauli operators square to the identity: (ˆσ ) = 1ˆ. 4 q Therefore, powers of the Hamiltonian can be expressed R = q=0 cq2 . Site j evolves conditionally on its neigh- P 2k (m) (n) 2k+1 bors: as Hˆ = cmnPˆ 1ˆPˆ and Hˆ = Hˆj for P j m,n j−1 j+1 j 4 k = 1, 2,... Hence the matrix exponential simplifies to x (q) P Hˆj =σ ˆ cq ˆ . (5) j Nj 1 q=0 (m) x (n) X Uˆj = Pˆ (1 cmn)1ˆ + cmn exp(iδtσˆ ) Pˆ j−1 − j j+1 ˆ (q) denotes the projector onto the q-totalistic subspace, m,n=0 j X inN which exactly q neighbors are in 1 ’s: 1 ˆ(m) x cmn ˆ(n) | i = Pj−1 exp(iδtσˆj ) Pj+1. (q) (Kq [0]) (Kq [1]) (Kq [2]) (Kq [3]) ˆ = Pˆ Pˆ Pˆ Pˆ . (6) m,n=0 Nj j−2 j−1 j+1 j+2 X perm(Kq ) X The last line has the form of the three-site update unitary Kq represents a sequence of q ones followed by 4 q in Eq. (3). The parallel suggests an activation unitary th − zeroes. Kq[i] 0, 1 denotes the sequence’s i element. exp(iδtσˆx). ∈ { } 4 The sum is over the q permutations of the bits in Kq. Qubits near the boundary do not have a complete set of neighbors. Any missing neighbors are fixed to 0 ’s. V. METHODS AND DEFINITIONS Other boundary conditions yielded similar results. | i We illustrate continuous-time evolution with five-site Throughout this work, we use various numerical and neighborhoods, the smallest we found to produce quan- statistical methods, quantifiers of dynamics, and com- tum entangled breathers. These emergent features sug- plex networks. This section details those methods and gest complexity greater than the three-site scheme’s. definitions. The three-site analog scheme, too, generates complexity but provides no information beyond the discrete-time scheme. Likewise, we explored diverse Hermitian activa- A. Well-known quantum states ˆ tion operators hj. They generate complexity when cou- ˆ pled with a nontrivial hj and initial state. By “nontriv- Figure 3 involves four well-known quantum states: ial,” we mean that the initial state is not an eigenstate the GHZ state, the W state, a random state, and a of the Hamiltonian. In trivial cases, the evolution only cluster state. The GHZ state is defined as G = introduces a global phase. 2−1/2( 00 ... 0 + 11 ... 1 ); and the W state, as |Wi = | i | i | i 9

2−L/2( 100 ... 0 + 010 ... 0 + + 00 ... 01 ). We de- consider cutting the lattice between sites ` and ` + 1. | i | i ··· | i bond,(α) fine the random state as R = C ξn n . The nor- The smaller subsystem has an entropy s called the | i n | i ` malization constant is denoted by C. The ξn are random bipartition bond entropy. The central cut’s bond entropy P complex numbers with real and imaginary parts drawn is denoted by sbond,(α). If the system has an odd number independently from a flat distribution over [0, 1). The n L/2 L of sites, the central bond lies between sites (L 1)/2 are the 2L computational-basis elements for an L-qubit| i and (L + 1)/2. − system. The α = 2 Rényi entropy is routinely measured exper- A cluster state is a quantum state defined by a regular imentally [49]. Therefore, Section III focused on α = 2, graph G(V,E). The set of vertices, or nodes, is denoted where we dropped the superscript (2)to simplify nota- by V ; and the set of edges, or links, by E. The num- tion. ber of nodes equals the number of qubits in the cluster Temporal fluctuations in entropy are captured with the state. We focus on planar rectangular grids of L vertices. absolute value of a second-order central finite difference. No matter the graph, the cluster state is constructed in [50] In discrete-time simulations, the time step has unit two steps. First, each qubit is initialized in 0 and trans- length. Hence the fluctuation at time step t is formed, by a Hadamard gate, to 2−1/2( 0 +| 1i ). Second, i | i | i a controlled-phase gate (α) 1 (α) (α) ∆s (ti) = s (ti+1) s (ti−1) . (11) A 2 A − A 1 0 0 0 0 1 0 0 For simplicity, we suppress the time dependence in the Cˆ (υ) = (8) phase 0 0 1 0  left-hand side’s notation. The absolute value highlights 0 0 0 eiυ the fluctuations’ magnitudes, suppressing the signs.     bond,(α) The set of all the bond entropies s` , for ` = 0 acts on every pair of nodes in the edge list E. The second to L 2, is called a Page curve. Consider averaging the step tends to generate entanglement along the graph’s α = 2− Page curve over the last half of the time evolution. links. We denote with an overbar the late-time average of any length-T time-series Q that includes only times greater T than t0 T/2: Q(ti) = Q(ti)/(T t0). The B. Local expectation values ti=t0 average Page∼ curve is fit with the ansatz [47]− P Let ˆj denote an observable of qubit j. The observ- bond,(2) 2(`+1)−L O s` = (`+1) log2(a) log2 1 + a +log2(K) . − able has a dynamical expectation value of ˆj (t) = O (12) a and log (K) denote free parameters. The first term Tr ρˆj(t) ˆ , whereρ ˆj(t) denotes the site-j reducedD E den- 2 O in Eq. (12) represents volume-law entanglement growth sity matrix, calculated by partially tracing over the with slope log (a). a = 1 signals an area law. When whole-system density matrixρ ˆ(t) = ψ(t) ψ(t) , i.e., 2 ` [0, L/2], the second term represents a deviation from ρˆ (t) = Tr ρˆ(t). | i h | j k6=j the∈ volume law. The deviation grows as the cut nears the lattice’s center. The third term represents an offset C. Entropy to the volume law. Figure 6(a) shows Eq. (12) for different a values. bond,(2) log2(K) is fixed so that s0 = 0. Nonintegrable Let denote a subsystem in the stateρ Â. The order-α RényientropyA is defined as systems are conjectured to have Page curves that tend towards Eq. (12); and integrable systems, to have Page 1 curves that deviate. More information appears in [47], s(α) = log [Tr(ˆρα )] (9) A 1 α 2 A whose conventions deviate from two of ours: (i) Our def- − inition of ` is shifted relative to theirs; hence the ` + 1 in for α [0, 1) (1, ). In the limit as α 1, the Rényi Eq. (12). (ii) Our entropies’ logarithms are base-2. entropy∈ becomes∪ the∞ von Neumann entropy:→

dim(A) (1) D. Entanglement-spectrum statistics s = Tr (ˆρA log ρÂ) = λi log (λi) . (10) A − 2 − 2 i=0 X Consider cutting the lattice down the center. Each The sum runs over the eigenvalues λi ofρ Â. The set subsystem’s reduced state has eigenvalues λi, which form { } λi of eigenvalues is called the entanglement spectrum the entanglement spectrum [51]. We index the eigenval- {of subsystem} . ues such that λ1 λ2 ... How many spacings λi+1 λi ≤ ≤ − We distinguishA three Rényi entropies. First, the local have size d? The density of size-d gaps forms a distribu- (α) th tion that we fit to the Brody ansatz [52], entropy sj is the j site’s entropy. Second, the two- point entropy, s(α) for j = k, is of a pair of sites. Third, (d) = β(η + 1)dη exp( βdη+1) . (13) j,k 6 D − 10

9 tween sites j and k = j is (a) 1.0 (b) 6 (α) 1 (α) (α) (α) 6 ) Mjk = sj + sk sjk . (14) d 2 − ( bond ` 0.5 s 3 D The 1/2 is unconventional but normalizes the mutual in-

formation such that 0 M (α) 1. We define M (α) = 0 0 0.0 ≤ jk ≤ jj and interpret M (α) as the adjacency matrix of a weighted, 0 8 16 0 2 4 jk undirected graph that lacks self-connections. The abso- Cut, ` Gap, d lute value in Eq. (14) is necessary only for α > 1. For α = 1, the mutual information is positive-definite. FIG. 6. (a) The Page-curve ansatz (12), for a = 2.0 (red) It has the physical interpretation of an upper bound on 1.5 (purple), and 1.1 (brown). log (K) is fixed, for all curves, (1) 2 every two-point correlator [54]. Hence a nonzero M such that the minimum entropy vanishes. The dashed lines jk reflects two-point correlations. Yet the order-2 Rényi en- represent the corresponding volume law, the first term in Eq. (12). Integrability can cause a high-energy Hamiltonian tropy is routinely measured in experiments, so we fo- (2) eigenstate’s Page curve to deviate from such a best fit [47]. cus on Mjk . The complexity outcomes for α = 1, 2 are (b) The Brody ansatz (13) for η = 0.0 (blue), 0.5 (orange), qualitatively similar, as in the context of quantum phase and 1.0 (green). Poisson (η = 0) statistics promote small transitions [42]. To simplify notation, we drop the (α) gaps; Wigner-Dyson (η = 1) statistics promote eigenvalue re- superscript in Section III. An adjacency matrix can be pulsion. The gap d is normalized to have a unit average. defined alternatively in terms of two-point correlators, but they capture complexity less than mutual information does [42]. η+1 η+2 Γ denotes the gamma function, β = Γ η+1 , and η [0, 1] is the Brody parameter. This ansatz interpo- F. Network measures lates∈ between Poisson statistics (η = 0), characteristic of integrable systems, and Wigner-Dyson (η = 1) statistics, characteristics of nonintegrable systems [Fig. 6(b)]. Network measures are functions of an adjacency matrix; they quantify the corresponding graph’s connec- The ansatz is derived under the assumption that d tivity. We evaluate two network measures on mutual- averages to one. This condition is often enforced via information adjacency matrices: the clustering coefficient a spectral unfolding procedure. We use the polyno- and the disparity. They capture complexity in the con- mial method [53]: We remove all λ < 10−10, then the i text of quantum phase transitions near quantum critical 10 smallest and 10 largest λ values. This clipping is i points [41, 42]. a standard preprocessing tool, eliminating outliers and First, the clustering coefficient quantifies local, λ 0. Next, we fit a generic ninth-order polynomial i small-scale community structure. ClusteringC is a key fea- f(x≈) = 9 b xn to the set of ordered pairs (λ , i). The n=0 n i ture of social and biological networks. It quantifies tran- unfolded spectrum is defined by the dimensionless vari- 0 P sitivity: Suppose that node A connects to node B, which able λi = f(λi). The rescaled spacings di are calculated 0 connects to node C. A likely connects to C if the network from the λi. This procedure ensures that di averages to 0 has high clustering. The clustering coefficient equals the 1 because λ i, so every di 1. i ≈ ≈ number of connected triangles, divided by the number Consider initializing the lattice with a 1 centered in | i of possible connected triangles. If M denotes an L L 0 ’s. Evolving with rule T6 leads to only 11 values of × | i −10 adjacency matrix, λi > 10 . A small sample size dooms any statistical measure, so we omit T from Fig. 3(a). Entropy fluctua- Tr M 3 6 (M) = . (15) tions distinguish integrability from nonintegrability alter- C L−1 2 j,k=0 [M ]jk natively, as we showed in work on the quantum ratchet’s j6=k many-body chaos [44]. P Clustering is known to be high for complex networks, e.g., Watts-Strogatz networks [3]. The second complex-network measure, disparity, quan- E. Mutual information tifies how much a network resembles a backbone. The disparity is defined as The mutual information quantifies networks’ complex- 1 L−1 L−1 (M )2 ity. A complex network is network, or graph—a collection k=0 jk (M) = 2 . (16) of nodes, or vertices, and links, or edges—that is neither Y L L−1 j=0 P Mjk regular nor random [3]. For example, completely con- X k=0 nected graphs and Erd˝os-Rényi graphs are not complex Two examples illuminate thisP definition. First, consider networks. a uniform network: Mjk = a, except along the diago- The order-α quantum Rényi mutual information be- nal, where Mjj = 0. This network has a disparity of 11

= 1/(L 1). Second, a 1D chain has an adjacency ma- Y − trix of Mjk = a(δj(k+1) + δj(k−1)) and so a disparity of = (L+2)/(2L). If L = 2, the two networks describe the 1.0 (a) ε = 4.39% sameY graph, so each has = 1. As L increases, the fully Y ε = 5.93% connected network’s disparity approaches 0, while the 1D chain’s approaches 0.5. Figure 3(b)-(g) show average 1 0.8 ε = 8.00% c− 2 clusterings and disparities for various quantum states. / L (1) Fluctuations in the disparity [3] indicate a lack of equi- b ˆ P libration under the Goldilocks rule T6: The network re- 0.6 peatedly forms and breaks a backbone. 0.4 G. Quantum-entangled-breather lifetime 0 100 200 300 400 Time, t Rule F4 generates a quantum entangled breather 1 [Fig. 2(g)-(h)]. The system is initialized with two cen- (b) (c) (d) tered 1 ’s separated by a 0 . Left unperturbed, the 64 state oscillates| i indefinitely.| Figurei 4(d) shows how the breather decays under perturbations of three types. First, we add the rule-F26 Hamiltonian, scaled by a 48 positive 1, to the F Hamiltonian. Rule 26 activates 4 t a site whenever its neighbors contain at least one 1 . Sec- ˆ (1) Pj ond, we perturb the QCA via Schmidt truncation:| i We 32 evolve the system with a matrix-product-state approx- Time, imation. The Hilbert space’s effective dimensionality is L/2 reduced from χ = 2 until the breather becomes unsta- 16 ble. This truncation time-adaptively caps the amount of entanglement in the state [55]. Third, imperfect initialization affects τ. The initial 1 ’s are replaced with copies 0 of eiφ 0 + √1 2 1 . The| i perturbation’s strength is 0 0 8 16 quantified| i by the− positive| i 1; φ turns out not to af- fect the breather’s stability. Consider the qubit one site Site, j leftward of the center, the qubit at site j = L/2 1 (since j = 0, 1,...,L 1). This qubit’s projectionb c onto − − FIG. 7. A quantum entangled breather’s robustness. (a) 1 is fitted with The breather is generated by the Goldilocks rule F . We per- | i 4 turb F4 with the non-Goldilocks rule F26 at strength ε. Black (1) 1 Pˆ (t) = 1 σˆz (t) = A + Be−t/τ . curves show the exponential fit of Eq. (17). Each curve is bL/2c−1 2 − bL/2c−1 (17) shifted by 0.1 from the curve below it, to ease visualization. D E h D E i (b)-(d) Space-time dependence of the average excitation den- From the fit, we extract the lifetime τ. Figure 7 illus- sity, defined through the projection of each site’s state onto trates the fitting procedure used to produce Fig. 4(d), |1i. The lifetime, defined as τ in Eq. (17), ends at double bars run on L = 17 sites. color-coded in accordance with (a). The quantum entangled breather is robust with respect to all three perturbations. The breather’s lifetime depends on the Hamiltonian perturbation as a slow inverse power law: τ ε−1.3. Under Schmidt truncation, the from this convention only when discussing physical im- lifetime exhibits∝ a sharp threshold at χ =8-10. Below plementations of QCA. If multiple energy scales must be the threshold, τ χ−7; above, τ exceeds the simulation specified, time units are presented. time, T = 1, 000.∝ For imperfect initialization, we find We have developed a custom QCA-simulation library, stability for 1/2, independently of φ: The breather used to generate most of the data analyzed in this Re- has an infinite≤ lifetime. However, increasing decreases port. The library simulates time evolution using opti- the visibility of the breather’s oscillations between 1 mized diagonalization code. Analog QCA propagators and 0 [Fig. 2(g)]. | i are represented exactly or are approximated to second | i or fourth order in the time-step length. We introduced the second-order approximation analytically when prov- H. Numerical and statistical methods ing the equivalence of our analog and digital time evolutions. The analog scheme involved a Hamiltonian Hˆ that In simulations, quantities are rescaled such that ~ = 1, generated a unitary Uˆ. Uˆ factorized into even and odd and energies and times are dimensionless. We deviate unitaries at order (δt)2. The fourth-order approximation 12

smoothed with a moving average whose temporal window equals the system size, L. The data shown come 1 10− from L = 19. Therefore, a 19-time-step window low- slope: 1.98 pass-filters the data. Numerical derivatives amplify high- 3 frequency fluctuations. Hence filtering is important 10− for observing lower-frequency trends. We visualize the smoothing effect with faint lines that show the raw data, 5 10− sampled every L time steps. Error

7 10− slope: 4.05

9 10− VI. PHYSICAL IMPLEMENTATION OF 2 1 0 QUANTUM CELLULAR AUTOMATA 10− 10− 10 Time step, dt We prescribe a scheme for implementing QCA physically, beginning with general results before narrow- FIG. 8. Convergence of numerical time-evolution algorithm ing to an example platform. First, we show that the for order-2 routine (squares) and order-4 routine (circles). We longitudinal-transverse quantum Ising model can, in a initialized an L=11-site lattice to a random state |Ri, then specific parameter regime, reproduce QCA dynamics. simulated one time step of the analog rule T6. Many experimental platforms can simulate this Ising model and so can, in principle, simulate QCA. Exam- ples include superconducting qubits, trapped ions, and partially updates even-j sites, fully updates odd-j sites, Rydberg atoms. then finishes updating even-j sites. By definition, the digital QCA are exact, to numerical precision. We focus on Rydberg atoms trapped in an optical- We define the error as 1 ψ˜ ψ 2. ψ denotes the tweezer lattice. The atoms interact through a van der state at the last time step of− evolution |h | i| by| i the full prop- Waals coupling that induces a Rydberg blockade: Two ˜ close-together atoms cannot be excited simultaneously. agator. ψ denotes the analogous state after evolution We exploit this blockade to condition a site’s evolution by the approximate E propagator. Figure 8 shows the er- on the site’s neighbors. Moreover, Rydberg atoms can ror after one time step, plotted against the time step’s be arranged in various geometries subject to short- and length. The data exhibit the anticipated order-2 and long-range interactions [56]. We exploit these freedoms order-4 scalings, as compared with the exact evolution. to engineer nearest-neighbor rules, with a linear atom We simulate F -type QCA using the OpenMPS library, chain and next-nearest-neighbor rules with a ladder-like which has been tested thoroughly. OpenMPS allows us to lattice [Fig. 9(h)]. represent state vectors compactly, with matrix product states [55]. This Schmidt truncation enables us to test Related work appeared in the literature recently: A the quantum entangled breather’s robustness [Fig. 4(d)]. Rydberg-chain realization of QCA was proposed in [57]. Wintermantel et al. focus on 3-site neighborhoods and a Throughout this work, fits are made via standard chi- spin-flip activation. The spin-flip QCA is closely related squared minimization routines. Such fits appear in the to classical cellular automata: Initial states on the Bloch Brody-parameter estimation of Fig. 4(a), the Page-curve sphere’s poles remain on the poles. However, [57] involves ansatz of Fig. 4(c), and the entangled-breather lifetimes applications to engineering entangled states. We present of Fig. 4(d). The standard error in the fit parameters is a general framework for Rydberg-atom QCA: Our pro- defined as the square-root of the diagonal entries of the posal encompasses 3-site and 5-site neighborhoods, as fit-parameter covariance matrices. This prescription was well as entanglement-generating activation unitaries. used to calculated error bars for the Brody parameter and the entangled-breather lifetime. In the Page-curve The present paper presents results about analog three- fits, we illustrated with shaded bands the temporal fluc- site-neighborhood QCA in only two sections; this is tuations of data used to calculate the long-time averages. one of the two. Usually, we feature digital three-site- The fluctuations are defined as the standard deviations neighborhood QCA (TR rules) and analog totalistic five- in the data used to calculate the averages. The shaded site-neighborhood QCA (FR rules). These choices nar- bands in Fig. 3(d)-(e) show the standard deviations in row down the plethora of possibilities to illustrative ex- the network-measure data used to calculate the long-time amples, though we tested many more possibilities and averages shown with data points. Often, the standard found similar results. Here, we discuss the analog three- deviations are smaller than the data points. site-neighborhood QCA, Eq. (7), to demonstrate our The bond-entropy fluctuations in Fig. 4(b) are physical implementation’s generality. 13

A. From QCA to the Ising model We map the Rydberg Hamiltonian onto the Ising Hamiltonian as follows. First, we set Ω = 2hx and ∆ = 2(hz 2rJ). 2r equals the number of neighbors In analog QCA, the Hermitian activation operator − ˆ x ˆ k Ωj. Second, we set Vj,k = 4J for the neighbors hj =σ ˆj . According to Eq. (2), hj contributes to the ∈ k Ωj and Vj,k = 0 for all other k. The second condi- Hamiltonian if the neighboring sites satisfy the condi- ∈ tions imposed by the projectors. Such a state change tion is easily satisfied in a linear chain, if Ωj consists of can be modeled with the Ising Hamiltonian. the nearest-neighbor sites. Next-nearest neighbors can be neglected due to the rapid decay, with separation, of Let hx denote the transverse field hx; and hz, the longitudinal field. The Ising Hamiltonian has the form the van der Waals interaction’s strength. Modifying the lattice geometry, we show later, enables us to include J next-nearest neighbors in Ω . Hˆ = h σˆx + h σˆz + σˆzσˆz . (18) j x j z j 2 j k We numerically simulated an L=17-atom chain evolv- j j j k∈Ω X X X Xj ing under the Hamiltonian Eq. (19). The simulations The interaction strength J is uniform across the neigh- show the equivalence of a Rydberg-atom simulator’s dy- bors k Ωj and vanishes outside. If hx J hz, namics and QCA. We focused on the analog versions spin j evolves∈ as though the other spins belonged ≈ to of rules T1, T6, and F4. The numerical parameters re- a static, classical external magnetic field, rather than flect the physics of 87Rb atoms with the excited state 6 to an interacting global quantum system. h serves as r = 70S1/2 , for which C6 = 863 GHz µm . We x | i the jth qubit’s gap frequency. h + Jm serves as the fixed Ω = 2 MHz and Vj,j+1 = 36 MHz, which corre- z effective external field’s frequency. The magnetization spond to an interatomic distance of a 5.4 µm, such ' z that Vj,j+1 Ω. m = k∈Ω σˆk counts the 0 ’s in Ωj and subtracts j | i We simulated interactions within each QCA neighbor- off theD number ofE 1 ’s. P | i hood, as well as interactions with the closest neighbors Under each of several TR rules, the activation operates outside the neighborhood. The latter, we checked, do only if a certain number of neighbors occupy 1 ’s: m | i not significantly alter the dynamics. The QCA, recall, must have a particular value. (Under each of the other has boundary sites set to 0 ’s. We implemented these TR rules, the activation operates if m has one of several boundary conditions by tailoring| i the external field ∆ ex- activating values.) Examples include T1, T6, and T8. The perienced by the boundary sites. Ising model can implement these rules if the longitudinal The numerical results are shown in the top row in field’s to hz = mJ. The central spin’s oscillations will − Fig 9. The Rydberg-atom dynamics agree strongly with be resonant with the effective external field at the de- the QCA. Thus, Rydberg atoms offer promise for quan- sired m value. If m does not have the desired value, the tum simulation of QCA and for studying QCA’s robust- central spin’s oscillations are strongly off-resonant. This ness with respect to physical-implementation details. off-resonance effects the projectors in Eq. (2). The central spin’s maximum probability, across a Rabi cycle, of 2 2 flipping is hx/ hx + (hz + Jm) . Rules that activate at C. Analog T -type implementation multiple m values would require additional fields and res- p onances [57]. Examples include rule T14, which activates Here, we introduce implementations of several analog when m = 0, 2. TR rules: T6, T1 and T8. Under these rules, a neighborhood’s central spin is activated if m = 2, 0, 2, respec- tively. Consider a linear atom chain with− interatomic B. From the Ising Model to the Rydberg-atom distance a = a j in the Rydberg-blockade regime quantum simulator j,j+1 [Fig. 9(g)]. Appropriately∀ setting the detuning ∆ can enhance the oscillations undergone by a neighborhood’s Each atom has a ground state g coupled to a highly | i central spin, conditionally on the spin’s neighbors. excited Rydberg state r of principle quantum number Figure 9 shows the numerically computed evolution. 1. Atoms j and k = |j iexperience the isotropic van der 6 6 The system was initialized with one 1 centered in 0 ’s. Waals interaction Vj,k = C6/a . C6 denotes an experi- | i | i j,k Rule-T1 simulations are depicted in Figs. 9(a)-(b); and mental parameter; and aj,k, the interatomic separation. rule-T6 simulations, in Figs. 9(c)-(d). The QCA sim- Rydberg atoms evolve under the effective Hamiltonian ulations (depicted on the left-hand side of each figure Ω 1 pair) agree qualitatively with the Rydberg simulations Hˆ = σˆx + ∆ nˆ + V nˆ nˆ . (19) Ryd. 2 j j 2 j,k j k (depicted on the right-hand side). j j j,k X X X The analog version of rule T1 evolves the system under the Hamiltonian Ω denotes the Rabi frequency of the coupling between the ground and excited states. ∆ denotes a detuning in ˆ ˆ(0) x ˆ(0) H = Pj−1σˆj Pj+1 . (20) the natural Hamiltonian that is transformed and approx- j imated to yield Eq. (19). The detuning affects all atoms X uniformly. The projectors’ action is implemented with the Ising 14

Hamiltonian when the resonance condition is imposed between 000 and 010 . In each of these states, | iΩj | iΩj two neighbors occupy 0 ; two neighbors point upward; the magnetization m =| 2.i Hence the resonance condition implies that hz = 2J, or ∆ = 2Vj,j+1. Rule T8 eﬀects − − the Hamiltonian Eq. (20), except each Pˆ(0) is replaced (1) with a Pˆ . Hence m = 2, and hz = 2J, i.e., ∆ = 0. − The Goldilocks rule T6 evolves the system under the Hamiltonian

ˆ ˆ(0) x ˆ(1) ˆ(1) x ˆ(0) H = Pj−1σˆj Pj+1 + Pj−1σˆj Pj+1 . (21) j X The resonance must be enhanced when the neighboring sites satisfy m = 0. Hence we set hz = 0 or, equivalently, ∆ = Vj,j+1 in the Rydberg Hamiltonian. −

D. F -type implementation

Simulating the F -rules with Rydberg atoms requires that excited next-nearest-neighbor atoms have the same interaction energy as nearest-neighbor atoms. One can meet this condition by forming a zigzag lattice, as shown in Fig. 9(h). The distance ann between next-nearest-neighbor atoms equals the distance between next-nearest-neighbor atoms. Hence the interaction energies are Vj,j+1 = Vnn V . ≡ We discuss, as an example, the F4 rule’s implementation. A neighborhood’s central spin must oscillate if two of its four neighbors are excited. As for T6, m = 0, leading to the resonance condition ∆ = 2V . Figure 9(e)- (f) depict the numerical results, and− Fig. 9(h) depicts the atomic arrangement. The plots confirm strong qual- itative agreement between the QCA dynamics and the quantum-simulator dynamics. FIG. 9. Upper panel: Comparison of QCA evolution and Rydberg-atom-quantum-simulator dynamics. We study the VII. CONCLUSIONS rules T1 (a)-(b) and T6 (c)-(d), whose dynamics are generated for a linear atom chain (g) with an appropriate detuning. We study also rule F (e)-(f), for which the lattice has We have discovered a physically potent feature of en- 4 the shape in (h): Two parallel chains of atoms, represented tangled quantum cellular automata: the emergence of by circles, are trapped in a plane. The nearest-neighbor dis- complexity under Goldilocks rules. Goldilocks rules bal- tance a (red lines) equals the next-nearest-neighbor distance ance activity and inactivity. This tradeoff produces new, nn (orange lines), so Va = Vnn. A local detuning is applied highly entangled, yet highly structured, quantum states. at the boundaries to simulate the QCA boundary conditions. These states are persistently dynamic and neither uni- The higher-order effects of next-nearest-neighbor interactions form nor random. The two Goldilocks rules focused (for rules T1 and T6) and next-next-nearest-neighbor interac- on here—T6 for the 3-site-neighborhood QCA and F4 tions (for rule F4) are simulated. Parameters: L = 17. In the for the 5-site-neighborhood QCA—activate a neighbor- Rydberg simulations, Ω = 2 MHz, and Va = 36 MHz. Time hood’s center site when the neighbors are half in 1 ’s is expressed in units of 2π µs. and half in 0 ’s. Complexity signatures of Goldilocks| i rules include| complex-networki structure, high clustering and high disparity fluctuations; persistent entropy fluc- some persistent dynamical features. However, the con- tuations; and robust emergent features, such as quan- fluence of several complexity signatures distinguishes the tum entangled breathers. Non-Goldilocks rules still dis- Goldilocks rules. play interesting dynamical features. Indeed, the PXP Moreover, we have demonstrated that our QCA time- model, equivalent to rule T1, has received much atten- evolution protocols are implementable in extant digital tion in the literature and serves as an edge case, showing and analog quantum computers. In particular, we have 15 shown that Rydberg atom arrays can be arranged to Appendix: Cluster-state mutual information engineer five-site QCA neighborhoods, large enough to support a robust quantum entangled breather. Digital A cluster state’s mutual-information adjacency matrix quantum computers, programmed with discrete gate se- forms a graph that can have more links than the original quences, can also simulate QCA dynamics that support graph that defined the cluster state. For example, con- emergent complexity even with three-site neighborhoods. sider a three-qubit cluster state defined by a 3 1 grid Our work uncovers a direction for quantum computation: (a 1D chain). The state vector is × to demonstrate that the rich features of biological and social complexity manifest in abiological quantum systems. 1 1  1  1 eiυ ψ =   . 3/2   | i 2  1   1     eiυ    e2iυ     The 1-site and 2-site reduced density matrices are 1 2 1 + e−iυ ρˆ =ρ ˆ = , 0 2 4 1 + eiυ 2

1 4 (1 + e−iυ)2 ρˆ = , 1 8 (1 + eiυ)2 4 ACKNOWLEDGMENTS

2 1 + e−iυ 2 e−iυ + e−2iυ 1 1 + eiυ 2 1 + eiυ 2e−iυ ρˆ0,1 =  −iυ −iυ −2iυ , The authors thank Clarisa Benett, Haley Cole, Daniel 8 2 1 + e 2 e + e Jaschke, Eliot Kapit, Evgeny Mozgunov, and Pedram eiυ + e2iυ 2eiυ eiυ + e2iυ 2    Roushan for useful conversations. This work was per-   formed in part with support by the NSF under grants and, OAC-1740130, CCF-1839232, PHY-1806372, and PHY- 2 1 + e−iυ 1 + e−iυ 1 + e−2iυ 1748958; and in conjunction with the QSUM program, 1 1 + eiυ 2 2 1 + e−iυ which is supported by the Engineering and Physical Sci- ρˆ0,2 =  iυ −iυ  . 8 1 + e 2 2 1 + e ences Research Council grant EP/P01058X/1. This work 1 + e2iυ 1 + eiυ 1 + eiυ 2 . is partially supported by the Italian PRIN 2017, the Hori-    (A.1) zon 2020 research and innovation programme under grant   We index rows and columns such that (0, 0) appears in agreement No 817482 (PASQuanS). NYH is grateful for the upper left-hand corner. The reduced density matrix funding from the Institute for Quantum Information and ρˆ equals the result of takingρ ˆ , swapping row 1 with Matter, an NSF Physics Frontiers Center (NSF Grant 1,2 0,1 row 2, and swapping column 1 with column 2. Diago- PHY-1125565) with support of the Gordon and Betty nalizing these density matrices yields the entanglement Moore Foundation (GBMF-2644), for a Barbara Groce spectra λ forρ ˆ andρ ˆ , as well as λ , 0, 0 forρ ˆ Graduate Fellowship, and for an NSF grant for the In- ± 0 2 ± 0,1 andρ ˆ .{ The} λ = [1 cos(υ/2) ]/2. Zero-valued{ } eigen- stitute for Theoretical Atomic, Molecular, and Optical 1,2 ± values do not contribute±| to the entropy,| so Physics at Harvard University and the Smithsonian As- trophysical Observatory. s(2) = s(2) = s(2) = s(2) = 2 log (3 + cos υ) , (A.2) 0 2 0,1 1,2 − 2

wherein υ [0, 2π). The entanglement spectrum ofρ ˆ1 ∈ Conceptualization (LDC); Data Curation (LEH, MTJ, is λ1, λ2 = (1 cos υ)/4, (3 + cos υ)/4 , and the en- { } { − } DLV, PR); Formal Analysis (LEH, MTJ, DLV, PR, NYH, tanglement spectrum ofρ ˆ0,2 is λ1, λ2 0, 0 . Hence the { } SN, SM, LDC); Funding Acquisition (SM, LDC); Investi- remaining entropies are gation (all authors); Methodology (LEH, NYH, NB, SM, s(2) = s(2) = 3 log 5 + 2 cos υ + cos2 υ . (A.3) LDC); Project Administration (LDC); Resources (LDC); 1 0,2 − 2 Software (LEH, MTJ, DLV, PR); Supervision (NYH, SM, The mutual-information adjacency matrices’ nonzero en- NB, LDC); Validation (LEH, LDC); Writing - original (2) (2) (2) draft (LEH, LDC); Writing - review & editing (LEH, tropies can be expressed as M0,1 = M1,0 = M1,2 = MTJ, PR, NYH, NB, SN, SM, LDC). M (2) = s(2)/2 and M (2) = M (2) = s(2) s(2)/2. 2,1 1 0,2 2,0 0 − 1 16

The graph that deﬁnes the cluster state lacked a link mation, however: M0,2 = 0. For the same reason, the between nodes 0 and 2. The nodes share mutual infor- 1 19 cluster state of Fig.6 3 has a large clustering and an× intuitively large disparity.

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