diff --git "a/FtAzT4oBgHgl3EQfi_1S/content/tmp_files/load_file.txt" "b/FtAzT4oBgHgl3EQfi_1S/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/FtAzT4oBgHgl3EQfi_1S/content/tmp_files/load_file.txt" @@ -0,0 +1,2612 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf,len=2611 +page_content='Functional completeness of planar Rydberg structures Simon Stastny,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Hans Peter B¨uchler,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' and Nicolai Lang∗ Institute for Theoretical Physics III and Center for Integrated Quantum Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' University of Stuttgart,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 70550 Stuttgart,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Germany (Dated: January 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' We introduce a versatile framework, together with a set of provably minimal logic primitives as building blocks, to implement these constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' As examples, we present lattice realizations of the string-net Hilbert spaces that underlie the surface code and the Fibonacci anyon model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' We discuss possible optimizations of planar Rydberg structures to increase their geometrical robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' The rationale of geometric programming is the solu- tion of algorithmic problems by encoding them into the geometry of the atomic array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' finding MIS is a long-known optimization problem in graph theory that has been shown to be NP- hard [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' This makes the computation of ground state ∗ nicolai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content='lang@itp3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content='uni-stuttgart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content='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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' A very different application of the Rydberg blockade mechanism is the engineering of synthetic quantum matter on the single-atom level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' The potential of this approach has been demonstrated recently by Verresen et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' [33] (related results were reported by Samajdar et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' [34]), who proposed the realization of topological spin liquids on delicately designed lattice structures of atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' First experimental results were reported shortly after [36], accompanied by a theoretical study of the used quasiadiabatic preparation scheme [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' This paper is written from and motivated by the syn- thetic quantum matter perspective, but its results apply to geometric programming as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Our starting point is the question whether other local constraints (besides the dimer constraint) can be realized on the Rydberg plat- form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' While our proof is constructive, it does typically not yield optimal (= small) solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' We therefore expand on our main result and compile a comprehensive list of provably minimal structures that realize all important primitives of Boolean logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Together with a structure that facilitates the cross- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=', the charge-free sector of the toric code [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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]”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' a system that, with added quantum fluc- tuations, could support non-abelian Fibonacci anyons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Note added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' When finalizing this manuscript we be- came aware of related results reported by Nguyen et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' al in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' The authors follow the rationale of geomet- ric programming, so that their motivation, approach and framework differ from ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' RATIONALE AND OUTLINE Here we illustrate the rationale of the paper and pro- vide a brief outline of its main results without technical overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Readers interested in the details can skip for- ward to Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Readers only interested in specific applications can read this section first and then skip to Section VII or Section IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' We focus on systems without quantum fluctuations, where the ground states are determined by local detunings and Rydberg blockade interactions (Sec- tion III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' There can be different configurations that minimize the energy, hence the ground state manifold is typically degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' A simple example is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 1a where the po- sition of the atoms is shown in (i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' the two atoms are constrained by the Rydberg blockade (gray circles) and Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Rationale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' (a) Structure of two atoms (i) with local detunings ∆ (blue vertices) that are in Rydberg blockade (gray circles);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' the blockade is indicated by a black edge connecting the atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' The ground state manifold (ii) is given by patterns of excited atoms (orange) that minimize the energy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' here it is two-fold degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' The two ground state configurations realize the truth table (iii) of a NOT-gate Q = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' (b) Structure of five atoms (i) with local detunings ∆ (blue) and 2∆ (green) in a ring-like Rydberg blockade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' The ground state manifold (ii) is four-fold degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' This construction is called amalgamation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' cannot be excited simultaneously (indicated by the black edge connecting them).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' The color of the atoms encodes their detuning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' here both atoms lower the energy of the system by ∆ when excited into the Rydberg state (blue nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' In (ii) we show the two excitation patterns that minimize the energy (orange nodes denote excited atoms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Note that the atoms cannot be excited simultaneously due to the Rydberg blockade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Here we interpret one of the atoms as “input” (A) and the other as “output” (Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' This concept generalizes to more complicated Boolean gates (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 1b): Consider the five atoms in a ring-like blockade (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Three of the atoms (blue) lower the energy by ∆, two (green) by 2∆ when excited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' By inspection one finds the four degenerate ground state configurations in (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' This is promising as truth tables of Boolean gates that operate on two bits have four rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' However, they only have three columns (two for the inputs of the gate and one for its output).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' We call atomic structures with designated input/output atoms Rydberg complexes (Section V A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' To leverage this decomposition, we need a method to combine “gate complexes” to form larger “circuit complexes”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' we call this procedure amalgamation (Sec- tion V B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' A simple example is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 1c where we attach the NOT-gate from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 1a to the output of the NOR- gate in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 1b (note that the detunings of the atoms that are joined add up).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Using the detunings and blockades in (i) yields the four degenerate ground state configurations in (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' This functional completeness is our first main result and motivates the title of the paper (Section VI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' For instance, the existence of a structure that realizes the truth table of an OR-gate is a corollary of functional completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' However, the specific construction as the combination of a NOR-gate and a NOT-gate in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' The answer to the first question is negative: There are geometrically different structures that realize the same truth table in their ground state manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Note that the func- tional completeness implies the existences of structures for all common gates of Boolean logic (such as AND, XOR, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' We take this as motivation to construct provably minimal structures for all these primitives (Sections VII and VIII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Together with the procedure of amalgama- tion, these equip our versatile toolbox to engineer more complicated structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Our second important contribution is an application of the functional completeness as a tool to engineer synthetic quantum matter (Section IX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Many interesting quantum phases in two dimension are characterized by hidden pat- terns of long-range entanglement, known as topological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' These patterns can give rise to anyonic excitations which make such systems potential substrates for quantum memories and even quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' A large class of entanglement patterns can be understood as condensates of extended objects (like strings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' However, in experiments, we typically start from Hilbert spaces with a local tensor product structure (for example, an array of two-level atoms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Our only hope is to make the extended objects emerge due to interactions in the low-energy sector of a suitably designed physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' This often boils down to enforce local gauge sym- metries which single out states that can be interpreted in terms of extended objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' It is then just a matter of copying and joining these structures in a translational invariant way to tessellate the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' The truth tables realized by the ground states of all atomic structures presented in this paper depend on the positions of the atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' (Because these positions define which pairs are in blockade and which atoms can be ex- cited simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=') However, the exact placement is often ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' For example, consider the structure in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 1a (i) which realizes the NOT-gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' For the complexes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 1, this translates into the geomet- ric objective to maximize the distances between nodes and gray circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' We conclude the paper with an outline of open ques- tions, directions for further research (Section XI), and a brief summary (Section XII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 4 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' PHYSICAL SETTING We consider planar arrangements of trapped atoms with repulsive van der Waals interactions when excited into the Rydberg state [2, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Every atom is assigned an index i ∈ V = {1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfi_1S/content/2301.01508v1.pdf'} +page_content=' Here we are mainly interested in the regime Ωi → 0 where the Hamiltonian reduces to H[C] = − � i ∆ini + � i