diff --git "a/FtAzT4oBgHgl3EQfi_1S/content/tmp_files/2301.01508v1.pdf.txt" "b/FtAzT4oBgHgl3EQfi_1S/content/tmp_files/2301.01508v1.pdf.txt" new file mode 100644--- /dev/null +++ "b/FtAzT4oBgHgl3EQfi_1S/content/tmp_files/2301.01508v1.pdf.txt" @@ -0,0 +1,4036 @@ +Functional completeness of planar Rydberg structures +Simon Stastny, Hans Peter B¨uchler, and Nicolai Lang∗ +Institute for Theoretical Physics III and Center for Integrated Quantum Science and Technology, +University of Stuttgart, 70550 Stuttgart, Germany +(Dated: January 5, 2023) +The construction of Hilbert spaces that are characterized by local constraints as the low-energy +sectors of microscopic models is an important step towards the realization of a wide range of quantum +phases with long-range entanglement and emergent gauge fields. Here we show that planar structures +of trapped atoms in the Rydberg blockade regime are functionally complete: Their ground state +manifold can realize any Hilbert space that can be characterized by local constraints in the product +basis. We introduce a versatile framework, together with a set of provably minimal logic primitives +as building blocks, to implement these constraints. As examples, we present lattice realizations of +the string-net Hilbert spaces that underlie the surface code and the Fibonacci anyon model. We +discuss possible optimizations of planar Rydberg structures to increase their geometrical robustness. +I. +INTRODUCTION +Recent advances in the control of single atoms and their +coherent manipulation [1–5] are the technological founda- +tion for applications such as quantum simulation [6–9], +high-precision metrology [10, 11] and, hopefully, future +quantum computers [12–15]. For any of these applica- +tions, suitable platforms must offer a fine-grained control +over of their degrees of freedom, dynamically tunable +interactions, and the possibility to decouple the environ- +ment. Promising in this regard are arrays of individually +trapped, neutral atoms that can be manipulated by opti- +cal tweezers [1, 3] and excited into Rydberg states [16, 17]. +These exhibit strong interactions which lead to the Ry- +dberg blockade mechanism where excited atoms prevent +their neighbors within a tunable radius from being excited +[18–22]. In this paper, we study on very general grounds +the theoretical capabilities of the Rydberg platform in +the blockade regime and demonstrate its versatility by +constructing the gauge-invariant Hilbert spaces of two +models with abelian and non-abelian topological order. +Encouraged by the fast development of the Rydberg +platform, there has been increased interest in identifying +promising near-term applications for the NISQ era [23]. +Among the many applications of two-dimensional arrays +of Rydberg atoms, the field of geometric programming and +the design of synthetic quantum matter have been identi- +fied as promising candidates to leverage the capabilities +of available and upcoming NISQ platforms. +The rationale of geometric programming is the solu- +tion of algorithmic problems by encoding them into the +geometry of the atomic array. This direction of research +is founded on the insight that due to the Rydberg block- +ade, the ground states of these systems naturally map to +maximum independent sets (MIS) on so called unit disk +graphs [24]; finding MIS is a long-known optimization +problem in graph theory that has been shown to be NP- +hard [25]. This makes the computation of ground state +∗ nicolai.lang@itp3.uni-stuttgart.de +energies of Rydberg arrangements NP-hard as well [26], +but also opens the possibility to tackle a variety of other +hard optimization problems [27, 28] by polynomial-time +reductions to the MIS problem [29]. First solutions of MIS +instances on various graphs in two and three dimensions +have been demonstrated in experiments recently [30–32], +and a quantitative comparison of experimental solutions +with classical algorithms suggest a superlinear quantum +speedup for some classes of graphs [32]. +A very different application of the Rydberg blockade +mechanism is the engineering of synthetic quantum matter +on the single-atom level. The potential of this approach +has been demonstrated recently by Verresen et.al. [33] +(related results were reported by Samajdar et.al. [34]), +who proposed the realization of topological spin liquids on +delicately designed lattice structures of atoms. In this sce- +nario, the Rydberg blockade enforces a dimer constraint +(the local gauge constraint of an odd Z2 lattice gauge +theory [35]) which, in combination with quantum fluctua- +tions, can give rise to long-range entangled many-body +states with abelian topological order. First experimental +results were reported shortly after [36], accompanied by a +theoretical study of the used quasiadiabatic preparation +scheme [37]. +This paper is written from and motivated by the syn- +thetic quantum matter perspective, but its results apply +to geometric programming as well. Our starting point is +the question whether other local constraints (besides the +dimer constraint) can be realized on the Rydberg plat- +form. To find an answer, we first formalize the problem +and then use this formulation to derive our main result, +namely that every local constraint that can be encoded +by a Boolean function can be implemented in the ground +state manifold of a planar arrangement of atoms in the +blockade regime. Crucial for this result is the existence of +a structure that implements the truth table of a NOR-gate +(“Not OR”) in its ground state manifold. While our proof +is constructive, it does typically not yield optimal (= +small) solutions. We therefore expand on our main result +and compile a comprehensive list of provably minimal +structures that realize all important primitives of Boolean +logic. Together with a structure that facilitates the cross- +arXiv:2301.01508v1 [quant-ph] 4 Jan 2023 + +2 +ing of two “wires” within the plane, these primitives +provide a toolbox to build structures that satisfy more +complicated constraints. As an example, we construct a +system with a ground state manifold that is locally iso- +morphic to the gauge-invariant Hilbert space of an even +Z2 lattice gauge theory, i.e., the charge-free sector of the +toric code [38]. With a similar construction, we tailor a +pattern of atoms with a ground state manifold isomorphic +to the string-net Hilbert space of the “golden string-net +model [39]”; a system that, with added quantum fluc- +tuations, could support non-abelian Fibonacci anyons. +Having constructed all these structures, we briefly discuss +possibilities to numerically optimize their geometries to +make them more robust against geometric imperfections +and the effects of long-range van der Waals interactions. +Note added. When finalizing this manuscript we be- +came aware of related results reported by Nguyen et. al +in Ref. [40]. The authors focus on optimization problems +on non-planar graphs and find the same structures for +some of the primitives discussed in this paper (especially +the implementation of the ring-shaped NOR-gate and the +crossing). The authors follow the rationale of geomet- +ric programming, so that their motivation, approach and +framework differ from ours. +II. +RATIONALE AND OUTLINE +Here we illustrate the rationale of the paper and pro- +vide a brief outline of its main results without technical +overhead. Readers interested in the details can skip for- +ward to Section III. Readers only interested in specific +applications can read this section first and then skip to +Section VII or Section IX. +In this paper, we consider two-dimensional arrange- +ments of trapped atoms that can either be in their elec- +tronic ground state or excited into a Rydberg state (Ryd- +berg structures). We focus on systems without quantum +fluctuations, where the ground states are determined by +local detunings and Rydberg blockade interactions (Sec- +tion III). The detunings lower the energy for atoms in +the Rydberg state by an atom-specific amount, and the +Rydberg blockade interaction forbids atoms closer than a +specific distance to be excited simultaneously. The inter- +play of these two contributions singles out ground states +that are characterized by excitations patterns where no +additional atom can be excited without violating the Ryd- +berg blockade, and where the sum of the detunings of the +excited atoms is maximal (so called maximum-weight inde- +pendent sets). There can be different configurations that +minimize the energy, hence the ground state manifold is +typically degenerate. In this paper, we ask which ground +state manifolds such structures can realize and, conversely, +how to tailor structures that realize a prescribed ground +state manifold (Section IV). +A simple example is given in Fig. 1a where the po- +sition of the atoms is shown in (i); the two atoms are +constrained by the Rydberg blockade (gray circles) and +Figure 1. Rationale. (a) Structure of two atoms (i) with local +detunings ∆ (blue vertices) that are in Rydberg blockade (gray +circles); the blockade is indicated by a black edge connecting +the atoms. The ground state manifold (ii) is given by patterns +of excited atoms (orange) that minimize the energy; here it +is two-fold degenerate. The two ground state configurations +realize the truth table (iii) of a NOT-gate Q = A. (b) Structure +of five atoms (i) with local detunings ∆ (blue) and 2∆ (green) +in a ring-like Rydberg blockade. The ground state manifold (ii) +is four-fold degenerate. If one selects the three labeled atoms +and identifies them with the columns of the table in (iii), the +four ground state configurations realize the truth table of a +NOR-gate Q = A ↓ B = A ∨ B. (c) Joining the output atom of +the NOR-gate with the input atom of the NOT-gate (and adding +their detunings) yields a new structure that realizes the truth +table of an OR-gate: Q = A ↓ B = A ∨ B. This construction is +called amalgamation. +cannot be excited simultaneously (indicated by the black +edge connecting them). The color of the atoms encodes +their detuning; here both atoms lower the energy of the +system by ∆ when excited into the Rydberg state (blue +nodes). In (ii) we show the two excitation patterns that +minimize the energy (orange nodes denote excited atoms). +Note that the atoms cannot be excited simultaneously +due to the Rydberg blockade. If one lists the ground state +configurations in a table, where each column corresponds +to an atom and each row to a ground state configuration, +we find the “truth table” of a Boolean NOT-gate Q = A. +Here we interpret one of the atoms as “input” (A) and +the other as “output” (Q). +This concept generalizes to more complicated Boolean +gates (Fig. 1b): Consider the five atoms in a ring-like +blockade (i). Three of the atoms (blue) lower the energy +by ∆, two (green) by 2∆ when excited. By inspection +one finds the four degenerate ground state configurations +in (ii). This is promising as truth tables of Boolean gates +that operate on two bits have four rows. However, they +only have three columns (two for the inputs of the gate +and one for its output). We therefore select three of the +five atoms by assigning labels to them: A and B play the + +3 +role of the inputs and Q is the output. We call atomic +structures with designated input/output atoms Rydberg +complexes (Section V A). If we list the four ground state +configurations of these three atoms, we find the truth +table of a NOR-gate Q = A ↓ B = A ∨ B in (iii). Note +that the remaining two atoms (we call them ancillas)— +while not contributing independent degrees of freedom— +are still necessary to realize this specific ground state +manifold. At this point things get interesting because it +is a well-known fact of Boolean algebra that the NOR-gate +is functionally complete (just like the NAND-gate): Every +Boolean function can be decomposed into a circuit build +from NOR-gates only. +To leverage this decomposition, we need a method +to combine “gate complexes” to form larger “circuit +complexes”; we call this procedure amalgamation (Sec- +tion V B). A simple example is shown in Fig. 1c where we +attach the NOT-gate from Fig. 1a to the output of the NOR- +gate in Fig. 1b (note that the detunings of the atoms that +are joined add up). Using the detunings and blockades in +(i) yields the four degenerate ground state configurations +in (ii). When we label the inputs of the NOR-gate again by +A and B, and now focus on the output Q of the attached +NOT-gate, we find indeed the truth table of an OR-gate +Q = A ↓ B = A ∨ B in (iii). Thus we can parallel the log- +ical composition of gates by a geometrical combination of +atomic structures such that the relation between ground +state configurations and truth tables remains intact. In +combination with the insight that every Boolean circuit +can be drawn in the plane without crossing lines (after +suitable augmentations), this allows us to show that the +truth table of any Boolean function can be realized as +the ground state manifold of a suitably designed atomic +structure. This functional completeness is our first main +result and motivates the title of the paper (Section VI). +For instance, the existence of a structure that realizes +the truth table of an OR-gate is a corollary of functional +completeness. However, the specific construction as the +combination of a NOR-gate and a NOT-gate in Fig. 1c raises +the questions whether this particular realization with +six atoms is unique and whether it is minimal (in the +sense that the same truth table could not be realized +with fewer atoms). The answer to the first question is +negative: There are geometrically different structures +that realize the same truth table in their ground state +manifold. The answer to the second question is positive, +though: We show that it is impossible to implement this +truth table with less than six atoms. Note that the func- +tional completeness implies the existences of structures +for all common gates of Boolean logic (such as AND, XOR, +etc.). We take this as motivation to construct provably +minimal structures for all these primitives (Sections VII +and VIII). Together with the procedure of amalgama- +tion, these equip our versatile toolbox to engineer more +complicated structures. +Our second important contribution is an application of +the functional completeness as a tool to engineer synthetic +quantum matter (Section IX). Many interesting quantum +phases in two dimension are characterized by hidden pat- +terns of long-range entanglement, known as topological +order. These patterns can give rise to anyonic excitations +which make such systems potential substrates for quantum +memories and even quantum computers. A large class of +entanglement patterns can be understood as condensates +of extended objects (like strings). A crucial first step for +the realization of these phases is therefore the prepara- +tion of Hilbert spaces spanned by states of such extended +objects. However, in experiments, we typically start from +Hilbert spaces with a local tensor product structure (for +example, an array of two-level atoms). Our only hope is +to make the extended objects emerge due to interactions +in the low-energy sector of a suitably designed physical +system. This often boils down to enforce local gauge sym- +metries which single out states that can be interpreted in +terms of extended objects. Such local constraints can be +reformulated as Boolean functions that must be satisfied +by the states of the local degrees of freedom of the underly- +ing system. For any constraint of this form, our functional +completeness result ensures the existence of a structure of +atoms, interacting via the Rydberg blockade mechanism, +that realizes this constraint in its ground state space. It is +then just a matter of copying and joining these structures +in a translational invariant way to tessellate the plane. +The ground state manifolds of such tessellations can there- +fore implement a large class of non-trivial Hilbert spaces +on which condensation (driven by quantum fluctuations) +might lead to topologically ordered many-body quantum +phases. Using our toolbox developed in the first part of +the paper, we demonstrate this construction explicitly +for the abelian toric code phase (Section IX A) and the +non-abelian, computationally universal Fibonacci anyon +model (Section IX B). +The truth tables realized by the ground states of all +atomic structures presented in this paper depend on the +positions of the atoms. (Because these positions define +which pairs are in blockade and which atoms can be ex- +cited simultaneously.) However, the exact placement is +often ambiguous. For example, consider the structure in +Fig. 1a (i) which realizes the NOT-gate. It is clear that the +blockade constraint (black edge) does not change if the +atoms are slightly shifted, as long as the blockade radii +(gray circles) encompass both atoms. We refer to the set +of atom positions as the geometry of a structure and argue +that “robust” geometries should avoid distances between +atoms that are close to the critical blockade distance. For +the complexes in Fig. 1, this translates into the geomet- +ric objective to maximize the distances between nodes +and gray circles. We formalize this notion by assigning a +number to geometries that quantifies their “robustness” +(Section X A) and numerically construct optimized geome- +tries that maximize this number (Section X B). +We conclude the paper with an outline of open ques- +tions, directions for further research (Section XI), and a +brief summary (Section XII). + +4 +III. +PHYSICAL SETTING +We consider planar arrangements of trapped atoms +with repulsive van der Waals interactions when excited +into the Rydberg state [2, 41]. Every atom is assigned +an index i ∈ V = {1 . . . N}, placed at position ri ∈ R2, +and described by a two-level system |n⟩i where n = 0 +corresponds to the electronic ground state and n = 1 the +excited Rydberg state. +The quantum dynamics of such systems is achieved +by coupling the electronic ground state to the Rydberg +state by external laser fields with Rabi frequency Ωi and +detuning ∆i for each atom [42–44]. Here we are mainly +interested in the regime Ωi → 0 where the Hamiltonian +reduces to +H[C] = − +� +i +∆ini + +� +i 0 the coupling strength of the van der Waals +interaction; we refer to H[C] with U = UvdW as the van +der Waals (vdW) model. However, in many situations +a simplified model U = U∞ with U∞(r ≥ rB) = 0 and +U∞(r < rB) = ∞ with blockade radius rB is a reasonable +approximation for the low-energy physics of Eq. (1); we +refer to H[C] with U = U∞ as the PXP model [33, 47]. In +this paper, we use the PXP model unless stated otherwise. +We discuss valid choices for the blockade radius rB in +Section X A where we optimize the geometry of structures +to limit the effects of residual van der Waals interactions. +In the PXP model, the effect of the van der Waals +interactions is approximated by a kinematic constraint +that is completely encoded by a blockade graph B = +(V, E), where an edge e = (i, j) ∈ E between atoms i, j ∈ +V indicates that they are in blockade, i.e., their distance +is smaller than the blockade radius rB. An abstract graph +that can be realized in this way is called a unit disk +graph (not every graph has this property); conversely, +a geometry GC that realizes a prescribed graph as its +blockade graph is a unit disk embedding of this graph (the +“unit” here is the blockade radius rB). Throughout the +paper, the blockade graph of a structure will be drawn +by black edges connecting atoms that are in blockade. +IV. +DEFINITION OF THE PROBLEM +The primary goal of this paper is to find structures C +such that there is a well-separated low-energy eigenspace +H0[C] of H[C] where H0[C] satisfies certain prescribed +Figure 2. Setting & Objective. A two-dimensional structure +C = (ri, ∆i)i∈V of atoms i ∈ V with position ri and detuning +∆i is governed by the Hamiltonian H[C] that describes the +Rydberg blockade interaction with blockade radius rB. The +Hamiltonian gives rise to a low-energy eigenspace H0[C] < H +of width δE, separated from the excited states by a gap ∆E. +The objective of this paper is the construction of a structure C +from a given target Hilbert space HT such that H0[C] ≃ HT. +properties that we describe in detail below. We quantify +the separation of H0[C] by its spectral width δE and +its gap ∆E to the rest of the spectrum (Fig. 2). Note +that the experimental prerequisites for the construction of +arbitrary structures C are already in place [4, 45, 46, 48]. +If one would switch on a weak drive δE < Ωi ≪ ∆E, this +would induce quantum fluctuations between the states of +the Hilbert space H0[C], potentially giving rise to many- +body states with interesting properties. In this paper, +we do not study such quantum effects but focus on the +implementation of the subspace H0[C]. We specify the +eigenspace to construct in terms of a target Hilbert space +HT: +H0[C] +!≃ HT . +(2) +Informally speaking, our goals is to “solve” this equation +for structures C for given HT. To make this possible, the +target Hilbert space HT must be specifiable in a form +that we define in the remainder of this section. +Formal languages. +Throughout the paper we make +use of the notion of (formal) languages [49] on the bi- +nary alphabet F2 = {0, 1}. A word x ≡ (x1x2 . . . xn) ≡ +x1x2 . . . xn ∈ F∗ +2 is a finite string of letters xi ∈ F2 (the +set of all such finite strings is denoted F∗ +2). +A (for- +mal) language L is then simply a collection of words: +L ⊆ F∗ +2. +Here we only consider uniform languages +with words that have all the same length. For exam- +ple, LCPY := {000, 111} ⊂ F∗ +2 is a uniform language of +words with length n = 3, x = (111) is a word in LCPY and +x1 = 1 is the first letter of x. The words y = (011) ∈ F∗ +2 +and z = (0000) ∈ F∗ +2 are not in this language: y, z /∈ LCPY. +The subscript “CPY” stands for “copy” and hints at the +role this language will play later. +Other examples are the class of languages generated by +the truth tables of Boolean functions. Let w : Fn−1 +2 +→ F2 +be an arbitrary Boolean function of n − 1 variables; then +L[w] := {x1 . . . xn−1y | y = w(x1, . . . , xn−1)} ⊂ F∗ +2 +(3) + +5 +Figure 3. +Tessellated language & target Hilbert space. +A +tessellated target Hilbert space HT is a subspace of the full +Hilbert space of K qubits placed on each edge of a square +lattice L; it is spanned by product states |x⟩ of bit patterns +x ∈ LL[fT]. The tessellated language LL[fT] comprises all bit +patterns x ∈ F∗ +2 that locally satisfy the Boolean check function +fT : Fg +2 → F2. The g = 4K arguments of the check function +on each site s are singled out by the bit-projector us. +is the language generated from the rows of the truth table +of w, where the first n−1 letters of each word correspond +to the input x and the last letter encodes the output +w(x). A language of this class always has 2n−1 words of +uniform length n. Note that the “copy” language LCPY is +not of the form Eq. (3). +Another special class is given by tessellated languages +on lattices. In the following, we introduce the concept ex- +emplarily for a finite square lattice L with periodic bound- +aries; the generalization to other lattices and boundary +conditions is straightforward. Start by associating K clas- +sical bits to every edge e ∈ E(L) of the lattice (Fig. 3). A +bit configuration of the system x ∈ XL = FK|E(L)| +2 +⊂ F∗ +2 +assigns every bit a Boolean value xi +e (i = 1 . . . K). We +focus on the family of uniform languages L ⊆ XL that +can be characterized by a Boolean function that is local +in the following sense: For a site s ∈ V (L) of the square +lattice, let the bit-projector us(x) = (x1 +e1, . . . , xK +e4) single +out the (ordered) set of g = 4K bits on the four edges +ei emanating from s. Let f : Fg +2 → F2 be an arbitrary +Boolean function of g arguments, henceforth referred to +as check function. The tessellated language of bit patterns +on L generated by f is then defined as +LL[f] := {x ∈ XL | ∀s ∈ V (L) : f(us(x)) = 1} . +(4) +In words: LL[f] is the set of bit patterns on the lattice L +that locally satisfy the constraints imposed by f. +Target Hilbert spaces. +To any uniform language +L ⊆ Fn +2 we can naturally associate the linear subspace of +states on n qubits (or spin-1/2) +H(L) := span { |x⟩ | x ∈ L } ⊆ (C2)⊗n . +(5) +For example, H(LCPY) = span { |000⟩ , |111⟩ } is the two- +dimensional subspace on three qubits spanned by product +states with configurations in LCPY = {000, 111}. By con- +trast, the Hilbert space H′ = span +� +(|000⟩ + |111⟩)/ +√ +2 +� +is not of the form (5). +We require the target Hilbert space HT, that we aim +to realize as ground state manifold H0[C], to be specified +by a language LT according to Eq. (5): +HT = H(LT) . +(6) +We are particularly interested in the special class of tes- +sellated target Hilbert spaces given in terms of tessellated +languages that are generated by a check function (Fig. 3): +HT = HL[fT] := H(LL[fT]) . +(7) +Recall that these languages come equipped with a spatial +structure (in the sense that the bits are located on the +edges of a lattice L). This spatial structure is inherited +by the Hilbert space HL[fT] viewed as state space of a +system where K qubits are placed on every edge of L. +For example, the Hilbert space HZ2 of the even Z2 +lattice gauge theory is a particular subspace of a Hilbert +space that describes a system of qubits on the edges of +a square lattice (i.e., K = 1 and g = 4). HZ2 is spanned +by the product states of patterns of qubits in the state +|1⟩ that form closed loops [50]. +HZ2 is an admissible +tessellated target Hilbert space because we can realize +HZ2 = HL[fZ2] with the check function +fZ2(x1, x2, x3, x4) = 1 ⊕ x1 ⊕ x2 ⊕ x3 ⊕ x4 +(8) +where ⊕ denotes modulo-2 addition (Exclusive-OR or +XOR); the bit-projector us(x) simply singles out the four +bits on edges emanating from site s: +us +� +� +� +� +� +� +� +� = (x1 +e1, x1 +e2, x1 +e3, x1 +e4) . +(9) +Physically, Eq. (8) enforces Gauss’s law on a charge-free +background by forbidding strings of qubits in state |1⟩ +to end on a site. Further examples for tessellated target +Hilbert spaces are the more general “string-net” Hilbert +spaces that can describe a large variety of topological +orders and deconfined gauge theories [39]. +V. +RYDBERG COMPLEXES +Before we can tackle our main goal, namely the con- +struction of tessellated Rydberg structures C with H0[C] ≃ +HT = HL[fT] for a given check function fT, we first need +to specify the notion of a finite Rydberg complex as a pre- +liminary step. Specific examples for Rydberg complexes +can be found throughout the remainder of the paper. + +6 +A. +From structures to complexes +Consider the language LCPY = {000, 111} and let +HCPY = H(LCPY) = span { |000⟩ , |111⟩ } be our target +Hilbert space. Our goal is to realize HCPY as the ground +state manifold H0[CCPY] of a structure CCPY of n = 3 atoms. +This, however, is impossible: Since |111⟩ ∈ HCPY, none of +the three atoms can be in blockade with each other. Con- +sequently, H0[CCPY] cannot contain only the states |000⟩ +and |111⟩ (Appendix A 1). This problem is not specific +to the language LCPY but shared by many (though not all) +languages. The solution is to consider larger structures +of N ≥ n atoms and to identify the letters of words with +a subset of n distinguished atoms (we call them ports); +the remaining N − n atoms play then the role of ancillas. +A structure together with a distinguished set of ports will +be referred to as a (Rydberg) complex. +Let us formalize this notion. Consider a structure C +of N atoms and a language L ⊆ Fn +2 of words of uniform +length n ≤ N. Let L = {A, B, . . . } denote a set of n labels +where each label is associated with a fixed letter position +of words in L. (If one prints all words of L as rows of a +table, the labels correspond to the column headers.) Let +ℓ : L → V be an injective label function that assigns a +label to a subset of n atoms (the ports); the N − n atoms +without labels are the ancillas. We refer to the structure +C together with the labeling ℓ as a (Rydberg) L-complex +CL if the states that span H0[C] can be identified by the +configurations of the ports alone: +H0[CL] ≡ H0[C] = span { |x, a(x)⟩ ∈ H | x ∈ L } . +(10) +In |x, a(x)⟩, the state of ports is given by the first n bits x +(in some fixed order) and the state of ancillas by a N − n +bit-valued function a : L → FN−n +2 +. The ground state +space H0[CL] will be referred to as an L-manifold. An +important aspect of this definition is that the ancillas do +not introduce additional low-energy degrees of freedom; +they are only needed to unleash the full potential of the +blockade interactions. In this sense, we say that a complex +CT ≡ CLT realizes a target Hilbert space HT = H(LT) and +write +(C2)⊗N ⊇ H0[CT] ≃ HT = H(LT) ⊆ (C2)⊗n +(11) +with the isomorphism ≃ given by |x, a(x)⟩ ↔ |x⟩. If we +say that a complex realizes a language L, we mean that +it realizes the target Hilbert space HT = H(L) defined by +this language. +As an example, consider the again the “copy” language +LCPY = {000, 111} with n = 3; the ground state manifold +of a LCPY-complex CCPY ≡ CLCPY must be two-dimensional +(since |LCPY| = 2) and characterized by the property that +three distinguished atoms (the ones assigned labels by ℓ) +are always forced to be in the same state: +H0[CCPY] = span { |000, a(000)⟩ , |111, a(111)⟩ } . +(12) +Such a complex will be one of our primitives to implement +check functions for tessellated target Hilbert spaces. We +will discuss a specific realization CCPY that requires a single +ancilla in Section VI; that is, with N = n = 3 atoms the +target Hilbert space HCPY cannot be realized, whereas +with N = 4 it can. +As another example, consider the logical XOR-gate +wXOR(x1, x2) = x1 ⊕ x2 which may be needed as a prim- +itive for a check function like Eq. (8). We can ask for +a complex CXOR that realizes the target Hilbert space +HXOR = H(LXOR) given by the language LXOR ≡ L[wXOR] = +{000, 011, 101, 110} that is generated by this Boolean gate. +The ground state manifold of such a complex must be +spanned by four states, +H0[CXOR] = span +� +|000, a(000)⟩ , |011, a(011)⟩ , +|101, a(101)⟩ , |110, a(110)⟩ +� +(13) +where the configurations of potential ancillas are deter- +mined by the configurations of the three ports. We will +introduce a specific realization CXOR in Section VII; it re- +quires N = 7 atoms of which four are ancillas, and we +show that this is indeed the smallest complex that can +realize the language of a XOR-gate. +Since LXOR = L[wXOR] is generated from a Boolean gate, +we refer to complexes that realize a language of this form +as gates, too. Furthermore, we denote the atoms that map +to the input bits of the gate as input ports, and the atom +that corresponds to the output bit as the output port. We +also extend this nomenclature to Boolean functions w on +more than two inputs. Let us stress that these terms are +only inspired from the usual role played by such functions +as parts of Boolean circuits. In the present context, there +is no time evolution or dynamics involved (there is no +information “flowing into” the input ports, although it +might be sometimes helpful to use this picture). +The construction of an L-complex for a given language +L with word length n can be split into two steps: First, +one has to find a structure C on at least n atoms with an +|L|-fold degenerate ground state manifold. Then, one has +to identify a labeling ℓ of n atoms such that their states +in the ground state manifold map one-to-one to words +in L. The structure C together with the labeling then +yields an L-complex. Note that the same structure can be +interpreted as different complexes for different languages +by choosing different label functions. Furthermore, not +every structure with |L|-fold degenerate ground state +manifold allows for a valid labeling that realizes L. Hence +the construction is a quite non-trivial task in general. +This makes a reductionist approach seem most promising, +where one starts with a finite set of small “primitive” +complexes and constructs larger complexes by “gluing” +them together. +B. +Amalgamation +The process of combining two complexes by joining +(some of) their ports is referred to amalgamation. To +define the process formally, we first need a new concept +to combine two languages. + +7 +Consider two uniform languages L1 and L2 of words +of length n1 and n2, respectively. Let γ ⊆ {(p1, p2) | pi ∈ +{1, . . . , ni}} be a set of disjoint [51] pairs of letter positions +and set γi := {pi | p ∈ γ}. For a word x ∈ Li, let xγi +denote the word with all letters at positions in γi deleted. +Then, the γ-intersection of L1 and L2 is defined as +L1 +γ +∩L2 := +� +x yγ2 | x ∈ L1, y ∈ L2, ∀(a,b)∈γ xa = yb +� +which is a language of words of length n1 + n2 − |γ|. +L1 +γ +∩L2 is the set of concatenations of words from L1 and +L2 where the letters at the positions indicated by pairs +in γ coincide, and where the second copy of these letters +has been deleted. Analogously, we define the reduced +γ-intersection as +L1 +γ +∩ L2 := +� +xγ1 yγ2 | x ∈ L1, y ∈ L2, ∀(a,b)∈γ xa = yb +� +, +only that now both copies of identified letters are deleted; +hence this is a language of words with length n1+n2−2|γ|. +As an example, consider again the XOR-language +LXOR = {000, 011, 101, 110} and the CPY-language LCPY = +{000, 111}. We would like to copy the output of the XOR- +gate. To do this, we intersect the output bit (letter 3) of +the XOR-language with one of the bits (say letter 1) of the +CPY-language: γ = {(3, 1)}. The γ-intersection is the new +language +LXOR +γ +∩LCPY = {00000, 01111, 10111, 11000} +(14) +with words of length 3 + 3 − 1 = 5. The underscores indi- +cate the letters that derive from words of both languages. +If one drops these letters as well (by using the reduced +γ-intersection), the language describes a XOR-gate with +fan-out of two: +LXOR +γ +∩ LCPY = {0000, 0111, 1011, 1100} . +(15) +The above definitions on the level of languages are +useful because they are paralleled by a combination of +complexes called amalgamation: Consider two complexes +CL1 and CL2 that realize the languages L1 and L2 with +N1 and N2 atoms, respectively. Fix a set of pairs of ports +γ such that L′ = L1 +γ +∩L2 ̸= ∅, and then combine the two +complexes by identifying the atoms in γ: +CL′ = CL1 +γ +⊗ CL2 := += +. +(16) +The new complex CL′ has N1 + N2 − |γ| atoms. For this +construction, we assume that the ports that belong to +pairs in γ are located on the boundary of their complex +(we will show in Section VI why this is possible). The +Hamiltonian of the new complex is +H[CL′] = (H[CL1] + H[CL2] + δH) /γ +(17) +where the formal quotient •/γ indicates that pairs of +atoms in γ are identified; δH denotes additional interac- +tions between the two subcomplexes CLi that vanish in +the PXP model (in the vdW model they are finite but +strongly suppressed due to the quick decay of UvdW). +In a nutshell: H[CL′] is the sum of the Hamiltonians +of the original two complexes were the detunings of the +ports that are identified by γ add up. For example, let +n(1) and n(2) describe ports of CL1 and CL2, respectively, +and let γ identify these two ports. Then H[CL1] contains +a term −∆(1)n(1) and H[CL2] contains a term −∆(2)n(2). +The Hamiltonian (17) of the amalgamation contains the +term (−∆(1)n(1) − ∆(2)n(2))/γ = −(∆(1) + ∆(2))n′ where +n′ = n(1)/γ = n(2)/γ describes the atom that corresponds +to the identification of the two ports. +With δH = 0, it is straightforward to verify that the +amalgamation CL′ realizes the language L′ = L1 +γ +∩L2. +This is so because the ground state energy of H[CL′] is +lower-bounded by the sum of the ground state energies +of the summands H[CLi]; but this lower bound is realized +by configurations in L′ ̸= ∅. +The ports identified by +γ can be interpreted as ancillas of the new complex if +|L1 +γ +∩L2| = |L1 +γ +∩ L2|, i.e., if the states of these atoms +provide redundant information about the ground state +manifold; in this case, one would define L′ = L1 +γ +∩ L2 +instead. +An important special case of the above construction +is the amalgamation of gates where the input ports of +one gate are identified with the output ports of others. +For example, let w(x1, x2) and w′(x′ +1, x′ +2) be two Boolean +gates that are concatenated into the circuit on three inputs +˜w(x′ +1, x1, x2) := w′(x′ +1, w(x1, x2)). It is easy to see that +L[ ˜w] = L[w] +γ +∩ L[w′] with γ = {(3, 2)} where 3 labels the +third letter of words in L[w], which encodes the output +y = w(x1, x2), and 2 labels the second letter of words in +L[w′], which encodes the input x′ +2. Note that for Boolean +circuits without redundancies it is always |L[w] +γ +∩L[w′]| = +|L[w] +γ +∩ L[w′]| because all words are identified by the input +bits. This example demonstrates that the amalgamation +of gates is a crucial ingredient for the decomposition of +complex Boolean circuits into a small set of simple gates. +VI. +FUNCTIONAL COMPLETENESS +We have now all concepts and tools in place to formulate +the main result of this paper: +Theorem 1 (Functional completeness). For every tes- +sellated target Hilbert space HT = HL[fT] on some lattice +L that is generated by a check function fT, there exists a +structure CT in the PXP model such that +HT +loc +≃ H0[CT] , +(18) +with finite gap ∆E > 0 and perfect degeneracy δE = 0. + +8 +Figure 4. Decomposition of Boolean functions. (a) Any Boolean function fT can be represented by a graph GfT (a “Boolean +circuit”) with dedicated input vertices (blue squares), one output vertex (red square), and trivalent vertices (circles) of two +types (b): NOR-gates with two incoming and one outgoing edge (orange circles) and CPY-vertices with one incoming and two +outgoing edges (black circles); the edges themselves can be interpreted as trivial single-bit gates, here referred to as LNK-gates +(black edges). If the inputs (A,B) and outputs (Q,R) of all three primitives are assigned Boolean values that satisfy the truth +tables in (b), the value at the output vertex is y = fT(x1, . . . ) by construction. (c) The embedding Γ(GfT) (“drawing”) of the +abstract graph GfT in the plane R2 typically involves crossings (whenever GfT is non-planar); furthermore, input and output +vertices may lie in the interior of the graph. Since a crossing of wires can be implemented with the available vertices (d), the +graph can always be enhanced such that it becomes planar and input/output vertices lie on the perimeter of the embedding. (e) +Locally, the embedding Γ(GfT) decomposes into three primitives, namely the structures referred to as NOR, CPY, and LNK that +are functionally defined by the truth tables in (b) and geometrically by the sketches in (e). +In Eq. (18), +loc +≃ denotes an isomorphism of Hilbert +spaces like Eq. (11) that, in addition, preservers the lo- +cality structure: it maps local unitaries on HT to local +unitaries on H0[CT] and vice versa. +Here the locality +structure of H0[CT] is induced by the locality structure +of H which reflects the physical realization of the system. +The locality structure of HT = HL[fT] derives from the +lattice L and the bit-projector us that was used to define +the tessellated language LL[fT]; it is therefore part of +the defining properties of the Hilbert space HT. This +local isomorphism will be explicit for the examples in +Section IX. +Proof. The proof of Theorem 1 is constructive in principle +and best split into several steps: Steps 1 to 4 deal with the +construction of a Rydberg complex CfT=1 that implements +the constraint of the check function on a single site of +the lattice. In the final Step 5, the structure CT is then +constructed as the amalgamation of copies of CfT=1 on +the full lattice. +Step 1: Decomposition of fT. +The first goal is to con- +vert the check function fT : Fg +2 → F2 on g binary inputs +into a finite set of Boolean gates as “building blocks.” +There are many universal gate sets to choose from [52] +but the one that is most natural to the Rydberg platform +is the singleton {NOR} that contains only the NOR-gate [53] +A ↓ B := A ∨ B . +(19) +The idea behind this choice is simple: placing three atoms +A, C, B in a row such that the pairs (A, C) and (C, B) are +in blockade but the pair (A, B) is not naturally gives rise +to a constraint akin to C = A ↓ B (we discuss the details +below). The functional completeness of {NOR} allows us +to write +fT(x1, . . . , xg) = (. . . (xi ↓ xj) . . . (xk ↓ xl) . . . ) +(20) +where the expression on the right can be any (recursive) +combination of expressions built from the input variables +paired by NOR-gates. On an abstract level, this is a neat +result; however, in reality one has to be more careful +because variables can be used multiple times at different +locations in the NOR-expansion of fT. +To identify the true physical building blocks needed to +cast Eq. (20) into a structure of atoms, it is advisable +to translate the NOR-expansion into a graph GfT that +represents the underlying Boolean circuit and uses the +inputs xi only once at dedicated “input vertices” and +outputs the result fT(x1, . . . , xg) at a dedicated “output +vertex” (Fig. 4a). Otherwise, GfT is a trivalent graph with +two types of vertices, corresponding to CPY-operations +that copy a bit and NOR-gates that combine two bits +according to Eq. (19). If we assign arrows to the edges +to highlight the information flow, the two vertices are +distinguished by the number of in- and outgoing edges +(CPY: 1 in and 2 out, NOR: 2 in and 1 out). Furthermore, +we can interpret the edges themselves as trivial single-bit +gates (“LNK-gates”). If we assign Boolean values to the +inputs and outputs of these three primitives according to +the truth tables in Fig. 4b, the value of the output vertex +is given by y = fT(x1, . . . , xg). Without loss of generality, +we consider only circuits without redundancy, i.e., for +a given input {x1, . . . , xg} the state of the inputs and + +9 +outputs of all its primitives is uniquely determined. This +implies that there are exactly 2g such assignments that +are parametrized by the g inputs {x1, . . . , xg} (this can +be seen as a boundary condition; in a dynamical circuit, +one would call it an initial condition). +Step 2: Embedding of GfT. +The graph GfT represents +the Boolean circuit of fT on an abstract level (only the +connectivity of GfT is relevant). Our final goal is to trans- +late this graph into a functionally equivalent structure of +atoms in the plane. Thus we have to find an embedding +Γ(GfT) of GfT in R2; this embedding should be planar, +i.e., without crossing edges to avoid unwanted interac- +tions. Here we skip a formal definition of Γ(GfT) and +appeal to the intuition of the reader: Γ(GfT) describes a +drawing of GfT in the plane without crossing edges and +with well-separated vertices (Fig. 4c). Of course not every +graph GfT is planar, i.e., can be drawn without crossing +edges in the plane. However, it has been shown long +ago that every Boolean circuit can be made planar by +augmenting it with “crossover sub-circuits” whenever two +lines cross [54]. This crossover can be constructed with +various gate sets, including the NOR-singleton (Fig. 4d). +The embedding of the crossover then uses only the three +available primitives in Fig. 4b so that we can, without +loss of generality, assume Γ(GfT) to be planar. +Note +that the existence of a crossover also implies that we can +assume the input and output vertices to be located on +the perimeter of the embedding (as realized in Fig. 4c). +Translated into complexes, this will prove our claim in +Section V that we can assume the ports to sit on the +perimeter of a complex. +While Γ(GfT) may look very convoluted on a larger +scale, locally it decomposes into the three simple primi- +tives depicted in Fig. 4e, namely CPY, NOR, and LNK. The +next step is then to implement these three primitives as +complexes both geometrically (i.e., following the geome- +try in Fig. 4e) and functionally (i.e., following the truth +tables in Fig. 4b). An fT-complex can then be obtained +by amalgamation of these primitives according to the +geometric blueprint provided by Γ(GfT). +Step 3a: Implementing the LNK-complex. +The LNK- +complex is the physical counterpart of the “wires” in the +drawing of the circuit Γ(GfT). Logically, it corresponds to +the trivial gate w(x) = x with language LLNK = {00, 11}. +On the level of pure Boolean logic, wires are not entities +of their own but on the physical level, sending a bit from +one location to another requires dedicated machinery. +Before we discuss its construction, it is useful to intro- +duce a more fundamental complex that can be used to +construct two of the three primitives: the NOT-gate with +defining language L¬ = {01, 10}; it realizes the single-bit +gate w(x) = x and formalizes the core concept of the +Rydberg blockade. In the PXP model, it can be realized +naturally without ancillas by the Hamiltonian +H¬ = −∆(nA + nQ) +(21) +with a complex C¬ where |rA − rQ| < rB. The subscripts +denote the labels of the ports assigned by ℓ (we reserve +A, B, . . . for input ports and Q, R, . . . for output ports). +The ground state manifold is H0[C¬] = span { |01⟩ , |10⟩ } +with degeneracy δE¬ = 0 and gap ∆E¬ = ∆ > 0. +The elementary LNK-complex that translates a bit in +space can then be constructed as the amalgamation of +two NOT-gates (Fig. 5a) with Hamiltonian +HLNK = −∆nA − 2∆˜n1 − ∆nQ , +(22) +where adjacent atoms are in blockade but next-nearest +neighbors are not. Above and in the following we label +ancillas with a tilde and assign them numerical indices. +As for the NOT-gate, it is δELNK = 0 and ∆ELNK = ∆ with +the LNK-manifold +H0[CLNK] = span { |0(1)0⟩ , |1(0)1⟩ } . +(23) +Here and in the following we mark the states of ancillas by +parentheses. Repeated amalgamation of elementary LNK- +complexes results in LNK-complexes of arbitrary length +(always composed of an odd number of atoms and with +halved detuning at the endpoints). The two states in +H0[CLNK] of such chains correspond to the two ground +states of an antiferromagnetic Ising chain. +Step 3b: Implementing the CPY-complex. +The purpose +of the CPY-complex is to copy classical bits; it is defined by +the “copy” language LCPY = {000, 111}. The CPY-complex +is necessary because expansions in universal gates can +reuse inputs multiple times. Furthermore, circuits can +be simplified dramatically if intermediate results can be +reused. In conventional drawings of Boolean circuits, the +possibility to copy bits is silently assumed whenever one +splits up wires. Again, in a physical implementation one +has to provide the means to do so. +The implementation of the CPY-complex is detailed in +Fig. 5b. +It is easy to see (Appendix A 1) that there +cannot be a CPY-complex without ancillas because the +configuration 111 excludes a Rydberg blockade between +any of the three ports (which would automatically render +them completely uncorrelated). Adding a single ancilla +does the trick because the amalgamation of three NOT- +complexes on a single atom yields the desired complex +by construction. The four atoms are described by the +Hamiltonian +HCPY = −∆(nA + nQ + nR) − 3∆ ˜n1 , +(24) +and the geometry of the complex CCPY is chosen so that +the ancilla is in blockade with the three ports, but these +are not within blockade of each other. In combination +with Eq. (24), this implements the CPY-manifold +H0[CCPY] = span { |000(1)⟩ , |111(0)⟩ } +(25) +with δECPY = 0 and ∆ECPY = ∆ > 0. +Step 3c: Implementing the NOR-complex. +The NOR- +complex is crucial as it realizes a functionally complete +two-bit gate; it is specified by the language LNOR = +{001, 010, 100, 110}. +In contrast to the LNK- and CPY- +complexes, the NOR-complex cannot be bootstrapped from +the NOT-complex but must be constructed from scratch. + +10 +Figure 5. Complete set of logic primitives. (a) The (elementary) LNK-complex CLNK can be realized by a chain of three atoms +where adjacent atoms are in blockade (black edges). The detuning of the ports ∆ (blue squares, labeled by ℓ) is half that of the +ancilla 2∆ (green circle) in the bulk. The width δE and gap ∆E are shown together with a schematic spectrum that highlights +the logical manifold H0[CLNK] and one of the states orthogonal to H0[CLNK] that define the gap. The state of ancillas is shown +in parentheses. (b) The CPY-complex CCPY can be realized with a central ancilla (red circle) that is in blockade with the three +surrounding atoms (blue squares). To make the two logical states degenerate, the ancilla has a detuning of 3∆ if the other atoms +are detuned by ∆. (c) The NOR-complex CNOR can be realized with two ancillas (blue and green circles) that form a ring-like +blockade with the three ports (blue and green squares). To make the four logical states unique and degenerate, the detunings +cannot be chosen uniformly but must break the reflection symmetry about the axis through the output port Q. +In Appendix A 2 we show that a NOR-complex cannot be +realized with less than two ancillas in the PXP model. One +implementation of a NOR-complex is detailed in Fig. 5c. +The five atoms are governed by the Hamiltonian +HNOR = −∆(nA + nQ + ˜n1) − 2∆(nB + ˜n2) +(26) +which gives rise to the NOR-manifold +H0[CNOR] = span +� |001(01)⟩ , |010(10)⟩ , +|100(01)⟩ , |110(00)⟩ +� +(27) +with δENOR = 0 and ∆ENOR = ∆; this requires that the +atoms are arranged in a ring-like blockade, as depicted in +Fig. 5c. Note that the two ancillas are only necessary to +enforce the degeneracy of the logical states 010 and 100 +with 110. All remaining constraints come for free with the +Rydberg blockade. As we will show in Section VII, the +NOR-complex in Fig. 5c is not unique. We will also see that +the only fundamental Boolean gate that can be realized +with as few as five atoms is the NOR-gate, confirming our +intuition in Step 1 that the NOR-gate is the most natural +on the Rydberg platform. +Step 4: Constructing the fT-complex. +To construct +a complex CfT that implements the check function fT +(more precisely: the language L[fT]), one combines the +three primitives above according to an embedding Γ(GfT). +Since all vertices are (at most) trivalent, it is easy to +check that an amalgamation in the PXP model is possible +without geometrical obstructions, and that this procedure +yields an fT-complex with δEfT = 0 and ∆EfT ≥ ∆ > 0. +At this point, we have a complex with g = 4K input +ports on its boundary that outputs y = fT(x1 +e1, . . . ) on a +dedicated output port (also on its boundary, but this is +not important in the following): +(28) +To enforce the constraint fT(x1 +e1, . . . ) +!= 1, we only have +to add a local detuning on the output port to lower the +energy of valid configurations and gap out invalid ones. +This boils down to a simple modification of the check +function complex, +→ +(29) +where the output port is detuned and downgraded to +an ancilla. The ground state manifold of the modified +complex CfT=1 consists of all input configurations for +which fT(x1 +e1, . . . ) = 1. +Step 5: Constructing CT. +The complex CfT=1 enforces +the local constraint of the check function on a single site +of the lattice on which the tessellated target Hilbert space +HT = HL[fT] is defined. To construct CT for the full +system, place a copy CfT=1 �→ Cs +fT=1 on every site s ∈ +V (L) of the lattice, and amalgamate adjacent complexes +at the corresponding ports (possibly using LNK-complexes +to avoid unwanted interactions): + +11 +CT := +(30) +By construction, the ground states of this complex are in +one-to-one correspondence with words x ∈ LL[fT] (using +the ports on the edges denoted by blue squares). Note +that here we show the construction for a square lattice L; +the generalization to other lattices is straightforward. +This concludes the construction of CT such that HT +loc +≃ +H0[CT] in the PXP approximation. Note that the ancillas +do not introduce additional degrees of freedom in this +subspace and local unitaries on HT map to local unitaries +on H0[CT] (the latter involve the ancillas of the CfT=1 +complexes and can therefore be very complicated—but +they remain local on H). +■ +We conclude this section with a few remarks. First, +while the proof above is constructive, one should not +expect the resulting structures to be useful in real-world +applications, except for simple special cases. In particular, +we established no claims about optimality (in any sense) +of the constructed fT-complexes; on this we focus in the +next Section VII. Second, the modification in Eq. (29) +to construct CfT=1 from CfT is often straightforward to +implement and can simplify the complex considerably: +When there are no blockades between the output port and +some of the input ports, one simply deletes the output +port along with all ancillas that are in blockade with +it. This removes all configurations of input ports from +the ground state manifold where the output was not +excited (see Appendix B). And finally, the removal of the +output port may not be necessary at all if the constraint +fT(x1 +e1, . . . ) +!= 1 can be rewritten as an equality of the +form +f1(x1 +e1, . . . , x1 +e2, . . . ) +!= f2(x1 +e3, . . . , x1 +e4, . . . ) , +(31) +with Boolean functions f1,2 that take only 2K inputs each. +Then CfT=1 = Cf1 +γ +⊗ Cf2 where the two complexes are +amalgamated at their output ports: += +(32) +An example for this construction can be found in Sec- +tion IX A. +VII. +LOGIC PRIMITIVES +A crucial step of the proof in the previous section is to +show that every Boolean function f can be realized by a +Rydberg complex Cf in the sense that the language L[f] +of its truth table can be realized as ground state manifold. +As mentioned above, the complexes that arise from the +decomposition of f into LNK-, CPY- and NOR-primitives +are typically large and convoluted. +For example, the +decomposition of a simple AND-gate (∧) into NOR-gates +reads +A ∧ B = (A ↓ A) ↓ (B ↓ B) , +(33) +which would require two CPY- and three NOR-complexes, +wired together by a bunch of LNK-complexes so that the +resulting complex requires more than 20 atoms. As this +is way too much overhead for a simple gate, the question +arises whether important primitives of Boolean logic can +be realized by complexes that are much smaller than the +ones described by the NOR-decomposition in Section VI. +The answer is positive: In the following, we discuss +provably minimal complexes for the most important gates +of Boolean logic, all of which improve signi��cantly over +the na¨ıve NOR-decomposition. Besides the usual gates of +Boolean algebra, NOT (¬ or •), AND (∧), and OR (∨), we +search for minimal complexes that realize the following +common logic gates (given in disjunctive normal form): +NOR: +A ↓ B = A ∧ B +(34a) +NAND: +A ↑ B := A ∨ B +(34b) +XOR: +A ⊕ B := (A ∧ B) ∨ (A ∧ B) +(34c) +XNOR: +A ⊙ B := (A ∧ B) ∨ (A ∧ B) . +(34d) +Of these gates, only NOR and NAND are universal on their +own. The following identities show that some of these +gates are simply inverted versions of others (we will use +this below): +A ∧ B = A ↑ B +(35a) +A ∨ B = A ↓ B +(35b) +A ⊕ B = A ⊙ B . +(35c) +Of the gates {¬, ∨, ∧, ↑, ↓, ⊕, ⊙}, we already know min- +imal complexes for NOT (2 atoms) and NOR (5 atoms), +recall Section VI. + +12 +A +Q +R +A +Q +R +1 +A +Q +R +2 +CPY +A Q R +1 +2 +1 +1 +1 +0 +0 +0 +A +Q +R +CPY +A +Q +A +Q +1 +A +Q +2 +LNK +A Q +1 +2 +0 +0 +1 +1 +A +Q +LNK +A +Q +A +Q +1 +A +Q +2 +NOT +A Q +1 +2 +0 +1 +1 +0 +A +Q +NOT (¬) +B +Q +A +B +Q +A +1 +B +Q +A +2 +B +Q +A +3 +B +Q +A +4 +AND +A B Q +1 +2 +3 +4 +1 +1 +1 +0 +0 +0 +0 +1 +0 +1 +0 +0 +A +B +Q +AND (∧) +A +B +Q +A +B +Q +1 +A +B +Q +2 +A +B +Q +3 +A +B +Q +4 +OR +A B Q +1 +2 +3 +4 +0 +1 +1 +1 +0 +1 +1 +1 +1 +0 +0 +0 +A +B +Q +OR (∨) +Q +A +B +Q +A +B +1 +Q +A +B +2 +Q +A +B +3 +Q +A +B +4 +NOR +A B Q +1 +2 +3 +4 +0 +1 +0 +1 +0 +0 +1 +1 +0 +0 +0 +1 +A +B +Q +NOR (↓) +A +B +Q +A +B +Q +1 +A +B +Q +2 +A +B +Q +3 +A +B +Q +4 +XOR +A B Q +1 +2 +3 +4 +0 +1 +1 +1 +0 +1 +1 +1 +0 +0 +0 +0 +A +B +Q +XOR (⊕) +B +A +Q +B +A +Q +1 +B +A +Q +2 +B +A +Q +3 +B +A +Q +4 +NAND +A B Q +1 +2 +3 +4 +1 +1 +0 +0 +0 +1 +0 +1 +1 +1 +0 +1 +A +B +Q +NAND (↑) +A +B +Q +A +B +Q +1 +A +B +Q +2 +A +B +Q +3 +A +B +Q +4 +XNOR +A B Q +1 +2 +3 +4 +0 +1 +0 +1 +0 +0 +1 +1 +1 +0 +0 +1 +A +B +Q +XNOR (⊙) +Figure 6. Common logic primitives. Rydberg complexes for the most common primitives of Boolean circuits. All complexes +are provably minimal, see Appendix A. Note that minimal complexes are not necessarily unique; e.g. the shown NOR-gate is +an alternative to the one in Fig. 5c, both of which are minimal. For each complex we show (1) the geometry with blockade +radii (gray dashed circles), (2) the complete ground state manifold (orange: |1⟩i, black: |0⟩i), and (3) the truth table of the +ports (labeled atoms) in the ground state manifold. The rows of the truth tables correspond to the numbered ground state +configurations. Colors of ancillas and ports in the geometry encode the detuning (see key). Atoms in blockade are connected by +black solid lines. + +13 +Using Eq. (35b), we can immediately construct an +OR-complex with six atoms by amalgamation of a NOT- +complex to the output port of a NOR-complex (remember +Fig. 1). +However, it is unclear whether this complex +is minimal, i.e., cannot be realized with fewer atoms. +Therefore we systematically devised proofs that a given +truth table cannot be realized with a given number N of +atoms, starting at N = 3 for each gate, and increasing the +number incrementally until the proof fails, i.e., realizations +can no longer be excluded. These arguments are quite +technical and can be found in Appendix A. However, this +approach has two benefits: First, it provides rigorous +lower bounds on how many atoms are needed to realize +a given gate, and second, it often provides a blueprint +for the construction of a minimal complex that saturates +this bound by carefully observing why one cannot exclude +realizations with a given number of atoms. +To complement this rigorous approach, we conducted a +brute force search on a computer that exhaustively scans +for (small) complexes that realize a given truth table. In +accordance with our proofs, we found solutions with the +minimal atom number for a given truth table (in addition, +we also found non-minimal complexes). +Interestingly, +there were alternative minimal solutions that we missed +in our manual approach; so minimal complexes are not +necessarily unique. +A selection of provably minimal complexes for all im- +portant Boolean primitives is shown in Fig. 6 (for the +sake of completeness, we include the NOT-, LNK- and CPY- +complexes discussed in Section VI). There are a few com- +ments in order. First, an example of non-unique minimal +complexes is the depicted NOR-complex built from five +atoms arranged in a triangular structure (cf. the ring- +shaped structure in Fig. 5c). Second, the six-atom OR- +complex we proposed above indeed is minimal, though not +unique either. Third, the selection of minimal complexes +in Fig. 6 for {∨, ∧, ↑, ↓, ⊕, ⊙} all build around the triangle- +based core of the NOR-complex, once again emphasizing +its central role in the context of Rydberg complexes. Fi- +nally, it turns out that the relations (35) are all reflected +in the minimal complexes, e.g., the amalgamation of a +NOT-complex and a XNOR-complex yields a minimal XOR- +complex; similar constructions hold for NAND and AND as +well as NOR and OR. If we recall the relation between NOT +and the minimal LNK-complex, the general picture emerges +that inverting complexes are simpler (by one atom) than +non-inverting ones. This is understandable in so far as +inversion is the most basic operation the Rydberg block- +ade is capable of, thus leading to the simplest complexes. +This is in contrast to the notation for Boolean circuits +known from electrical engineering where inverting gates +are represented by more complicated symbols than their +non-inverting counterparts (Fig. 6). +VIII. +CROSSING +The crossing complex realizes the somewhat surprising +feature of intersecting information channels in a strictly +two-dimensional setup of strongly interacting information +carriers (recall Step 2 in Section VI). The possibility +to realize such a planar crossing in a circuit with the +three primitives LNK, CPY and NOR was crucial for the +proof of Theorem 1. Note that the existence of such a +complex followed immediately from the existence of the +three aforementioned complexes and the well-known fact +that Boolean circuits can be made planar [54]. However, +just as for the Boolean gates in Section VII, the NOR-based +implementation of the circuit crossing in Ref. [54] is of +low practical value as it requires seven NOR-gates (if we +implement NOT-gates directly, Fig. 4d); even a simpler +crossing based on only three minimal XNOR-gates requires +∼ 27 atoms, see Fig. 7a. Thus we are again tasked with +finding a minimal complex that realizes the same function. +By systematically excluding the existence of crossing +complexes for N = 4, . . . , 9 atoms, we finally find the +minimal complex CCRS depicted in Fig. 7b comprising 10 +atoms. The proof for its minimality is very technical and +more complicated than for the logic primitives because +geometric constraints must be taken into account for the +crossing [55]. The structure with two dangling ports (Q +and R) immediately suggests the inverted crossing CICRS +in Fig. 7c with eight atoms, i.e., a complex that allows +two signals to pass each other while inverting both at +the same time. The minimality of the inverted crossing +complex CICRS with eight atoms follows as a corollary +from the minimality of the non-inverted crossing CCRS +with 10 atoms as the latter can be obtained from the +former by amalgamation of two NOT-complexes (thereby +adding two atoms). In line with our comment at the +end of the previous Section VII, the inverted variant of +the crossing is smaller than its non-inverted counterpart. +We note that the inverted crossing CICRS has also been +described in Ref. [40] were it plays an important role +in mapping non-planar optimization problems to planar +Rydberg structures. +IX. +EXAMPLES: SPIN LIQUID PRIMITIVES +In this part, we focus on our motivation outlined in the +introduction, namely the implementation of tessellated +target Hilbert spaces of systems that are characterized by +local gauge constraints. We discuss two models exemplar- +ily: the surface code with abelian Z2 topological order +and the non-abelian Fibonacci model. For the surface +code, we will be able to utilize the Boolean primitives +discussed in Section VII; by contrast, for the Fibonacci +model such a reduction will not be useful. + +14 +(a) +A +B +R +Q +(b) +CRS +A +Q +R +B +B +A +Q +R +B +A +Q +R +1 +B +A +Q +R +2 +B +A +Q +R +3 +B +A +Q +R +4 +CRS +A B Q R +1 +2 +3 +4 +0 +0 +0 +0 +0 +1 +0 +1 +1 +0 +1 +0 +1 +1 +1 +1 +(c) +ICRS +A +Q +R +B +B +A +R +Q +B +A +R +Q +1 +B +A +R +Q +2 +B +A +R +Q +3 +B +A +R +Q +4 +ICRS +A B Q R +1 +2 +3 +4 +0 +0 +1 +1 +0 +1 +1 +0 +1 +0 +0 +1 +1 +1 +0 +0 +Figure 7. Crossing. (a) The crossing constructed from the Boolean circuit crossing based on XNOR-gates (see Ref. [54] and Fig. 6); +it is an amalgamation of LNK-, CPY-, and XNOR-complexes. The ground state manifold (not shown) is 4-fold degenerate and +ensures A = Q and B = R. The complex requires ∼ 27 atoms and is therefore of no practical relevance. (b) By contrast, the +minimal crossing CCRS requires only 10 atoms; it was constructed by systematically excluding functionally equivalent complexes +with fewer atoms. The shown data is explained in the caption of Fig. 6. (c) The minimal inverted crossing CICRS is smaller than +the non-inverted crossing and requires only eight atoms. To construct CCRS from CICRS, two NOT-complexes must be amalgamated +to adjacent ports. This is a recurring scheme due to the inverting nature of the Rydberg blockade. +A. +Surface code +The toric code [38] is the prime example for a spin +liquid in two dimensions with long-range entangled ground +states that do not break any symmetries but instead +feature topological order. The toric code is referred to +as surface code if realized on surfaces with boundaries +[56]; we will stick to this name in the following. The +surface code describes a gapped phase with Z2 topological +order that is described by the mechanism of string-net +condensation [39]. It allows for localized excitations that +are abelian anyons [57] which, in turn, leads to ground +state degeneracies on topologically non-trivial surfaces +(including flat surfaces with non-trivial boundaries). As +a consequence, surface codes are promising candidates +for quantum memories that encode logical qubits reliably +into delocalized degrees of freedom [58]. This makes the +implementation of systems with this kind of topological +order interesting both from an academic and an applied +perspective [36, 59, 60]. +Here we consider the surface code on a finite square +lattice with “rough” boundaries (like the gray background +lattice in Fig. 8d); “rough” boundaries are terminated by +dangling edges that attach to four-valent vertices. The +Hamiltonian +H = −JA +� +Sites s +As − JB +� +Faces p +Bp +(36) +operates on qubits that live on the edges e of the square +lattice. The operators +As = +� +e∈s +σz +e +and +Bp = +� +e∈p +σx +e +(37) +are referred to as star and plaquette operators, respec- +tively. +Here, e ∈ s denotes edges that emanate from +site s and e ∈ p denotes sites that bound face p; σα +e +are Pauli matrices for α = x, y, z acting on the qubit +on edge e. +Since [As, Bp] = 0, the Hamiltonian (36) +is frustration-free and its ground state |G⟩ is character- +ized by As |G⟩ = Bp |G⟩ = |G⟩ for all sites s and faces +p (assuming JA, JB > 0). Due to the uniform “rough” +boundaries there is no ground state degeneracy and |G⟩ +is unique. +The construction of |G⟩ is straightforward: To satisfy +the constraint As |G⟩ = |G⟩ on sites s, one can choose +the product state |0⟩ with σz +e |0⟩ = |0⟩ for all edges. This +state does not satisfy the constraint Bp |G⟩ = |G⟩ on faces, +though. To fix this, one defines the multiplicative group +B = ⟨{Bp | Faces p}⟩ generated by all plaquette operators +(note that B2 +p = 1), and constructs the superposition +|G⟩ ∝ +� +C∈B +C |0⟩ . +(38) +The state |G⟩ is invariant under any Bp by construc- +tion since B is left-invariant under any Bp by defini- +tion. Furthermore, since [As, Bp] = 0, the site-constraint +As |G⟩ = |G⟩ is still satisfied. Thus Eq. (38) describes, +up to normalization, the unique ground state of Eq. (36). +The states |C⟩ ≡ C |0⟩ have a peculiar structure: each +C can be described as a collection of closed loops on the +lattice where the σx +e of products of Bp operators act (loops +that terminate on dangling edges at the boundary are +considered closed); this loop structure is then imprinted +on |0⟩ so that |C⟩ is a product state with a loop pattern +C of flipped qubits |1⟩. The ground state Eq. (38) is +therefore given by the equal-weight superposition of all +closed loop configurations on the square lattice—which +makes it an example of a string-net condensate [39] with +a non-trivial pattern of long-range entanglement [61, 62]. + +15 +(a) +C +A +D +B +CSCU +(c) +C +A +D +B +1 +C +A +D +B +2 +C +A +D +B +3 +C +A +D +B +4 +C +A +D +B +5 +C +A +D +B +6 +C +A +D +B +7 +C +A +D +B +8 +(b) +A B C D +1 +2 +3 +4 +5 +6 +7 +8 +0 +1 +0 +1 +1 +0 +0 +1 +0 +1 +1 +0 +1 +0 +1 +0 +0 +0 +0 +0 +1 +1 +0 +0 +0 +0 +1 +1 +1 +1 +1 +1 +(d) +CLoop +Figure 8. Surface code. (a) Unit cell/vertex complex CSCU for the surface code (Z2 topological order). The complex is the +amalgamation and deformation of two XNOR-complexes [see Fig. 6 and Eq. (43)] and implements the check function constraint +fLoop = 1 defined in Eq. (42). The deformations are necessary to prevent an unwanted blockade of ancillas in the amalgamation. +Black edges denote blockades between atoms, gray edges illustrate the underlying square lattice. (b,c) Truth table and ground +state manifold of the complex. The manifold contains all configurations with an even number of labeled atoms excited, thereby +realizing Gauss’s law on the site (colored edges). This provides the local isomorphism between HT = HLoop and H0[CLoop]. (d) +Periodic tessellation CLoop of the vertex complex CSCU. The copies overlap on the edges and are amalgamated at these ports +(which makes the detunings uniform in the bulk). +To prepare this state in a real system, one could try to +implement the Hamiltonian (36) and cool the system into +its ground state. This is a challenging task due to the +four-body interactions (37) which are notoriously hard +to realize. On the Rydberg platform, an alternative and +more promising approach goes as follows: In a first step, +one prepares only the subspace +HLoop := { |Ψ⟩ | ∀ Sites s : As |Ψ⟩ = |Ψ⟩ } += span { |C⟩ | C ∈ B } +(39) +as the low-energy manifold of a suitably designed struc- +ture of atoms. (HLoop is the Hilbert space of a Z2 lattice +gauge theory with charge-free background [50]. The local +constraint As |Ψ⟩ = |Ψ⟩ corresponds to the gauge sym- +metry of this theory and is known as Gauss’s law.) The +Bp-terms in Eq. (36) induce quantum fluctuations on +this subspace which give rise to the string-net condensed +ground state in Eq. (38). On the Rydberg platform, quan- +tum fluctuations can be induced perturbatively by ramping +up the Rabi frequency Ωi. Such fluctuations can give rise +to interesting quantum phases, as shown in Ref. [33] for +a different model. This motivates the construction of a +Rydberg complex CLoop with +H0[CLoop] +loc +≃ HT = HLoop = span { |C⟩ | C ∈ B } , +(40) +i.e., a Rydberg complex the degenerate ground states +of which can be locally mapped one-to-one to loop con- +figurations on the square lattice. +H0[CLoop] is then a +subspace with dimension dim H0[CLoop] ∼ 2M where M +denotes the number of unit cells of the square lattice. +Note that H0[CLoop] cannot be decomposed into factors +of local Hilbert spaces (like, e.g., the full Hilbert space +H = (C2)⊗2M can). +To this end, we assign bits x1 +e to the edges of the square +lattice L (K = 1). Our goal is to specify the tessellated +“loop language” LL[fLoop]—which contains all bit patterns +that trace out closed loop configurations on the lattice +(closed in the sense defined above)—in terms of a local +check function fLoop and a local bit-projector us on each +site s of the square lattice. +The bit-projector simply +selects the four bits on edges adjacent to s, +us +� +� +� +� +� +� +� +� = (x1 +e1, x1 +e2, x1 +e3, x1 +e4) +(41) +and the check function reads +fLoop(x1, x2, x3, x4) = (x1 ⊙ x2) ⊙ (x3 ⊙ x4) +(42) +with the XNOR-gate ⊙ defined in Eq. (34d), that is, A⊙B = +1 iff A = B. It is easy to verify by inspection that fLoop = 1 +if and only if the number of active bits is even, thereby +enforcing Gauss’s law on every site of the lattice (because +loops cannot terminate there). + +16 +We could now construct a complex as discussed in +Section V, using the minimal XNOR-complex depicted in +Fig. 6. For this construction, we would amalgamate three +of these complexes according to Eq. (42) and detune the +final output to enforce fLoop = 1; this would require at +least 16 atoms per site. We can do much better, though, +by rewriting the constraint as an equality: +fLoop = 1 +⇔ +x1 ⊙ x2 = x3 ⊙ x4 . +(43) +Indeed, Eq. (43) evaluates to true iff x1 + x2 + x3 + x4 +is even. In general, an implementation of an equality +constraint f1 = f2 of two functions on separate inputs is +achieved by amalgamation of their complexes Cf1 and Cf2 +at their output ports, as noted at the end of Section VI. +Therefore, the vertex complex CSCU ≡ CfLoop=1 (“Surface +Code Unit cell”) that realizes the constraint Eq. (43) is +that of only two XNOR-gates amalgamated at their outputs +(Fig. 8a) which requires only 11 atoms. Surprisingly, it +turns out that this realization is also minimal, see Ap- +pendix C 1 for a proof. (Note that typically the construc- +tion of larger complexes from minimal primitives does not +yield minimal complexes.) The two XNOR-complexes that +make up the vertex complex are geometrically deformed +variants of the XNOR-complex shown in Fig. 6. This is +necessary to prevent unwanted blockades between ancillas +in the amalgamation. +In Fig. 8b we show the configurations of the four la- +beled ports (A, B, C, and D) of the complex in the 8-fold +degenerate ground state manifold. In Fig. 8c we illus- +trate the excitation patterns of these eight ground states +(atoms excited to the Rydberg state are colored orange). +Highlighting the edges of the square lattice whenever the +labeled ports associated to them are excited yields the +local mapping (40) to the loop structure of states in HLoop. +Note that the ancillas do not add additional degrees of +freedom in the ground state manifold. +For the tessellation (Fig. 8d) the vertex complex is +copied and shifted periodically along the basis vectors of +the square lattice. The labeled ports are then amalga- +mated to the corresponding ports of complexes on adjacent +sites. Quite remarkably, due to the amalgamation, the +detunings in the bulk become uniform, which makes this +tessellation interesting under the constraints of current +platforms [32, 36]. (Note that imposing periodic bound- +ary conditions on the lattice, i.e., going back to the toric +code, would render the detunings completely uniform.) +Finally, we briefly comment on the modifications of +the surface code patch in Fig. 8d that would be neces- +sary to use it as a quantum code. It is well-known [56] +that a surface code patch encodes a single logical qubit +if its four sides alternate in boundary types: top and +bottom remain “rough” but left and right are modified to +“smooth” boundaries by cutting of the dangling edges of +the square lattice. On these boundaries, the sites become +trivalent “T”-shaped with the same Gauss’s law (i.e., the +number of active edges must be even). On these sites, the +4-valent complex in Fig. 8a must be replaced by a trivalent +one. Conveniently enough, this is just the XOR-complex +in Fig. 6 as the truth table of XOR contains exactly the +four assignments of three Boolean variables such that +x1 + x2 + x3 is even. As a bonus, closing of the left and +right sides of the patch with XOR-complexes leads to com- +pletely uniform detunings along these boundaries. The +simplicity of the vertex complex on trivalent sites suggests +a definition of the surface code on the Honeycomb lattice +(which is perfectly possible [39]). However, because of the +two sites per unit cell, this does not reduce the number +of required atoms per unit cell to implement the check +function. Indeed, the realizations with minimal Rydberg +complexes on both lattices are essentially equivalent, as +can be seen in Fig. 8d by rotating the tessellation by 45°. +B. +Fibonacci model +The surface code only supports abelian anyons, which +are not sufficient for universal topological quantum com- +putation, where gates are implemented fault tolerantly by +braiding of localized excitations and measurements corre- +spond to their fusion [63–65]. The simplest anyon model +that supports universal computation by braiding is known +as Fibonacci model due to the role the Fibonacci numbers +play in the fusion rules [66–68]; it may be realized in some +fractional quantum Hall states [69, 70]. As quasiparticles, +the properties of Fibonacci anyons are a consequence of +and encoded in the entanglement pattern of the ground +state on which they live. The latter turns out to have +a representation as a string-net condensate with weights +and “string-net” patterns that differ from the surface code +[cf. Eq. (38)]. If we consider a Honeycomb lattice with +qubits on its edges, the fixed-point ground state of the +Fibonacci model has the form [39] +|G⟩ = +� +S +Φ(S) |S⟩ , +(44) +where the sum goes over all patterns (“string-nets”) S of +flipped qubits |1⟩ on the edges of the Honeycomb lattice +where no single string ends on a vertex. That is, in con- +trast to the loop patterns C of the surface code, vertices +with three fusing strings are allowed. The coefficients +Φ(S) of the superposition are non-trivial functions of the +pattern S, so that the condensate is no longer an equal- +weight superposition [39, 71, 72]. It is possible to write +down a solvable, local Hamiltonian like Eq. (36) with the +exact ground state (44) which is, however, so complicated +that it is essentially useless for implementations [39]. This +complication, together with the potential usefulness of +the model for quantum computation, motivates again the +construction of a Rydberg complex CFib that implements +the tessellated target Hilbert space +H0[CFib] +loc +≃ HT = span { |S⟩ | String-net S } +(45) +which has the dimension dim H0[CFib] ∼ (1 + ϕ2)M + +(1 + ϕ−2)M where M is the number of unit cells of the + +17 +(a) +B +A +E +D +C +CFMU +CfFib=1 +(c) +B +A +E +D +C +1 +B +A +E +D +C +2 +B +A +E +D +C +3 +B +A +E +D +C +4 +B +A +E +D +C +5 +B +A +E +D +C +6 +B +A +E +D +C +7 +B +A +E +D +C +8 +B +A +E +D +C +9 +B +A +E +D +C +10 +B +A +E +D +C +11 +B +A +E +D +C +12 +B +A +E +D +C +13 +(b) +A B C D E +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +1 +1 +1 +1 +1 +1 +1 +0 +1 +1 +1 +1 +1 +0 +1 +0 +1 +1 +1 +1 +0 +1 +0 +1 +1 +0 +1 +1 +0 +1 +1 +0 +1 +1 +1 +1 +0 +0 +1 +1 +1 +0 +1 +0 +1 +0 +0 +0 +0 +0 +0 +0 +1 +1 +0 +1 +1 +0 +0 +0 +1 +1 +1 +1 +0 +(d) +CFib +Figure 9. Fibonacci model. (a) Unit cell complex CFMU for the Fibonacci model that implements two copies of the single-site +check function constraint fFib = 1 defined in Eq. (47). The complex is the amalgamation of two equivalent 8-atom complexes +CfFib=1 on the two trivalent sites that make up the basis of the honeycomb unit cell. Black edges denote blockades between +atoms, gray edges illustrate the underlying Honeycomb lattice. (b,c) Truth table and ground state manifold of the unit cell +complex. The manifold contains all configurations with closed strings and, in addition, configurations with three strings fusing +on a site. This provides the local isomorphism between the string-net Hilbert space HT and H0[CFib]. (d) Periodic tessellation +CFib of the complex CFMU. The copies overlap on the edges and are amalgamated at the corresponding ports. +Honeycomb lattice and ϕ = (1+ +√ +5)/2 is the golden ratio +[73, 74]. As for the surface code, H0[CFib] is a Hilbert +space that cannot be decomposed into factors of local +Hilbert spaces. +Since the Honeycomb sites are trivalent, the bit- +projector takes now the form +us +� +� +� +� +� +� = (x1 +e1, x1 +e2, x1 +e3) +(46) +and the check function that specifies the allowed string- +nets can be written in the compact form +fFib(x1, x2, x3) = (x1 ⊕ x2 ≡ x3) ∨ (x1 ∧ x2 ∧ x3) +(47) +where the first clause (x1 ⊕ x2 ≡ x3) realizes the loop +constraint (≡ denotes the logical equivalence which is +equivalent to the XNOR-gate as a connective) and the +second clause (x1 ∧ x2 ∧ x3) allows for the fusion of three +strings. Note that without the second clause we fall back +to the loop constraint of the surface code (now on the +honeycomb lattice). +Since there are five assignments with fFib = 1, this +check function cannot be realized by a single logic gate +(despite having three ports) but must be decomposed +into a circuit. Furthermore, since the amalgamation of +two logic gates always results in a complex with an even +number of ports, at least three gates would be necessary +to realize the Fibonacci constraint. This already leads +into the territory of ≳ 15 atoms which we deem too much +overhead for a single site. Therefore we follow the same +approach as for the logic primitives in Section VII: We +systematically exclude the existence of complexes CfFib=1 +for N = 3, 4, . . . , 7 atoms (Appendix C 2). The approach +fails for N = 8 and we find the minimal complex in Fig. 9a +(dashed box). The amalgamation of two of the complexes, +one mirrored horizontally, yields the complex CFMU (“Fi- +bonacci Model Unit cell”) for the two-site unit cell of +the Honeycomb lattice, which can then be tessellated as +shown in Fig. 9d. In contrast to the surface code, the +detunings are not uniform in this case. The full ground +state manifold of the unit cell is shown in Fig. 9b. The +colored edges in Fig. 9c for each ground state configura- +tion establish the local mapping in Eq. (45). Note how +all string-net configurations are allowed except for single +strings terminating at a site. +Finally, let us mention that the complex for the hexag- +onal unit cell with 15 atoms in Fig. 9a can be interpreted +as the complex on a tilted square lattice (by virtually +contracting the vertical edges of the honeycomb lattice). +This complex, however, is not minimal as we know of a 12 +atom complex that realizes the Fibonacci check function +constraint on 4-valent sites. + +18 +X. +GEOMETRIC OPTIMIZATION +So far we optimized complexes only in terms of their +size (number of atoms) for a given language. As a result, +we ended up with minimal complexes that are defined +by their blockade graph B, local detunings {∆i}, and an +assignment of ports ℓ, i.e., atoms that realize the desired +language in the ground state manifold. Remember that +in a blockade graph B = (V, E) an edge e = (i, j) ∈ E +between atoms i, j ∈ V indicates that they are in blockade, +i.e., cannot be excited simultaneously. An abstract graph +that can be realized in this way by placing atoms in the +plane which are in blockade if and only if their distance +is smaller than some blockade radius rB is called a unit +disk graph, and a geometry GC that realizes a prescribed +graph as its blockade graph is a unit disk embedding +of this graph. So far, the actual geometry GC of our +minimal complexes was only taken into account insofar +as a unit disk embedding of the required blockade graph +B must exist. (Note that there are graphs that cannot be +realized as blockade graphs of planar geometries, so that +this “geometric realizability” is a non-trivial condition; +deciding whether a given graph can be realized in this +way is unfortunately NP-hard [75].) +Whenever there exists a planar geometry GC += +(ri)i∈V ∈ R2N ≡ CN that realizes a prescribed block- +ade graph, there typically exist many such geometries: +In most cases, there is a bit of “wiggle room” around a +given geometry without changing the blockade graph. In +addition, there can be geometrically distinct realizations +of the same blockage graph that cannot be continuously +deformed into each other without violating the blockade +constraints. For example: +This can lead to disconnected regions in the configuration +space CN that realize a given blockade graph. +To optimize the geometry of a complex in CN, we have +to quantify what we mean by a “good” complex. To this +end, we define an objective function Γ : CN → R that +quantifies the quality of the complex and that we seek to +minimize. One example is +˜Γ(GC) = δE +∆E +(48) +where δE and ∆E are the width of the ground state +manifold and the gap (recall Fig. 2). The problem with +Eq. (48) is that its evaluation scales exponentially with the +number of atoms N because the computation of δE and +∆E in principle requires access to the complete spectrum +of Eq. (1) (which is in general an NP-hard problem [26]). +While this is feasible for small complexes, it becomes +quickly a bottleneck as ˜Γ must be evaluated repeatedly +when iteratively optimizing a geometry. Furthermore, in +the PXP approximation, interaction energies are either +infinite or zero so that ˜Γ vanishes whenever the blockade +constraints are satisfied. Thus we need a simpler, heuristic +quantity that can be directly computed from the geometry +of the complex. +A. +Geometric robustness +To motivate the quantity we propose as objective func- +tion below, we first have to review the role of the blockade +radius rB in the PXP model. In the limit of vanishing driv- +ing, the blockade radius rB is the distance from an atom +where the van der Waals interaction matches its detuning: +C6 rB +−6 +i +!= ∆i. As the detunings can vary from atom to +atom in a generic structure C, so does the blockade radius +rBi (this dependence is quite weak, though). However, +as outlined in Section III, we would like to work in the +approximate framework of the PXP model with a unique +blockade radius rB, because then the effects of interactions +between atoms simplify to kinematic constraints encoded +in a blockade graph. In the following, we interpret a given +blockade graph B as the encoding of the constraints we +would like to realize with a structure C of yet unknown +geometry GC. +We can now introduce two dimensionless quantities. +First, the robustness of a structure w.r.t. a given blockade +graph B = (V, E) is defined as +ξB(C) := +min +(i,j)/∈E d(ri, rj) − max +(i,j)∈E d(ri, rj) +min +(i,j)/∈E d(ri, rj) + max +(i,j)∈E d(ri, rj) , +(49) +where d(ri, rj) denotes the Euclidean distance. The ro- +bustness is a scale-invariant, finite number ξB(C) ∈ [−1, 1] +where ξB(C) > 0 indicates a valid unit disk embedding GC +that realizes the prescribed blockade graph B for block- +ade radii in some finite interval. Larger positive values of +ξB(C) indicate more robust embeddings with more “wiggle +room” around the positions without changing the block- +ade graph, or, equivalently, a wider range of blockade +radii that yield the same blockade graph. If ξB(C) < 0, +the unit disk graph induced by GC does not match the +prescribed blockade graph B. +Similarly, the spread of a structure C is defined as +s(C) := maxi rBi − mini rBi +maxi rBi + mini rBi += (maxi ∆i)1/6 − (mini ∆i)1/6 +(maxi ∆i)1/6 + (mini ∆i)1/6 . +(50) +The spread s ∈ [0, 1] quantifies the relative variations +in blockade radii of a structure (a system with uniform +detuning ∆i ≡ ∆ has vanishing spread). Just as Eq. (49) +does not depend on the length scale, Eq. (50) is inde- +pendent of the C6 coefficient, i.e., the strength of the +interaction. + +19 +(a) +ξ(Copt +NOR△) = 0.268 +ξ(CNOR△) = 0.088 +s(C(opt) +NOR△ ) = 0.058 +ξ(Copt +NOR◦) = 0.236 +ξ(CNOR◦) = 0.111 +s(C(opt) +NOR◦ ) = 0.058 +(b) +ξ(Copt +SCU ) = 0.133 +ξ(CSCU) = 0.117 +s(C(opt) +SCU +) = 0.058 +Figure 10. +Optimization (Examples). +(a) Comparison of +perturbed (black) and optimized (red) geometries for the two +minimal NOR-complexes. Maximum distance blockades are +highlighted yellow, minimum distances of unblocked atoms +are indicated by dashed blue edges. The optimal geometries +are highly symmetric and match the manually constructed +ones in Fig. 6 and Fig. 5c. The robustness for each complex is +printed below the geometries and the spread on the bottom +of each column (we omit blockade graph indices). Note that +ξ(Copt +NOR△) > ξ(Copt +NOR◦) which makes the triangular version NOR△ +potentially more robust than the ring-shaped NOR◦. For all +geometries the validity constraint s(C) < ξ(C) is satisfied. (b) +Comparison of the optimized geometry for the vertex complex +CSCU of the surface code (red) and the manually constructed +geometry (black) from Fig. 8a; the robustness increases by +∆ξ = 0.016. Due to unconstrained atoms, the optimization can +break the symmetry and produce slightly skewed geometries. +We can now take into account the variability of the +blockade radius without abandoning the PXP model as +follows. We call a structure C a valid implementation of +a blockade graph B if +s(C) < ξB(C) . +(51) +This condition ensures that the geometry GC can be scaled +such that all distances of atoms that should (not) be in +blockade according to B, are smaller (larger) than the +smallest (largest) blockade radius of the structure C. As +this condition is scale-invariant, we do not have to specify +rB in the following. Note that all structures presented in +this paper are valid in the sense of Eq. (51). +B. +Numerical optimization +These considerations suggest the robustness ξB as a +measure for the quality of geometries. We therefore set +Γ = −ξB to maximize this quantity by minimizing Γ. +The blockade graph B and the detunings {∆i} are fixed +and define the functional properties of the complex; in +particular, the spread s(C) is constant. Thus we optimize +for geometries that satisfy the validity constraint (51) +with a maximal margin between robustness and spread. +We call a complex C globally (locally) optimal if ξB(C) > +0 and its geometry is a global (local) minimum of Γ in +CN. To minimize Γ on the high-dimensional space CN, +we employ the SciPy implementation [76] of generalized +simulated annealing [77, 78] in combination with a local +optimization based on the Nelder-Mead algorithm [79, 80], +see Appendix D for details. Remember that the robustness +is a scale-invariant quantity, so that the scale of the +optimized geometry is arbitrary. For normalization, we +rescale the geometries by setting the blockade radius +rB := 1 +2 +� +max +(i,j)∈E d(ri, rj) + min +(i,j)/∈E d(ri, rj) +� +!= 1 . +(52) +First, we initialized the algorithm with the hand-crafted +geometries of all primitives in Sections VII and VIII and +the vertex complexes in Section IX to optimize their ro- +bustness (we believe the results to be globally optimal but +we did not prove this). With these initial configurations, +the optimizer already started with a valid unit disk embed- +ding of B (ξB > 0) and tried to maximize the robustness +further. The results were typically only slightly deformed +versions of the manually constructed complexes, confirm- +ing our intuition. Some of the primitives (in particular the +ring-shaped NOR-complex in Fig. 5c) were already optimal +due of their high symmetry. In Fig. 10a we demonstrate +this by comparing slightly perturbed geometries (black) +to the subsequently optimized versions (red) for both +minimal realizations of the NOR-complex. In particular, +we find +ξBNOR△(Copt +NOR△) = 0.268 > 0.236 = ξBNOR◦(Copt +NOR◦) +(53) +and conclude that the triangular version NOR△ (Fig. 6) +is potentially more robust than the ring-shaped NOR◦ +(Fig. 5c). For both, the validity constraint (51) is safely +satisfied (x ∈ {◦, △}): +s(C(opt) +NORx ) = 0.058 < ξBNORx(Copt +NORx) . +(54) +Since the robustness depends only on the maximum (min- +imum) distance of atoms that are (not) in blockade, there +can be atoms with positions that are unconstrained in +small regions of the plane. These positions can be chosen +by the optimization algorithm at will, leading to slightly +skewed geometries that break the natural symmetry of +the complex; an example is given by the optimized surface +code unit cell complex in Fig. 10b. This is an artifact of +our particular objective function that can be eliminated +by more sophisticated choices for Γ (e.g. motivated by +specific experimental requirements). All optimized com- +plexes are accessible online [81], normalized according to +Eq. (52). +In a second run, we went one step further and initial- +ized the optimization with geometries that violated the +prescribed blockade graphs (by placing the atoms ran- +domly). In this case, the algorithm started with ξB < 0 +and first had to identify valid unit disk embeddings by +stochastic jumps in the configuration space. These runs + +20 +typically rediscovered the geometries we already knew. +In some cases, alternative geometries were found (which +turned out to be local maxima of robustness, though). +We conclude that it is not only possible to optimize given +geometries but also to find them (if they exist), at least +for small complexes. +As a final remark, we stress that geometric optimization +is in general not reducible, i.e., optimizing the primitives +of a larger circuit does not necessarily optimize the whole +circuit as constraints between primitives are not taken +into account by this approach. This is particularly im- +portant for tessellated complexes of quantum phases like +the spin liquids in Section IX, where one should optimize +the complete tessellation to minimize unwanted residual +interactions that are not present in the optimization of a +single-site or unit cell complex. +XI. +OUTLOOK +We conclude with a few comments on open questions +and directions for future research. +Minimality. +To find and prove the minimality of +complexes we systematically excluded realizations with +fewer atoms. While this approach is more efficient than +a brute force search (by exploiting constraints from the +language, the detunings, and the planar geometry), it +is still far from trivial and cannot be easily automated. +It would be both interesting and useful to develop an +algorithm that, given a uniform language, constructs a +minimal graph with weighted nodes, and a labeled node +for each letter position of the language, such that each +maximum-weight independent set [82] is in one-to-one +correspondence with a word of the language. We are +neither aware of such an algorithm nor of statements on +the complexity to find minimal solutions. (Note that a +solution of this problem might not even be a unit disk +graph, i.e., realizable by the blockade graph of a planar +Rydberg complex.) +Optimization. +It is clear that our treatment of op- +timization in Section X only scratches the surface. First, +our choice of the objective function Γ is heuristic and +other functions may be more appropriate for specific ex- +perimental settings. This would change the “optimal” +geometries of complexes, of course. Second, there is a +plethora of alternative numerical algorithms available that +could be used to minimize the objective function more ef- +ficiently. In particular the existence of distinct geometries +that are separated by complexes that violate the block- +ade graph may require more sophisticated algorithms to +escape locally optimal configurations and find the global +optimum. The algorithms also should scale well with +the size of the complex because, as mentioned previously, +tessellations should be optimized as a whole to take into +account constraints between its primitives. +If we go one step further and ask for an algorithm that +constructs geometries from a given blockade graph, we +quickly enter complexity hell: Deciding whether a given +blockade graph can be realized as a unit disk graph is +known to be NP-hard [75]. Even if we are promised to +be given a unit disk graph as blockade graph, there is +no efficient algorithm that outputs the geometry of a +complex that realizes it. This is so because there are +unit disk graphs that require exponentially many bits to +specify the positions of the nodes [83]. To add insult to +injury, even finding certain approximations of unit disk +graph embeddings are known to be NP-hard [84]. None +of these statements prevent us from looking for heuristic +algorithms to solve these problems for specific cases, of +course (as we demonstrated in Section X). +Uniformity. +Most of the complexes discussed in this +paper make use of atom-specific detunings (e.g. Fig. 6 +and Fig. 9d). Only the surface code tessellation in Fig. 8d +is uniform in detunings, at least in the bulk. While it is +possible to realize atom-specific detunings [45, 46], single- +site addressability adds significant experimental overhead. +Thus it is reasonable to ask whether complexes with +non-uniform detunings can be replaced by (potentially +larger) complexes with uniform detunings (without adding +additional degrees of freedom). For instance, there is a +third minimal NOR-complex with uniform detuning ∆i ≡ +∆. However, in amalgamated circuits this uniformity is +often destroyed—on the contrary, it is the non-uniformity +of the XNOR-complex (Fig. 6) that made the bulk of the +surface code uniform (Fig. 8d). The quest for uniformity is +therefore best formulated on the level of complete circuits +or tessellations. +Beyond planarity. +We focused completely on planar +Rydberg complexes to comply with the restrictions of +current experimental platforms: For the addressability of +single atoms it is simply convenient to have a dimension +of unimpeded access. However, technologically, three- +dimensional structures of Rydberg atoms are possible and +have been experimentally demonstrated [5, 31]. Releasing +the planarity constraint drastically changes the rules for +the construction of Rydberg complexes. For instance, +ports that are located inside a 2D complex (and would +require expensive crossings to be routed to the perimeter) +can be directly accessed from the third dimension, possibly +simplifying certain functional primitives. Note, however, +that at least the logic primitives in Fig. 6 do not profit +from a third dimension. (This follows from the proofs in +Appendix A.) +Beyond the PXP approximation. +Our construc- +tion of Rydberg complexes was based on the assumption +that atoms within the blockade radius can never be simul- +taneously excited, while atoms separated by more than +the blockade radius do not interact at all; this “PXP +approximation” implements the dynamical effect of the + +21 +interactions as a kinematic constraint. In reality, how- +ever, the atoms interact via the van der Waals interaction +UvdW = C6 r−6 which contributes also beyond the block- +ade radius, can lift the degeneracy δE of the ground state +manifold, and reduce the gap ∆E that separates it from +excited states. +One therefore expects that complexes +with δE ≈ 0 in the vdW model are geometrically more +constrained than in the PXP model. This has an effect +on the geometrical optimization of complexes (see above) +and the appropriate choice of the objective function: To +take into account residual interactions properly, heuristic +functions like the robustness should be replaced by realis- +tic functions like Eq. (48), at least for small complexes +where they can be computed exactly. +We checked that the three primitives in Fig. 5 can +be realized with perfect degeneracy δE = 0 and gap +∆E > 0 in the vdW model by small adjustments of the +detunings to balance residual interactions. In principle, a +NOR-complex can even be realized with only three atoms, +arranged in a triangle with precisely defined shape. This +is possible, because the two ancillas in Fig. 5c were only +necessary to balance the energies of states with one and +two input ports active; in the vdW model, the same can +be achieved by exploiting the residual interaction between +the two input ports. Which version of the NOR-complex +is more useful for implementations is an open question. +Quantum phase diagrams. +In this paper, we only +studied the ground state manifold of the Hamiltonian +(1) without quantum fluctuations (Ωi = 0). As has been +demonstrated in Refs. [33, 34], the interplay of quantum +fluctuations (Ωi > 0) and the strong blockade interac- +tions can give rise to interesting many-body quantum +phases at zero temperature. Thus it seems natural to +explore the quantum phase diagrams of the proposed spin- +liquid tessellations in Section IX, for example numerically +using density matrix renormalization group (DMRG) tech- +niques. Analytically, one could derive the effective Hamil- +tonians on the constructed low-energy manifolds for finite +but small Rabi frequencies Ωi ≪ ∆E in perturbation +theory [85]. Note that in general one expects the relative +strengths of the effective terms to depend on the specific +complex used to implement the local constraints. This +raises the subsequent question whether these couplings +can be tuned by modifications of the used complexes. +Dynamical preparation. +In recent experiments +[36], dynamical preparation schemes have been used to +prepare long-range entangled many-body states out-of- +equilibrium [37]. The idea is to use “quasiadiabatic” pro- +tocols Ωi(t) and ∆i(t) where the detuning increases con- +tinuously to its target value while a finite Rabi frequency +ensures the coupling of different excitations patterns. This +allows for the preparation of non-trivial superpositions of +states in the low-energy subspace of the classical Hamilto- +nian (1). It would be interesting to explore the states of +the proposed tessellations that can be prepared by such +dynamical protocols numerically, and study the effects of +defects in the intended logic of the complexes due to local +excitations. Similar questions arise for the primitives in +Sections VII and VIII and circuits built from these by +amalgamation. +XII. +SUMMARY +In this paper, we developed a framework to design +planar structures of atoms which can be excited into Ryd- +berg states under the constraint of the Rydberg blockade +mechanism (“Rydberg complexes”). Our framework tar- +gets the preparation of degenerate ground state manifolds +that are characterized locally by arbitrary Boolean con- +straints. We proved that the truth table of an arbitrary +Boolean function can be realized as ground state manifold +by decomposing its circuit representation into three prim- +itives that leverage the Rydberg blockade. Motivated by +this existence claim, we then presented provably minimal +complexes that realize the most important primitives of +Boolean circuits, including a crossing complex that is +needed to embed non-planar circuits into the plane. As +an application of our framework, we constructed periodic +Rydberg complexes with degenerate ground state mani- +folds that map locally on the non-factorizable string-net +Hilbert spaces of the surface code (with abelian topolog- +ical order) and the Fibonacci model (with non-abelian +topological order). In combination with quantum fluc- +tuations, these structures may be the starting point to +prepare topologically ordered states in upcoming quantum +simulators. We concluded the paper with a discussion of +the geometric optimization of Rydberg complexes using +numerical algorithms to increase their robustness against +geometric imperfections and the effects of long-range van +der Waals interactions. +Our results highlight the versatility of planar struc- +tures of atoms that interact via the Rydberg blockade +mechanism. We provide a conceptual foundation for the +rationales of geometric programming, the encoding and +solution of problems by tailoring the geometry of atomic +systems, and synthetic quantum matter, the goal-driven +design of quantum materials on the atomic level. Due to +the noisiness of near-term experimental platforms, the lat- +ter seems particularly promising because quantum phases +come with an inherent robustness against a finite den- +sity of excitations. 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The crucial point is that irreducible languages can +only be implemented by complexes with connected blockade graphs. +Second, because we are only interested in GSM of PXP models, all detunings can be assumed to be strictly positive, +∆i > 0. Indeed, atoms with negative detuning cannot be excited in the GSM so that they can be deleted from the +complex without changing the GSM (and without closing the gap). The argument against atoms with vanishing +detuning is more subtle. If such an atom is not excited in any of the GSM states, it can be deleted without changing +the GSM. If it is excited in some of the GSM states, there is always an otherwise identical state in the GSM where it +is not excited. Such an atom therefore must be a port because as an ancilla it would add internal degrees of freedom +that are not accessible via the ports (this follows from our definition of a complex). The language that corresponds to +a complex with a zero-detuning port therefore has the property that for every word with a “1” at the corresponding +position, there must be an otherwise identical word with a “0”. (This does not imply that the language is reducible; +for example, L = {111, 011, 000} has this property for the first letter but is irreducible.) While such languages do +exist, they cannot be truth tables of Boolean functions because such a port cannot be used as an input or an output +(assuming we forbid “dummy” inputs that have no effect on the output). All languages discussed and implemented in +this paper (also the ones for the vertex complexes of spin liquids) do not have this property, hence we can assume +non-vanishing detunings. +Because of the positivity of all detunings, ground states are always given by maximal independent sets (MIS*) +of the blockade graph. [A maximal independent set is a subset of vertices such that (1) no two vertices of the set +are connected by an edge of the graph and (2) no vertex can be added to the set without violating (1). Maximum +independent sets (MIS) are the largest maximal independent sets.] The inverse is not necessarily true: Depending on +the detunings, not every MIS* describes a ground state configuration (an example is the ring-like NOR-complex). +1. +CPY-complex +Lemma 1. A CPY-complex cannot be realized with less than 4 atoms (1 ancilla). +Proof. Assume there is a complex without ancillas described by +H = −∆1n1 − ∆2n2 − ∆3n3 =: En1n2n3 . +(A1) +Since (n1n2n3) = (111) must be a ground state of the complex, none of the pairs of the atoms can be in blockade so +that there is no kinematic constraint on the configurations (n1n2n3). To be a CPY-complex, it must be +−(∆1 + ∆2 + ∆3) = E111 +!= E000 = 0 +and +En1n2n3 > 0 +for all +(n1n2n3) ̸= (000), (111) . +(A2) +The finite-gap condition requires in particular ∆i > 0 for all i = 1, 2, 3 which leads to −(∆1 + ∆2 + ∆3) < 0 and +thereby contradicts the degeneracy condition. +Alternative argument: The copy language LCPY = {000, 111} is irreducible. Since (111) must be in the GSM, the +only admissible blockade graph is the trivial graph on three vertices without edges: B = (V = {1, 2, 3}, E = ∅). But a +disconnected blockade graph cannot implement an irreducible language. +■ +2. +NOR-complex +Lemma 2. A NOR-complex cannot be realized with less than 5 atoms (2 ancillas). +Proof. We show that a NOR-complex cannot be realized with one ancilla or less. First, assume there is no ancilla so +that the Hamiltonian is again +H = −∆1n1 − ∆2n2 − ∆3n3 =: En1n2n3 , +(A3) + +26 +now with potential kinematic constraints due to the Rydberg blockade. The conditions for a NOR-complex demand the +equality of the following energies: +E001 = −∆3 +(A4a) +E010 = −∆2 +(A4b) +E100 = −∆1 +(A4c) +E110 = −∆1 − ∆2 . +(A4d) +It follows immediately ∆1 = ∆2 = ∆3 and ∆1 = 0 so that all detunings must vanish. But then (n1n2n3) = (000) +is—independent of the configuration and its implied kinematic constraints—degenerate with the four states that belong +to the NOR-manifold (which it must not be). +Alternative argument: The NOR-language LNOR = {001, 010, 100, 110} is irreducible and forbids a blockade between +the two input ports [because of (110)]. The only consistent blockade graph B is therefore the line graph of three +vertices. But this graph has only two maximal independent sets, whereas we need at least four to realize LNOR. +So let us assume a system with one additional ancilla, +H = −∆1n1 − ∆2n2 − ∆3n3 − ∆4˜n4 , +(A5) +and an arbitrary geometry that may lead to kinematic constraints on the allowed configurations. Let now ε(n1n2n3) +denote the minimal energy of the system without the contribution from the ports under the “boundary condition” that +these are in the state (n1n2n3) and under the kinematic constraints imposed by the Rydberg blockade; furthermore, set +En1n2n3 := −∆1n1−∆2n2−∆3n3+ε(n1n2n3). In the current situation with only one ancilla, it is either ε(n1n2n3) = 0 +if the minimum is obtained by ˜n4 = 0, or ε(n1n2n3) = −∆4 if ˜n4 = 1 minimizes the energy (and this is consistent with +the configuration (n1n2n3)). With this notation, the conditions to be a NOR-complex take the following form. First, +the degeneracy of the NOR-manifold demands the equivalence of the following expressions: +E001 = −∆3 + ε(001) +(A6a) +E010 = −∆2 + ε(010) +(A6b) +E100 = −∆1 + ε(100) +(A6c) +E110 = −∆1 − ∆2 + ε(110) , +(A6d) +which immediately implies +∆1 = ε(110) − ε(010) +(A7a) +∆2 = ε(110) − ε(100) +(A7b) +∆3 = ε(110) + ε(001) − ε(100) − ε(010) . +(A7c) +Second, the gap condition requires (among other conditions) +ε(000) = E000 +!> E100 = −∆1 + ε(100) = ε(010) − ε(110) + ε(100) +(A8a) +⇔ +ε(000) + ε(110) > ε(010) + ε(100) +(A8b) +because a state with (n1n2n3) = (000) is not allowed in the NOR-manifold. Note that the only kinematic constraints +on the ancilla in Eq. (A8b) can come from the two input vertices since n3 = 0 for all four terms. We show now that +Eq. (A8b) cannot be satisfied with a single ancilla. +Consider first the case where ∆4 ≤ 0. Then the minimal energy under any condition (n1n2n3) is reached by switching +the ancilla off, ˜n4 = 0 (this is possible for all kinematic constraints), so that 0 + 0 > 0 + 0 leads to a contradiction. +Thus we have to assume ∆4 > 0 (this we could have anticipated from the arguments above). Now the energy can be +lowered by switching the ancilla on, but this might be forbidden by the kinematic constraints for certain boundary +conditions (n1n2n3). We consider three cases: +(i) No blockade between the two inputs and the ancilla. In this case, the ancilla will be switched on in all four terms +of Eq. (A8b) so that −∆4 − ∆4 > −∆4 − ∆4 violates the gap condition. +(ii) The ancilla is in blockade with one of the inputs. W.l.o.g. let n1 be in blockade with ˜n4. Then Eq. (A8b) reads +−∆4 + 0 > −∆4 + 0 which again violates the gap condition. +(iii) The ancilla is in blockade with both inputs. Now Eq. (A8b) reads −∆4 + 0 > 0 + 0 which is in contradiction with +the assumption ∆4 > 0. + +27 +n1 +˜n2 +n3 +˜n4 +n5 +(n1n3n5) = (111) +(n1n3n5) = (001) +(n1n3n5) = (100) +(n1n3n5) = (000) +Figure 11. The line graph is the only connected blockade graph on five vertices with (at least) four maximal independent sets +(orange vertices), (at least) one of which has (at least) three vertices. To realize the state (111), the vertices {1, 3, 5} must be +chosen as ports, with 3 as output; the four maximal independent sets then realize the truth table of AND (these four states +cannot be made degenerate while maintaining a gap, see text). +In conclusion, we showed that it is impossible to satisfy the gap condition with a single ancilla. +Alternative argument: Of the six connected graphs on four vertices, only the “tetrahedron graph” has four maximal +independent sets (the others have at most three), which is necessary to realize the four words in LNOR. But none of +these four maximal independent sets contain more than one vertex [which would be necessary for (110)]. +■ +3. +AND-complex, OR-complex and XNOR-complex +Lemma 3. AND-, OR- and XNOR-complexes cannot be realized with less then 6 atoms (3 ancillas). +Proof. All these complexes contain the state (111) such that no two ports can be in blockade with each other. This +implies that no realization of these gates is possible with four or less atoms as the only connected blockade graph +which fulfills this constraint is the star graph of the CPY-complex (which has only two MIS*). +The number of vertices is still small enough to systematically screen the 21 connected graphs on five vertices and +select the 11 relevant ones with at least four maximal independent sets. One can check that only the chain graph has +a MIS* with (at least) three vertices, which is needed to realize the port configuration (111) (Fig. 11). This MIS* +contains the vertices {1, 3, 5} of the chain, which we therefore must choose as ports: (n1n3n5) = (111). With these +ports, the set of four MIS* then realizes the language L = {111, 100, 001, 000} which we identify as the truth table of +the AND-gate if we choose the port on the central atom 3 as output. This proves that the OR- and XNOR-complex cannot +be realized with five atoms (even if another port is declared as output). +So far the arguments were purely kinematic insofar as only the blockade constraints and the knowledge that the +GSM is generate by maximal independent sets were used. To exclude the AND-gate, this is not enough, and we have to +use energetic arguments by studying possible choices for detunings. The degeneracy of the GSM requires the following +four expressions to be equal: +E111 = −∆1 − ∆3 − ∆5 +(A9a) +E100 = −∆1 − ∆4 +(A9b) +E001 = −∆2 − ∆5 +(A9c) +E000 = −∆2 − ∆4 , +(A9d) +which immediately implies ∆4 = ∆5 and therefore ∆3 = 0, which is not allowed (remember that vanishing detunings +are forbidden). This proves that also the AND-complex cannot be realized with five atoms. +■ +4. +NAND-complex and XOR-complex +Lemma 4. NAND- and XOR-complexes cannot be realized with less than 7 atoms (4 ancillas). +Proof. The truth tables of both NAND and XOR contain the states (110), (101) and (011) so that no two ports can be in +blockade with each other. This excludes a realization with less than four atoms (see Appendix A 3). If two ancillas are +available, we can switch one of the input ports on; this switches (at least) one ancilla off. The remaining two ports and +(at most) one ancilla then must realize the NOT-language L¬ = {01, 10}. This is impossible since the two ports cannot +be directly connected and the only blockade graph with a single ancilla realizes the LNK-language LLNK = {00, 11}. So +let us assume that the complexes can be realized with three ports and three ancillas. For the following arguments, +only the edges between ports and ancillas are of importance; potential blockades between ancillas can be ignored. We +consider three cases: +(i) There is at least one port that connects to all three ancillas. If this port is on, all ancillas are off, hence the two +remaining ports must be on as well; but then at least two of the three states (110), (101) and (011) cannot be +realized in the GSM. + +28 +∆1 +˜∆1 +∆2 +˜∆3 +∆3 +˜∆2 +(a) +(b) +(c) +Figure 12. The three bipartite graphs between three ports (red) and three ancillas (blue) where all ports have degree two. Note +that these do not represent complete blockade graphs as we omit blockades between ancillas. The detunings in (c) are used in +Appendix A 5. +(ii) There is at least one port that connects to a single ancilla. This edge can be interpreted as an amalgamated +NOT-complex. If we delete the port, subtract its detuning from the connected ancilla, and declare the latter as a +new port, the new complex of five atoms realizes the truth table of the original complex with one column inverted +(w.l.o.g. the first one). For both gates, this new manifold contains the states (010), (001) and (111) (plus another +one that depends on the gate). The only blockade graph on five vertices with at least four MIS*, one of which +contains at least three vertices [needed for (111)], has been identified in Appendix A 3 as the line graph. There it +has also been shown that there is no assignment of detunings that realizes a four-fold degenerate GSM. +(iii) All inputs are connected with exactly two ancillas. There are three possibilities to connect three ports with +two ancillas each (Fig. 12). By inspection one shows that in all three cases there is a pair of ports that, when +activated, forces all ancillas connected to the third port to be off; as this forces the third port to be on, at least +one of the states (110), (101) and (011) cannot be realized in the GSM. +This proves that the NAND- and XOR-complex cannot be realized with six atoms. +■ +Note: Removing a NOT-complex by deleting the port, subtracting its detuning from its ancilla, and declaring +the ancilla as new port, is the inverse of amalgamation; let us call it amputation. One has to make sure that the +subtraction of the detuning of the port ∆p from the detuning of its adjacent ancilla ∆a does not lead to negative (or +vanishing) detunings on the ancilla (= new port). Indeed, if ∆p > ∆a, the port would be always on in all ground state +configurations; this makes the port superfluous and the language of the GSM reducible. If ∆p = ∆a, the language of +the original complex would have the property that for every word with a “0” at the corresponding position, there is a +otherwise identical word with a “1”. This is the dual property of the one discussed at the beginning of Appendix A +and no language discussed in this paper has this property. +5. +Uniqueness of the blockade graph of the minimal XNOR-complex +In contrast to the minimal NOR-complexes (for which there are different blockade graph realizations), there is only +one realization of the minimal XNOR-complex. This will be useful in Appendix C 2 to prove the minimality of the vertex +complex of the Fibonacci model. +Lemma 5. The blockade graph of the minimal XNOR-complex with 6 atoms (Fig. 6) is unique. +Proof. We showed in Appendix A 3 that a XNOR-complex needs at least six atoms; so let us assume we have six atoms +at our disposal. We now try to contrive a complex that realizes the language L⊙ = {001, 010, 100, 111} systematically: +(i) Assume there exists such a complex with at least one port that connect to only one ancilla. If this port is +amputated, the remaining 5 atoms realize the XOR-language L⊙ = {101, 110, 000, 011}, which is impossible as +shown in Appendix A 4. +(ii) Assume at least one port connects to all three ancillas. If this port is switched on, all ancillas are switched off and +therefore the other two ports must be active. This is inconsistent with one of the states (001), (010) and (100). +(iii) Because of (i) and (ii), only the case where all ports connect to two ancillas remains. There are three classes of +blockade graphs that satisfy this, Fig. 12. The first two graphs in Fig. 12 can be immediately excluded as they +are inconsistent with the states (001), (010) and (100) (= only one port activated). Only the “hexagon graph” in + +29 +Fig. 12 remains as a possible blockade structure between ports and ancillas. Without additional blockades between +the ancillas, the maximal independent sets of this graph allow for the states {000, 001, 010, 100, 111} ⊃ L⊙. +Let ∆1,2,3 denote the detunings of the three ports and ˜∆1,2,3 the detunings of the three ancillas (where ˜∆i +describes the ancilla opposite of port i, Fig. 12). In the state (100), only the first port is excited. So the opposite +ancilla must be excited as well to block the two other ports (if this ancilla were off, one could lower the energy by +switching the other two ports on). To balance this state energetically with the state (111), the detuning of the +ancilla must equal the sum of the detunings of its two adjacent ports. Due to the permutation symmetry of L⊙ +and the rotation symmetry of the “hexagon graph”, this argument is valid for all three ancillas: +˜∆1 = ∆2 + ∆3 , +˜∆2 = ∆1 + ∆3 , +and +˜∆3 = ∆1 + ∆2 . +(A10) +Because all detunings must be positive, this implies for any pair of ancillas +− ˜∆i − ˜∆j < −∆1 − ∆2 − ∆3 = E111 . +(A11) +Since (111) must be in the GSM (i.e., E111 must be the lowest allowed energy), there must be an additional +blockade between all pairs of ancillas to prevent them from being excited simultaneously. This yields the blockade +graph of the XNOR-complex depicted in Fig. 6. It has only four maximal independent sets that realize the language +L⊙ = {001, 010, 100, 111}. The choices of the port detunings ∆i > 0 are arbitrary; the ancilla detunings are then +given by Eq. (A10). +We conclude that the blockade graph of the minimal realization of a XNOR-complex with six atoms is unique (there is +only freedom in choosing the detunings). In addition, we proved that no strict superset of L⊙ can be realized by a +complex with six atoms or less (this is used in Appendix C 2). +■ +Appendix B: Constructing subcomplexes +Here we discuss a method to construct subcomplexes by fixing a port in the active state and deleting its adjacent +ancillas in the blockade graph. This method is used in the proofs of Appendix C and the final remark of Section VI. +Consider a complex C that realizes a language L with ports that are not in blockade with each other. We can select one +of the ports p and define the sublanguage Lp ⊂ L of words x ∈ L with xp = 1. Our goal is to construct a L′ +p-complex +C′ +p where L′ +p is obtained from Lp by deleting the constant letter at position p that corresponds to the fixed port. The +simplest solution is to keep the geometry of the complex C and increase the detuning of the fixed port ∆p, thereby +creating a gap between states of the original GSM where the port is on and states where it is off; the port can then be +downgraded to an ancilla. In all states of the new GSM this ancilla is active, while its adjacent ancillas are inactive. +This suggests that one can delete these atoms to obtain a smaller complex C′ +p that realizes the same language L′ +p: +Lemma 6. Let the finite complex C realize the irreducible language L with ports that are not in blockade with each +other (with δE = 0 and ���E > 0). Consider one of the ports p with detuning ∆p > 0 and let the languages Lp and L′ +p +be defined as above. Then the structure C′ +p obtained from C by deleting the port p and all its adjacent ancillas is a +L′ +p-complex if the ports of C′ +p are inherited from C in the natural way. +Proof. First, note that since L is irreducible, it is Lp ̸= ∅, i.e., there are configurations in the GSM of C where the port +p is active. We have to show two things: (a) the structure C′ +p together with the inherited ports is a complex (i.e., its +ground states can be labeled by the configurations of the ports), and (b) the language that describes this GSM is L′ +p. +Let the GSM of the new structure C′ +p be defined by δE = 0 (since the structure is finite, it is automatically ∆E > 0). +Every kinetically allowed (= admissible) configuration in this GSM can be extended to an admissible configuration +of C by setting the deleted ancillas to off and the port p to on. If E0(C′ +p) denotes the ground state energy of C′ +p +and E0(C) the same for C, this implies that E0(C) ≤ E0(C′ +p) − ∆p. Conversely, because Lp ̸= ∅, there are admissible +configurations in the GSM of C where the port p is on and, consequently, all adjacent ancillas are off. By truncating +the configurations of the adjacent ancillas and the port p, this yields a admissible configuration for C′ +p with energy +E0(C) + ∆p so that E0(C′ +p) ≤ E0(C) + ∆p. In combination, we have +E0(C′ +p) = E0(C) + ∆p +(B1) +for the ground state energy of the new structure C′ +p. Using this result and the mappings of extension and truncation, +we can draw two conclusions: +(1) Every configuration in the GSM of C′ +p can be extended to a configuration in the GSM of C which corresponds to a +word in Lp. We can immediately conclude two things: + +30 +(i) Since the extended configurations must be distinguishable by the ports of the complex C ignoring port p (this +port is always on for configurations in Lp), and because these ports are inherited by the structure C′ +p, we can +conclude that the configurations of the GSM of C′ +p can also be distinguished by these ports. This makes C′ +p a +complex that realizes some language L?. +(ii) Every word in L? is mapped by the extension to a word in Lp which implies L? ⊆ L′ +p. +(2) Conversely, every configuration in the GSM of C which corresponds to a word in Lp can be truncated to an +admissible configuration of C′ +p with energy E0(C) + ∆p = E0(C′ +p), which implies L′ +p ⊆ L?. +In conclusion, we showed that L? = L′ +p and therefore that C′ +p is indeed a L′ +p-complex. +■ +Appendix C: Minimality of spin liquid primitives +1. +Vertex/Unit cell complex for the surface code (CSCU) +Lemma 7. The vertex complex (unit cell complex) CSCU of the surface code on the square lattice cannot be realized +with less than 11 atoms. +Proof. Here we show that the vertex complex of the surface code on the square lattice requires at least 11 atoms; to +this end, we use and expand on the tricks introduced in Appendix A 4. First, note that the GSM is symmetric under +the permutation of ports (Fig. 8b) and includes the state (1111), i.e., no two ports can be in blockade with each other. +In addition, the GSM is symmetric under the simultaneous inversion of an even number of letters in all words (= +columns): +LSCU ≡ +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +1111 +1100 +0011 +1001 +0110 +0101 +1010 +0000 +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +inv. 4. letter +−−−−−−−−→ LSCU ≡ +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +1110 +1101 +0010 +1000 +0111 +0100 +1011 +0001 +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +inv. 3. letter +−−−−−−−−→ LSCU = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +1100 +1111 +0000 +1010 +0101 +0110 +1001 +0011 +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +inv. 2. letter +−−−−−−−−→ LSCU = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +1000 +1011 +0100 +1110 +0001 +0010 +1101 +0111 +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +· · · (C1) +Let us now systematically exclude the existence of surface code complexes with N ≤ 10 atoms: +• N < 8: If one fixes one port of a surface code complex as active, the remaining three ports realize a XNOR-complex +with at least two atoms less than the surface code complex (because the active port deactivates at least one +ancilla permanently). Since we proved in Appendix A 3 that XNOR-complexes require at least six atoms, this +implies immediately that the surface code complex cannot be realized with N < 8 atoms. +• N = 8: If there are at least two ports that are connected to only one ancilla each, we can consider these as +amalgamated NOT-complexes and amputate two of them (see the note in Appendix A 4), thereby creating a +complex with only six atoms that realizes the same GSM due to the inversion symmetry detailed in Eq. (C1). +Since this is not possible, there can be at most one port that connects to only one ancilla. Choose one of the +other ports that connect to at least two ancillas and again fix it in the active state (here we use the permutation +symmetry of LSCU). This produces a complex with at most five atoms (the fixed port plus at least two ancillas +are removed from the surface code complex) that realizes again the XNOR-manifold, which is impossible. Hence +the surface code complex cannot be realized with eight atoms. +• N = 9: To show that the complex cannot be realized with nine atoms we consider three cases: +(i) Assume there is at least one port connected to three or more ancillas. If this port is fixed as active, it blocks +at least three ancillas. The resulting XNOR-complex on the three remaining ports has at most five atoms, +which is impossible. +(ii) Assume at least one port connects to a single ancilla. Amputating this port yields a LSCU-complex with +eight atoms. If an arbitrary port of this complex is fixed as active, the resulting complex has at most six +atoms. Inspection of the language LSCU [Eq. (C1)] shows that this complex realizes the truth table of a +XOR-gate, which, however, requires at least seven atoms (as shown in Appendix A 4). + +31 +(a) +(b) +(c) +1 +2 +3 +4 +5 +6 +7 +8 +(d) +(e) +(f) +Figure 13. The remaining six classes of blockade graphs for the surface code vertex complex on nine atoms with ports of degree +2 and without disconnected ancillas. Ports (ancillas) are colored red (blue) and connections between ancillas are omitted. The +only class that cannot be excluded kinematically is the “cross” graph (c) with atom labels {1, . . . , 8} and ports {1, 2, 3, 4}, see +text. +(iii) Because of (i) and (ii), only the case that all ports connect to exactly two ancillas remains. There are six +non-isomorphic bipartite graphs that connect sets of four (ports) and five vertices (ancillas), where all ports +have degree 2, Fig. 13. We exclude graphs with disconnected ancillas because these are typically covered by +the analogous step for N = 8. (Above we omitted this step to simplify the prove, so in principle here one +has to check the graphs with disconnected ancillas too. The result is the same, though.) By inspection, one +shows that all these graphs (except for the “cross” in Fig. 13c) allow for a pair of ports that, when activated, +block all ancillas of a third port (which then must be switched on as well). This, however, is inconsistent +with the language LSCU which includes for all triples of ports states where two are on and one is off. +The “cross” graph in Fig. 13c cannot be excluded with this type of kinematic reasoning because the set +of maximal independent sets (with the convention of ports shown in Fig. 13c) induces a superset of LSCU. +Therefore we have to use energetic arguments instead. With the atom indices shown in Fig. 13c, the gap +condition requires +E1111 = −∆1 − ∆2 − ∆3 − ∆4 +!< −∆1 − ∆2 − ∆3 − ∆8 = E1110 +⇒ +∆8 < ∆4 , +(C2a) +E1100 = −∆1 − ∆2 − ∆7 − ∆8 +!< −∆1 − ∆2 − ∆7 − ∆4 = E1101 +⇒ +∆8 > ∆4 . +(C2b) +Hence this graph cannot realize the LSCU-manifold. +• N = 10: To show that the surface code complex cannot be realized with N = 10 atoms, one follows the same +procedure as detailed above for the case of N = 9 atoms (here we only briefly summarize the necessary steps): +First, one excludes the case with ports that connect to a single ancilla (where one has to use that a N = 9 +realization of a LSCU-complex can have only ports that connect to at least two ancillas). Then, one excludes the +existence of ports that connect to at least four ancillas by using that XNOR-complexes cannot be realized with +five atoms or less. Finally, one must exclude blockade graphs with ports of degree three or two by the same +procedure as in Step (iii) above. In this case, there are 20 graph classes to cover of which 15 can be kinematically +excluded and 5 can be energetically ruled out. These arguments show that the surface code vertex complex +cannot be realized with 10 atoms. +■ +2. +Vertex complex for the Fibonacci model (CfFib) +Lemma 8. The vertex complex CfFib of the Fibonacci model on the Honeycomb lattice cannot be realized with less +than 8 atoms. + +32 +(a) +(b) +(c) +Figure 14. The remaining three classes of blockade graphs for the Fibonacci vertex complex on seven atoms with ports of +degree 2. Ports (ancillas) are colored red (blue) and connections between ancillas are omitted. Only the graph in (b) must be +energetically excluded. +Proof. The Fibonacci language LFib = {000, 011, 110, 101, 111} contains the three states (011), (110) and (101) which +we used in Appendix A 4 to show (with purely kinematic arguments) that XOR- and NAND-complexes cannot be realized +with less than seven atoms. As the argument only relied on these three states (and the existence of at least one other +state), it extends to the Fibonacci complex, which therefore also requires at least seven atoms. Furthermore, the three +states forbid blockades between any two ports. So assume a realization with seven atoms exists. We distinguish three +cases: +(i) At least one port connects to a single ancilla. Amputation (see the note in Appendix A 4) of this port yields a +complex with six atoms that realizes the manifold LFib = {100, 111, 010, 001, 011} obtained from LFib by inverting +the first letter (note that LFib is symmetric under permutations of ports). This language contains all XNOR states: +L⊙ ⊂ LFib. In Appendix A 5 we showed that there is only one blockade graph on 6 vertices that can realize these +states; this graph has only four maximal independent sets and therefore cannot realize the additional state (011) +in LFib. +(ii) At least one port connects to at least three ancillas. First, a port that connects to all four ancillas is inconsistent +with two of the three states (011), (110) and (101). So assume there is a port that connects to three of the four +ancillas. If this port is activated, it deactivates three ancillas and the remaining two ports (together with one +ancilla) realize the irreducible language L = {01, 10, 11} (the blockade graph of these three atoms must therefore +be connected). But there is no graph on three vertices with (at least) three maximal independent sets of which +(at least) one has (at least) two vertices. +(iii) Because of (i) and (ii), only the case where all ports connect to two ancillas remains. The possible classes of +bipartite graphs are shown in Fig. 14. With a similar line of arguments as used for the surface code [Case (iii) for +N = 9 in Appendix C 1], one can exclude two of the three graphs (a and c) with kinematic arguments [using the +states (011), (110) and (101)]. The set of maximal independent sets for the graph in Fig. 14b includes LFib as a +subset and can again be excluded by energetic arguments. +■ +Appendix D: Numerical approach for geometric optimization +To minimize the objective function Γ on the high-dimensional configuration space CN, we used the SciPy method +scipy.optimize.dual annealing [76, 88, 89] that implements generalized simulated annealing [77, 78] in combination +with a local optimization based on the Nelder-Mead algorithm [79, 80]. The stochastic algorithm starts from an initial +geometry (which can be chosen randomly), followed by iterations of jumps in CN with probabilities that depend on +the distance of the jump and the variation of the objective function Γ; following each random jump, the Nelder-Mead +algorithm optimizes the new configuration locally. After ≲ 2000 iterations we stop the algorithm and compute the +robustness of the final geometry. More technical details and all obtained optimal complexes can be found in Ref. [55]; +the data of the optimized complexes can also be accessed online [81]. +