diff --git "a/8dAyT4oBgHgl3EQfQvbW/content/tmp_files/load_file.txt" "b/8dAyT4oBgHgl3EQfQvbW/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/8dAyT4oBgHgl3EQfQvbW/content/tmp_files/load_file.txt" @@ -0,0 +1,673 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf,len=672 +page_content='Fault-tolerant error correction for a universal non-Abelian topological quantum computer at finite temperature Alexis Schotte,1, ∗ Lander Burgelman,2 and Guanyu Zhu1, 3, † 1IBM Quantum, IBM Almaden Research Center, San Jose, CA 95120, USA 2Department of Physics and Astronomy, Ghent University, Krijgslaan 281, 9000 Gent, Belgium 3IBM T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Watson Research Center, Yorktown Heights, NY 10598, USA We study fault-tolerant error correction in a quantum memory constructed as a two-dimensional model of Fi- bonacci anyons on a torus, in the presence of thermal noise represented by pair-creation processes and measure- ment errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The correction procedure is based on the cellular automaton decoders originating in the works of G´acs [1] and Harrington [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Through numerical simulations, we observe that this code behaves fault-tolerantly and that threshold behavior is likely present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Hence, we provide strong evidence for the existence of a fault- tolerant universal non-Abelian topological quantum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' INTRODUCTION Anyons are emergent quasi-particles that exist in two-dimensional condensed matter systems and whose exchange statistics generalize that of Bosons and Fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' These particles have spurred much interest due to their potential applications for quantum compu- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' In particular, it was found that with certain types of non-Abelian anyons, a universal quantum compu- tation can be performed by braiding and fusing these particles [3–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' An intriguing benefit of this paradigm is that, due to their topological nature, computations are intrinsically robust to perturbations at zero tem- perature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' At non-zero temperature, however, thermal anyonic excitations can corrupt the computation by performing non-trivial braids with the computational anyons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Since systems exhibiting anyonic excitations have a spectral gap ∆, this source of errors can be sup- pressed to some extent at temperatures T ≪ ∆/kB as the density of thermal anyons scales as e−∆/kBT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Alas, this passive protection does not suffice, because the presence of thermal anyons is unavoidable at non- zero temperatures when scaling up the size of com- putations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Therefore, proficient active error correction schemes for non-Abelian models are paramount for the realization of topological quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Besides their envisaged use for topological quantum computation, topologically ordered systems (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=', those that support anyonic excitations on top of their ground space) are also of much interest for quantum error cor- rection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' In particular, one of the characteristics of such systems is a robust ground space degeneracy, which al- ∗ alexis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content='schotte@posteo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content='net † guanyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content='zhu@ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content='com lows one to use their ground space as the code space of an error correcting code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' This realization led to the dis- covery of topological quantum error correcting codes, which encode logical quantum states in topologically ordered states of a system of qudits (typically arranged on a two-dimensional lattice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Since their discovery in the 90s, most research has focused exclusively on Abelian topological codes such as the surface code and the color code [5–14], which admit an elegant charac- terization in terms of the stabilizer formalism [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Due to their geometrical locality and high error thresholds, these codes are considered to be promising candidates for protecting quantum information from noise in error- corrected quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' One of the drawbacks of Abelian topological codes, however, is that they do not allow one to execute a universal set of logical gates in a protected fashion in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Hence, they must be supplemented with additional protocols such as magic state distillation [16] or code switching to higher-dimensional codes [17, 18] , which introduce a large space-time overhead [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Alternatively, there ex- ist non-Abelian topological codes which do not suffer from this inherent limitation, and are able to perform a universal gate set natively within their code space in two dimensions [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The trade-off is that such codes go beyond the stabilizer formalism and are therefore very hard to simulate classically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' While active error correction in Abelian anyon mod- els and Abelian topological codes has been studied extensively, quantum error correction based on non- Abelian anyon models has not enjoyed the same fo- cus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Nevertheless, important progress has been made over the last decade, including both analytical proofs and numerical demonstrations of threshold behavior for various non-Abelian topological error correcting codes [20–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Moreover, syndrome extraction circuits for such non-Abelian string-net codes have been devel- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content='00054v1 [quant-ph] 30 Dec 2022 2 oped in recent years [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' In addition, state prepa- ration for non-Abelian codes based on the Kitaev quan- tum double models via measurements has also been proposed recently for the experimental implementa- tion on qubit lattices [26, 27], although further devel- opment is still needed in the context of fault-tolerant state preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Notably, previous studies in this field already include codes based on the Fibonacci anyon model, which is universal for quantum computation [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' In particular, a quantum memory of qubits supporting doubled Fibonacci anyonic excitations was found to have a threshold that lies remarkably close to that of the surface code under similar assumptions [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' These results, however, all assume perfect syndrome measurements, which are topological charge measure- ments in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' As we aim to model more re- alistic scenarios, we must take faulty measurements into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Again, much is known in the case of Abelian topological codes [2, 7, 28–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' For their non-Abelian counterparts, one key result stands out: in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' [32] a proof was formulated that topological codes based on non-cyclic anyon models admit a error cor- rection thresholds with faulty topological charge mea- surements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' While this result is remarkable, non-cyclic anyon models are not universal for quantum computa- tion, and it remains an open question whether similar claims can be made for universal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' In this work, we take a step towards demonstrat- ing that fault-tolerance is indeed possible for universal non-Abelian topological codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' To this end, we define a quantum memory constructed as a two-dimensional model of Fibonacci anyons on a torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' We study ac- tive continuous quantum error correction on this model in the presence of thermal noise represented by pair- creation processes, and with faulty syndrome measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The correction procedure is based on the cel- lular automaton decoders originating in the works of G´acs [1] and Harrington [2], and further studied in the context of non-Abelian models in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Through numerical simulations, we study how the average mem- ory lifetime changes with the error rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The results in- dicate that this code is indeed fault-tolerant, which is strong evidence for the existence of fault-tolerant uni- versal non-Abelian codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The structure of this work is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' II we introduce the topological Fibonacci code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' We then describe the details of the noise model in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' III and introduce the cellular automaton decoder in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' We proceed by giving an outline of the numerical sim- ulations performed in this work in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Finally, we present the corresponding numerical results in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' VI and conclude with a discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' THE FIBONACCI CODE We consider a two-dimensional model comprised of hexagonal tiles laid out on the surface of a torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The resulting geometry can be represented as an L × L hexagonal lattice with periodic boundary conditions in both directions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Each of these hexagonal tiles can contain an excitation known as a Fibonacci anyon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Anyons are point-like quasi-particle excitations which can be characterized algebraically in terms of a unitary modular tensor category (UMTC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' A thorough description of anyon models using UMTCs goes be- yond the scope of this work, however, some details are given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' For now, it is sufficient to state that an anyon model specifies a set of anyon labels, also referred to as particle types, which can fuse according to a specific set of fusion rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The Fibonacci anyon model considered in this work contains two labels, 1 and τ, which obey the fusion rules 1×1 = 1 , 1×τ = τ×1 = τ , τ×τ = 1+τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' (1) In general, one can associate a vector space to a given set of anyons, where the basis vectors are la- beled by the different ways in which the anyons can fuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' This fusion space has a topological degeneracy, and can therefore be used to robustly encode quan- tum information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' In particular, for the Fibonacci anyon model the anyonic vacuum on a two-dimensional torus has a twofold degeneracy [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Starting from our two- dimensional model, we can therefore define an error correcting code whose code space is identified with the anyonic vacuum on the torus and which encodes a sin- gle logical qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' A basis for this code space can be defined using Wilson line operators along the homo- logically non-trivial cycles x and y shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' 1: W a x |1⟩x = Sa1 S11 |1⟩x , W a x |τ⟩x = Saτ S1τ |τ⟩x , a ∈ {1, τ} , (2) We note that a different basis, {|0⟩y , |1⟩y}, can be defined analogously by swapping the x and y labels above, where the two bases are related through the modular S matrix Sab = y⟨a|b⟩x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' For the Fibonacci anyon model its numerical values are S = 1 � 1 + φ2 � 1 φ φ −1 � , (3) where φ = 1 + √ 5 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The action of the mapping class group on the any- onic vacuum then corresponds to unitary operations on 3 x y Figure 1: The two homologically non-trivial cycles on a torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' the code space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' For the Fibonacci category, any logical unitary operator can be realized in this way, up to arbi- trary precision [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Therefore, the quantum error cor- recting code defined above natively supports universal quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Errors in this code appear as spurious anyonic ex- citations, which can corrupt the encoded informa- tion if their world lines between creation and re- annihilation are topologically non-trivial, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=', form a non-contractible cycle 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The objective of error cor- rection is then to systematically remove these spurious excitations, without corrupting the quantum memory in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' This correction is performed in an active and continuous manner, and can be broken down into a series of discrete steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' At each step, a suitable recov- ery operation is performed based on a measured list of positions and types of the excitations, called the error syndrome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' We conclude this section by noting that the numer- ical simulation of the error-correction process requires the introduction of some additional manipulations on fusion states of multiple Fibonacci anyons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' As these are technical details that do not contribute to the in- tuition of the procedure, we defer their definition to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' NOISE MODEL AND CORRECTABILITY Having defined our model and code space, we now turn to the description of the noise model used in our 1 Note that a pair of Fibonacci anyons can also fuse to a single non- trivial anyon when one member of the pair has been transported along a non-trivial cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Since the resulting state is no longer in the code space, this does not constitute a logical operation on the encoded information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' However, one can show that the encoded information is irrevocably lost in case of such an event [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' We model continuous active error cor- rection in our Fibonacci code as a sequence of time steps, where each time step itself consists of three parts: the application of pair-creation noise, faulty syndrome measurement, and error correction respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' At each time step, first, for each edge of the hexago- nal lattice graph a pair of anyons is created across this edge with a probability p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Immediately after each pair creation event, the resulting charge in the two affected tiles is sampled, effectively collapsing all superposi- tions of anyonic charge within each tile to either 1 or τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' After the pair creation noise has been applied, faulty syndrome extraction is simulated by generating a list of the anyon charge in all tiles, and flipping each outcome individually with a probability q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' In addition to the charges that are correctly detected, the resulting faulty syndrome can contain both “ghost defects” (indicating a non-trivial charge when none is truly present) and “missing defects” (failing to report a true non-trivial charge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Finally, this faulty syndrome is passed to a de- coder, introduced in the following section, which then performs a set of local operations based on the current (and past) syndrome information in an attempt to move the system back towards the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' After each time step, the current state of the sys- tem is copied and it is checked whether it is still cor- rectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' This is done by passing the copy to a clus- tering decoder [22–24] and simulating a decoding pro- cedure with perfect syndrome measurements starting from this given initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' If this perfect decoding is successful, the memory is considered intact and the simulation is continued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' If perfect decoding is unsuc- cessful, the memory is considered corrupted and the simulation is aborted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The memory lifetime is then de- fined as the number of time steps after which a perfect clustering decoder can no longer successfully restore the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' For a given pair of tiles which share an edge, the process of pair creation across this edge corresponds to the matrix elements ⟨a′, b′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' c′| Upc |a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' c⟩ = δc,c′F aa1 ττa′F a′τa b c b′ , (4) where we have used the F-symbols of the Fibonacci category, given in (A5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Here, |a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' c⟩ represents the state where the affected tiles have anyon charges a and b, respectively, with total charge c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' This then de- fines the probability distribution according to which outcomes a′ and b′ are sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Since our noise model does not allow any superposition in the any- onic charge of individual tiles, it should be considered semi-classical rather than fully quantum-mechanical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' 4 (a) (b) Figure 2: (a) Pair-creation events creating anyonic excitations in neighboring tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The dotted ellipse represents a collapse to the total charge of the anyons it contains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' A ghost defect is shown in blue, a missing defect is highlighted in orange with a cross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' (b) The outcome of this noise process represented on the decoding graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Note that the missing defect (highlighted in orange) will (by definition) not be visible in the syndrome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Note, however, that this does not render our model completely classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Indeed, superpositions in the total charge c of the affected tiles are an inherent part of the state evolution that cannot be captured faithfully by any classical probabilistic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Furthermore, while the extreme decoherence assumption for the anyon charge in individual tiles greatly simplifies the numerical sim- ulation outlined in this work, it was argued in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' [22] that this decoherence is unlikely to have any tangible influence on the observed memory lifetimes, as the es- sential topological nature of the noise processes is still captured correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' We note that this type of noise can be understood as originating from the connection to a thermal bath with inverse temperature β = 1/(kBT) determined by the error rate p through the relation p 1 − p = e−β∆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' (5) Here, ∆ represents the energy required to create a pair of anyonic excitations and place them in neighboring tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' To conclude this section, we emphasize that in the case of non-Abelian error correction, even decoding with perfect syndrome measurements is still an inher- ently stochastic procedure due to the indeterminacy of anyonic charge measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' This means that perfect decoding can sometimes either be successful or unsuc- cessful even starting from the same initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Our definition of the memory lifetime therefore simply cor- responds to a statistical estimate of the actual memory lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Furthermore, it is not known which decoder is optimal for the Fibonacci code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Hence, the choice for the clustering decoder to verify the correctability of states is, in a way, an arbitrary one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' This choice, how- ever, is motivated by the recent discovery that the clus- tering decoder yields high thresholds for a related er- ror correcting code exhibiting doubled Fibonacci any- onic excitations, and performed significantly better in this context than decoders based on a perfect matching strategy [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' In any case, one should keep in mind that the memory lifetime as defined above, does not repre- sent the true memory lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Instead the sub-optimal verification process entails that it merely provides us with a lower bound on the true value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' HARRINGTON’S CELLULAR AUTOMATON DECODER The model described above is paired with a decoder which is a straight-forward adaptation of the cellu- lar automaton decoder introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Previ- ously, this decoder has also been used for a similar phenomenological model of Ising anyons in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' [32], where the existence of an error correction threshold was proven analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' At each time step during the error correction simulation, based on the reported mea- surement outcomes in the faulty syndrome, the decod- ing algorithm will apply local transition rules to fuse neighboring anyons or to move anyons to neighboring tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Intuitively these transition rules work as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The lattice is divided into square colonies of size Q×Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' At each time step, the transition rules will attempt to fuse neighboring non-trivial anyons, as observed in the faulty syndrome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' If a non-trivial anyon has no neighbors, the transition rules will move it to the cen- ter of its colony.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' At larger timescales, higher-level transition rules are applied on a renormalized lattice where anyons located at colony centers will be fused with anyons at neighboring colony centers, or moved toward the center of their respective super-colonies, which consist of Q × Q colonies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' This renormalization scheme is then continued at higher levels until eventu- ally the Qn × Qn super-colony covers the entire lattice for some integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' To ensure the latter is possible, we will always assume that the linear lattice size satisfies L = Qn for some integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' An example of these pro- cesses is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' To describe the action of the decoding algorithm more precisely, we will define its action at different renormalization levels k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The level-0 transition rules 5 are those already discussed above and are applied at every time step based on the reported faulty syndrome obtained from the most recent round of faulty measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The transition rules are applied to one location at a time and take into consideration only the anyon content of that site and of its eight neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' A de- tailed definition of these rules is given in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' When an anyon is moved from a site l to a neighboring site l′, the (true) anyon content of site l is fused with that of site l′ and the resulting charge is placed on site l′ while the charge of site l is restored to the vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' This happens irrespective of whether or not the syn- drome for both sites was correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Hence, when the de- coder attempts to move a ghost defect (a trivial charge misidentified as a non-trivial one) to a neighboring site, this process does not create additional excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' This does not, however, mean that mistaking a trivial charge for a non-trivial one has no negative consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' In- deed, these wrong syndromes may cause the decoder to stretch out existing errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The level-1 transition rules are not applied in every time step, but only when t is a multiple of a param- eter called the work period, which we will denote by U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' We require that U = b2 for some positive integer b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' One should think of U as the time scale at which a coarse-graining is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Level-1 transition rules act at on a coarse-grained lattice where the sites corre- spond to the centers of the level-0 colonies, and these are grouped into level-1 colonies of size Q2 × Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Hence, the actions determined by the level-1 transition rules involve pairs of level-0 colony centers separated by a distance Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' An example of such a move is pro- vided in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' 4(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The transition rules themselves are nearly identical to the level-0 rules, but are based on two sets of level-1 syndromes s1,c and s1,n (defined below) rather than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' For a site l (which is a level- 0 colony center), the transition rules use s1,c(l) as the anyon content of site, while the anyon content of its neighbors (that is, the neighboring level-0 colony cen- ters) is taken to be s1,n(l′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The definitions of the level-1 syndromes s1,c and s1,n require a pair of variables fc, fn ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' In- tuitively these variables serve as detection thresholds for the level-1 syndromes by determining the fraction of measurements that must return a non-trivial out- come at a site before it qualifies as a non-trivial level-1 syndrome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The proper definition, however, is slightly more complicated and uses a coarse-grained counting method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Below, we give the precise definition of s1,c, the definition of s1,n is entirely analogous (using fn instead of fc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' We start by dividing the work period U = b2, into b intervals of b time steps each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' For each of these intervals, we say a non-trivial syndrome is present at a colony center l if a non-trivial charge was reported there for at least fcb of the b time steps in the interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' When at least fcb of the b intervals have a non-trivial syndrome, s1,c(l) is set to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' A visual example of this coarse-grained counting procedure is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' ≥ fc · b < fc · b ≥ fc · b Figure 3: A visual example of the coarse-grained counting procedure to determine the level-1 syndrome for a level-0 colony center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The row of dots represent U time steps, divided in b intervals of size b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The time steps during which a non-trivial measurement outcome was reported are indicated by the colored dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The crosses in the second row indicate in which intervals the fraction of non-trivial measurement outcomes is equal to or higher than fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The motivation for using two types of syndromes for k > 0 is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Suppose that an error spans across two neighboring colonies, which we will label ρ and ρ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The level-0 transition rules will transport all resulting anyons to the respective colony centers, where they can now be acted upon by level-1 transition rules at the end of the work period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Imagine that a non-trivial anyon is now present at both colony centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' When consid- ering the level-1 transition rules acting on ρ, there are four possible scenarios for the syndromes s1,c(ρ) and s1,n(ρ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' In case s1,c(ρ) = 0, the transition rules act trivially on ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' If both s1,c(ρ) = 1 and s1,n(ρ′) = 1 then the transition rules will be applied correctly and the anyons will be fused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' However, if s1,c(ρ) = 1 but s1,n(ρ′) = 0, the transition rules may move the anyon in ρ away from ρ′, thereby increasing the weight of the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Hence, it is desirable to set fc > fn to de- crease the odds that when a level-k syndrome reports an non-trivial anyon at a colony center, the level-k syn- drome for its neighbors are false negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' We must be careful not to set fc too high or fn too low, how- ever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' If we choose fc to high, low-weight errors could cause s1,c to never report any non-trivial charges, de- laying any necessary corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Similarly, setting fn too low will result in low-weight error triggering many false positives for s1,n, which can cause the decoder to make wrong decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' 6 Level-k transition rules are applied when t mod U k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' They operate on a renormalized lattice that uses the centers of level-(k − 1) colonies as sites, and groups these into level-k colonies of size Qk ×Qk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The level-k syndromes sk,c and sk,n are determined by the coarse-grained counting method described above, using bk intervals of bk time steps each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' For linear sys- tem size L, k ranges from 0 to kmax = logQ(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' It is important to note that non-Abelian anyons do not allow for instantaneous moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Indeed, while one can construct a unitary string operator for Abelian anyons, no such operator can be constructed for the non-Abelian case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' This discrepancy can be traced back to the fact that fusion outcomes are non-deterministic for non-Abelian anyons, implying it is not possible to move an non-Abelian anyon by annihilating it with one member of a particle-antiparticle pair (as is done in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=', the surface code).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Therefore, the actions determined by level-k transi- tion rules, for k > 0 cannot be applied withing a sin- gle time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Instead, they will be broken up into a sequence of moves involving only pairs of neighbor- ing sites which will be applied in Qk consecutive time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' We further limit the model by requiring that the number recovery operations affecting a single tile in the lattice (or site in the decoding graph), is no greater than one in each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' This allows all recovery opera- tions applied in one time step to be performed in par- allel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Hence, we must define a hierarchy determining which actions (moves or fusions between neighboring tiles) get prioritized based on the renormalization level from which they originated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' In our case, it was opted to always prioritize correction processes from the highest renormalization level 2 It was argued in [32] that the prohibition of instanta- neous corrections would likely not influence the thresh- old behavior other than slightly lowering the memory lifetimes relative to a hypothetical case where this re- striction is dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' We explicitly verify this claim for our Fibonacci model below in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' OUTLINE OF THE SIMULATION The goal of this work is to numerically determine a fault-tolerant error threshold for the error correcting 2 Note that if one were to prioritize the level-0 corrections, higher- level correction could never be completed, as they would be un- done immediately after their first action is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' (a) (b) (c) (d) Figure 4: Illustration of the transition rules on the decoding graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The gray disks represent non-trivial syndromes, and the blue arrows represent the actions suggested by the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The blue dotted lines represent the 3 × 3 colonies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' (a-c) show a sequence of level-0 transition rules and possible outcomes of those actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' In (d) non-trivial anyons have been transported to two neighboring colony centers, the blue arrow represent a level-1 transition which could be applied at the end of the work period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' code defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' II with pair-creation noise and measurement noise as outlined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' III, and with the cellular automaton decoder introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' This is achieved by performing Monte-Carlo simulations to determine the average memory lifetime for a range of system sizes and error rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' These results then allow one to estimate the value of the error threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' A single Monte-Carlo sample (with some fixed val- ues for the noise strength p and the measurement er- ror rate q) is obtained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' First, the state of the system is initialized as a ground state (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' : con- taining no anyons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Then a sequence of time steps is performed consisting of the application of pair-creation noise with rate p, a round of faulty syndrome measure- ments with error probability q, and finally a sequence of recovery operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' At the end of each time step, it is verified whether or not the state is considered cor- rectable, according to the criteria specified in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' 7 (a) (b) (c) Figure 5: (a) Level-0 colonies of size Q × Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' (b) Level-1 colonies defined as Q × Q level-0 colonies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' (c) Renormalized lattice used for the level-1 transition rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' This sequence of time steps is continued until one of the following three outcomes occurs: (1) The largest connected group of anyons grows too large, rendering its classical simulation intractable 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' (2) A noise pro- cess or recovery operation induces a logical error by fusing a pair of anyons along a path that forms a non- contractible loop when combined with their fusion tree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' (3) The verification procedure at the end of a time step fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The memory lifetime is then set as the number of time steps that were completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The course of a single Monte-Carlo sample in the simulation is summarized as pseudo-code in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' 3 Note that such cases are likely to correspond configurations in which the initial state cannot be recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Algorithm 1 Numerical simulation initialize state t = 0 while correctable with clustering decoder & no logical errors made do t ← t + 1 apply pair-creation noise perform faulty measurements for k = 0 : kmax do if t mod U k = 0 then update level-k syndromes apply level-k transition rules end if end for end while memory lifetime = t VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' NUMERICAL RESULTS The Monte Carlo simulation described above were performed for various system sizes with p = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The following parameters were used: Q = 3 , b = 7 , Fc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content='7 , Fn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The resulting average memory lifetimes for L = 3, L = 9 and L = 27 are shown below in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' These results clearly indicate that the code pre- sented in this work is indeed fault-tolerant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Further- more, while the current data is not sufficient to demon- strate a clear-cut fault-tolerant threshold, it still ex- hibits threshold behavior and is remarkably similar to the results previously obtained for the toric code [2] and the Ising topological code [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' We estimate that the fault-tolerant threshold for the Fibonacci topologi- cal with pair-creation noise and measurement noise lies between p = 10−4 and p = 5 · 10−4, which corre- sponds to an inverse temperature between β = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content='2/∆ and β = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content='6/∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' This is comparable to the threshold found for the Ising topological code [32], and only one order of magnitude below that for the surface code un- der similar circumstances [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' For physical error rates near p = q = 10−4, corresponding to a temperature 1/β one order of magnitude below the spectral gap, a code of linear size L = 27 yields logical error rates of the order 10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' 8 (a) Average memory lifetime in function of the error strength, with p = q, for various system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Each data point represents the average over 1000 Monte Carlo samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The blue line shows the coherence time of a single physical qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The average memory lifetimes for p ≤ 10−3 were fitted to a function of the form f(p) ∼ p−a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The results for L = 3, L = 9 and L = 27 are shown as the green, yellow and red lines respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' (b) Average lifetime in function of the error strength with p = q for various system sizes and (unphysical) instantaneous corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Each data point represents the average over 1000 Monte Carlo samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The blue line shows the coherence time of a single physical qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Figure 6 A second round of simulations was performed to de- termine average memory lifetimes with the assumption that all corrections happen instantaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' While this is akin to the Abelian topological codes, where dis- tant anyons can be fused using unitary string-operators, this scenario is unphysical for non-Abelian anyons as they do not admit unitary string-operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Never- theless, it is worth studying to which extend the re- sults in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' 6(a) are influenced by the restriction to non-instantaneous recovery operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' [32] it was conjectured that allowing instantaneous correc- tions does not significantly change the qualitative be- havior of the average memory lifetimes as a function of the error rate, but mostly just increases the memory lifetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Our results, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' 6(b), confirm this hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' DISCUSSION AND OUTLOOK The results presented in this work demonstrate that fault-tolerant error correction is possible for non- Abelian topological quantum error correcting codes supporting a universal logical gate set within their code space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' For a code consisting of Fibonacci anyons in hexagonal tiles on a two-dimensional torus, sub- jected to pair creation noise and measurement noise, we demonstrated that the cellular automaton decoder detailed in this work is fault-tolerant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' In particular, for physical error rates p ≤ 10−3, it was found that the logical memory lifetime surpasses the physical coher- ence time for all system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' When interpreting the pair-creation noise as resulting from a non-zero tem- perature, this pseudo-threshold corresponds to an in- verse temperature β = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content='9/∆, where ∆ is the energy required to create a pair of Fibonacci anyons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Further- more, our results suggest that this code admits a fault- tolerant quantum error correction threshold around p = 10−4, or β = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content='2/∆, which is similar to the fault- tolerant threshold found for the Ising topological code [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Several future research directions present them- selves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' First, more research on a possible fault-tolerant threshold is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Wile the numeric results pre- sented in this work provide a strong indication that a fault-tolerant error correction threshold exists, they do not conclusively prove its existence, nor do they pro- vide a precise estimate of its value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Hence, an impor- tant open problem is the formulation of a mathematical proof of its existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Such proofs were previously for- mulated for the toric code [2] and for non-cyclic non- Abelian anyon models such as in the Ising topological 9 code [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Due to the cyclic nature of Fibonacci anyons (or any universal anyon model), however, the existing proofs are not sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Second, it would be interesting to study different decoders in an identical setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' This includes both different cellular-automaton decoders such as those in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' [31, 34], as well as new decoders tailored to the Fibonacci topological code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Third, while this work demonstrates that the Fi- bonacci topological code can be operated fault- tolerantly as a quantum memory, results regarding its use for fault-tolerant quantum computing are still lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' We envisage that fault-tolerant topological quantum computing at non-zero temperatures could be achieved by combining the code and decoding proce- dure presented in this work with the scheme for per- forming Dehn twists presented in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' [35–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Alter- natively, one can also perform transversal logical gates in a folded Fibonacci code [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Finally, it would be of great interest to expand the current results to microscopic models for non-Abelian topological quantum error correction, such as the Fi- bonacci Turaev-Viro code [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors would like to thank Guillaume Dauphi- nais and Jim Harrington for enlightening discussions on the cellular automaton decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The computational resources (Stevin Supercomputer Infrastructure) and services used in this work were provided by the Flem- ish Supercomputer Center (VSC), funded by Ghent University, the Research Foundation Flanders (FWO), and the Flemish Government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' AS was supported by a fellowship of the Belgian American Educational Foun- dation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' LB was supported by a PhD fellowship from the FWO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' GZ was supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Department of Energy, Office of Science, National Quantum In- formation Science Research Centers, Co-design Center for Quantum Advantage (C2QA) under contract num- ber DE-SC0012704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Appendix A: Fibonacci anyons Below, we give with a brief overview of the topolog- ical aspects of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' A thorough exposition of the theory of anyon models [39–42] is beyond the scope of this work, and we restrict to a basic description of the aspects of the Fibonacci model that are required for the specific purpose of our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' We refer to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' [33] for an in-depth discussion of anyonic fusion states on the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The Fibonacci anyon model has two particle types, 1 and τ, that satisfy the fusion rules 1 × 1 = 1 , 1 × τ = τ × 1 = τ , τ × τ = 1 + τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' (A1) It is a non-Abelian anyon model, as the fusion of two τ-anyons can yield two distinct outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' In this case, the fusion space V c ab associated to the fusion of anyons a and b to c is one-dimensional, and is spanned by the state vector |a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' c⟩ which we will represent graphi- cally as |a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' c⟩ → b a c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' (A2) When fusing several anyons, their total charge is a col- lective property of the anyons that does not depend on the specific order in which they are fused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Mathemati- cally, this is expressed through the associativity of the fusion rules, (a × b) × c = a × (b × c) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' (A3) If we consider the case of three anyons a, b and c that fuse to a total charge d then this fusion may be car- ried out in two distinct ways, implying the existence of two equivalent decompositions of the associated fu- sion space V d abc in terms of fusion states (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' These equivalent decompositions are related through a uni- tary transformation called an F-move, which is repre- sented graphically as b c e d a = � f F abe cdf f d b c a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' (A4) The coefficients F abe cdf in this expression are called F- symbols of the anyon model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' For the Fibonacci model they are given by F ττ1 ττ1 = 1 φ , F τττ ττ1 = F ττ1 τττ = 1 √φ , F τττ τττ = − 1 φ , (A5) where all other F-symbols consistent with the fusion rules (A1) are equal to 1, and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' In addition to this recoupling one can also consider the exchange or braiding of pairs of anyons, which pre- serves their total charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' At the level of the fusion 10 space, such an exchange corresponds to a basis trans- formation to a basis associated to a different linear or- dering of the anyons4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Within V c ab there are two such possible basis transformations, b a c = Rab c c a b = � Rba c �∗ c b a , (A6) which we will refer to as a clockwise and a counter- clockwise swap respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' For the Fibonacci model the R-symbols appearing in these expressions are given by Rττ 1 = e 4πi 5 , Rττ τ = e− 3πi 5 , (A7) where all other Rab c allowed by the fusion rules are equal to 1, and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' As we are dealing with a system that allows for an extensive amount of anyonic excitations, we will be in- terested in fusion states of many anyons, a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=', an, with some total charge c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' This gives rise to an ex- ponentially large topological Hilbert space V c a1a2···an spanned basis states of the form a1 a2 c b1 a3 an−1 an b2 bn−3 bn−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' (A8) When dealing with anyonic fusion states on surfaces of higher genus, one must take into account additional de- grees of freedom in these states that are related to the anyonic charge that runs along non-contractible cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' On a torus, this results in two distinct descriptions of fusion states, which are related through a basis change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' These are known as the inside and outside bases, and are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' A detailed discussion can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' We will simply refer to these ad- ditional degrees of freedom as the handle labels of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' 4 As opposed to the more conventional view of braiding as an active transformation that maps between different fusion spaces, we have opted for the equivalent framework of braiding as a passive basis transformation, as the latter is more appropriate in view of our specific model and numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' (a) (b) Figure 7: Two possible basis choices for anyons on a torus: a) the inside basis, b) the outside basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' For our current purpose, we won’t need the full de- scription of said handle labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' It is sufficient for us to pick one basis, and subsequently set the total charge of all anyons to the vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' We are left with only a single label (1 or τ) representing the anyonic charge flowing along the non-contractible cycle associated with our basis choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' This is precisely the origin of the twofold degeneracy of the anyonic vacuum on the torus, which we have taken to be our code space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' In the this work we will always start from the code state corresponding to a trivial handle label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' As a change in handle label at any point during the error cor- rection procedure must be the consequence of a topo- logically non-trivial process which constitutes a logi- cal error, any simulation is aborted at the occurrence of such an event, meaning that all handle labels can be safely ignored in the remainder of our discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' A basic functionality required for the simulation of the error correction procedure is the ability to correctly sample a measurement of the total charge of a given set of anyons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Starting from a given fusion state, this can be achieved by first transforming to a basis in which the relevant anyons are fused sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' For a system of many anyons these reordering basis transformations are obtained by combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' (A4) and (A6), giving 11 rise to the clockwise swap aj b3 b2 b1 aj+1 = � b′ 2 Bb1ajb2 aj+1b3b′ 2 aj aj+1 b3 b2 b1 , (A9) and the counterclockwise swap aj b3 b2 b1 aj+1 = � b′ 2 � Bb1ajb2 aj+1b3b′ 2 �∗ aj aj+1 b3 b2 b1 , (A10) where Bb1ajb2 aj+1b3b′ 2 = � c F aj+1ajc b3b1b′ 2 Rajaj+1 c F b1ajb2 aj+1b3c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' (A11) By performing a certain set of these basis transforma- tions, the fusion order can always be made consistent with the group of anyons of which we want to mea- sure the total charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Subsequently, the fusion state is recoupled such that the relevant group of anyons is con- nected to the rest of the state by a single edge c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' For a charge measurement of a pair of anyons this recoupling takes the form aj b3 b2 b1 aj+1 = � c F b1ajb2 aj+1b3c aj b3 b1 aj+1 c , (A12) and charge measurements of larger groups of anyons simply require consecutive applications of the recou- pling identity (A4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Finally, the charge outcome is ob- tained sampling the total charge c from the probability distribution corresponding to the resulting superposi- tion of fusion states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Appendix B: Classical simulatbility It is well known that Fibonacci anyons are universal for quantum computation [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' One might therefore be tempted to conclude that the classical simulation of the topological Fibonacci code described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' II with pair-creation noise is unlikely to succeed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' However, as was noted in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' [23], the simulation of noise and error correction processes does not require the simula- tion of general anyon dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' In particular, individ- ual noise processes create distinct connected groups of anyons with vacuum total charge (or extend such exist- ing groups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' These groups correspond to anyons that have interacted at some point during their lifetime, and must thus only be merged whenever a noise or error correction process involves two members from discon- nected groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Since each connected group has a triv- ial total charge, braiding between disconnected groups is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Hence, the total fusion space factorizes into a tensor product of fusion spaces of individual connected groups, and we are only required to simulate anyon dy- namics within each of these groups separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' This factorization of the fusion space is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' The creation and subsequent merging of discon- nected groups of anyons by noise and recovery pro- cesses can be thought of as a kind of percolation pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Hence, below the percolation threshold, one expects that the size of the largest connected group scales as O(log(L)) (with variance O(1)), where L is the linear system size [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' As this is a probabilis- tic statement, there will be instances where the largest connected group has a size larger than O(log(L), but the probability of such events is suppressed exponen- tially with the system size L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' This logarithmic scal- ing of the largest cluster size s = O(log(L)) coun- ters the exponential scaling of the dimension of the fu- sion space d = O(exp(s)) for individual connected groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Therefore, the fusion spaces of individual con- nect groups will have dimension dim = O(poly(L)), meaning that the dynamics within connected groups can be simulated efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Exploiting the tensor product structure within the to- tal fusion space, requires the use of basis for the any- onic fusion space which reflects this structure and can be updated dynamically to keep track of noise and re- covery processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' This is best achieved by using the framework of curve diagrams, which were introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' [23] (also see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' [44] for a more rigorous treatment using the language of modular functors), and also discussed extensively in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Since these are merely a technical tool for keeping track of the most appropriate basis during the numerical simulations, we will refrain from discussing them here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Interested read- ers are referred to the aforementioned references for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' 12 V ab 1 ⊗ V cde 1 ⊗ V fg 1 V ab 1 ⊗ V cde 1 ⊗ V fg 1 (a) a b c d e f g x y V abcdefg 1 = ⊕x,y � V ab x ⊗ V x cde y ⊗ V y fg 1 � (b) V ab 1 ⊗ V cde 1 ⊗ V fg 1 a b c d e f g (c) Figure 8: (a) A set of noise processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' (b) A generic state in the full fusion space of all anyons created by the noise processes involves superpositions in the labels x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' (c) The factorized Hilbert space which is sufficient to represent the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' Appendix C: Transition rules for Q = 3 Below, we give a full definition of the transition rules for Q = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' It is possible to define more general tran- sition rules that apply for any colony size, examples of such rules can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' [2] and [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' However, since the numerical simulations performed in this work were performed to Q = 3 we have taken the freedom to tailor the transition rules to this case specifically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' North-West if sk,c(ρ) = 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (0, −1)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (−1, 0)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (0, 1)) ̸= 0, move east;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (1, 0)) ̸= 0, move south;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (1, −1)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (−1, −1)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (−1, 1)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else move south;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' North if sk,c(ρ) = 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (−1, 0)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (0, −1)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (0, 1)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (−1, 1)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (−1, −1)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else move south;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' North-East if sk,c(ρ) = 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (0, 1)) ̸= 0, move east;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (−1, 1)) ̸= 0, move north-east;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (1, 1)) ̸= 0, move south;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (−1, 0)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (0, −1)) ̸= 0, move west;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (1, 0)) ̸= 0, move south;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (−1, −1)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else move south-west;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' East if sk,c(ρ) = 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (0, 1)) ̸= 0, move east;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (−1, 1)) ̸= 0, move north-east;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (1, 1)) ̸= 0, move south;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (−1, 0)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (1, 0)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else move west;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' South-East if sk,c(ρ) = 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (0, 1)) ̸= 0, move east;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (1, 0)) ̸= 0, move south;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (−1, 1)) ̸= 0, move north-east;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (1, −1)) ̸= 0, move south-west;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (1, 1)) ̸= 0, move south;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (0, −1)) ̸= 0, move west;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else move north;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' 13 South if sk,c(ρ) = 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (1, 0)) ̸= 0, move south;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (1, −1)) ̸= 0, move south-west;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (1, 1)) ̸= 0, move east;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (0, −1)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (0, 1)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else move north;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' South-West if sk,c(ρ) = 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (1, 0)) ̸= 0, move south;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (1, −1)) ̸= 0, move south-west;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (1, 1)) ̸= 0, move east;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (0, −1)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (−1, 0)) ̸= 0, move north;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (0, 1)) ̸= 0, move east;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (−1, −1)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else move north-east;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' West if sk,c(ρ) = 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (0, −1)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (−1, 0)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (1, 0)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (1, −1)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else if sk,n(ρ + (−1, −1)) ̸= 0, do nothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' else move east;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' G´acs, Reliable computation with cellular automata, Journal of Computer and System Sciences 32, 15 (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfQvbW/content/2301.00054v1.pdf'} +page_content=' 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