Topological Codes Research Articles

Decoding Toric Codes on Three Dimensional Simplical Complexes

Three dimensional (3D) toric codes are a class of stabilizer codes with local checks and come under the umbrella of topological codes. While decoding algorithms have been proposed for the 3D toric code on a cubic lattice, there have been very few studies on the decoding of 3D toric codes over arbitrary lattices. Color codes in 3D can be mapped to toric codes. However, the resulting toric codes are not defined on cubic lattice. They are arbitrary lattices with triangular faces. Decoding toric codes over an arbitrary lattice will help in studying the performance of color codes. Furthermore, gauge color codes can also be decoded via 3D toric codes. Motivated by this, we propose an efficient algorithm to decode 3D toric codes on arbitrary lattices (with and without boundaries).

Cluster-Precursors and Self-Assembly Li36Ca4Sn24-oS64 and LiMgEu2Sn3-oS28 Crystalline Structures

Using computer methods (the ToposPro software package), the combinatorial-topological analysis and modeling of self-assembly of the crystal structure of Li36Ca4Sn24-oS64 (space group (sg) Cmcm, a = 4.640 A, b = 27.112 A, c = 11.491 A, V = 1445.5 A3) and LiMgEu2Sn3-oS28 (sg Cmcm, a = 4.782 A, b = 20.717 A, c = 7.743 A, V = 767.1 A3). For the Li36Ca4Sn24 intermetallic compound a new type of 11-atom cluster K11 is installed, formed from double pentagonal rings: K11 = 0@11 (Li5)Ca(Sn5). The maximum cluster symmetry K11 and primary strand of translationally related K11 clusters corresponds to noncrystallographic symmetry 5m. Primary chains of linked K11 clusters preserving only symmetry m are located in the direction [100] and the distance between the centers of the clusters determines the length of vector a = 4.640 A. The layer is formed when the primary chains located antiparallel are bound. The tetrahedral clusters are located between the primary chains K4 = 0@4 (Li3Sn), forming chains in the direction [100]. The symmetry and topological code of the self-assembly processes of the 3D structure of the LiMgEu intermetallic compound LiMgEu2Sn3-oS28 is reconstructed from clusters K4 = 0@4 (LiMgEuSn) and K3 = 0@3 (Sn2Eu). The layer is formed when the parallel chains of K4 + K3 are bound.

Cluster Self-Organization of Intermetallic Systems: New Two-Layer Nanocluster Precursors K64 = 0@8(Sn4Ba4)@56(Na4Sn52) and K47 = Na@Sn16@Na30 in the Crystal Structure of Na52Ba4Sn80 -cF540

Using computer methods (ToposPro software package), geometrical and topological analysis of the crystal structure of Na52Ba4Sn80-cF540 intermetallic compound (a = 25.053 A, V = 16 010.82 A3, space group F-43m) are carried out. Two new nanocluster precursors with the -43m symmetry are determined, more specifically, the K47 two-layer nanocluster of the Na@16Sn@30Na composition with an NaSn16 internal Friauf polyhedron and 30 Na atoms in the shell and a K64 two-layer nanocluster of the 0@8(Sn4Ba4)@56(Na4Sn52) composition with the Sn4Ba4 internal polyhedron and 56 (4Na + 52Sn) atoms in the shell. The symmetrical and topological codes of self-assembly processes of 3D structures from K64 and K47 cluster precursors are reconstructed in the following form: primary chain → microlayer → microframework. The Na4 and Sn8 clusters with the -43m symmetry and Na atoms are determined as the spacers occupying the voids in the 3D framework from the K64 and K47 nanoclusters.

Cluster Self-Organization of Intermetallic Systems: New Three-Layer Cluster Precursor K136 = 0@Zn12@32(Mg20Zn12)@92(Zr12Zn80) and a New Two-Layer Cluster Precursor K30 = 0@Zn6@Zn24 in the Crystal Structure of Zr6Mg20Zn128-cP154

Using computer methods (ToposPro software package), the geometrical and topological analysis of the crystal structure of the Zr6Mg20Zn128-cP154 intermetallic compound with cubic cell parameters a = 13.709 A, V = 2576.42 A3, and space group Pm-3 is carried out. Two new nanocluster precursors with the ‑43m symmetry are determined: a three-layer K136 nanocluster of the 0@Zn12@32(Mg20Zn12@92(Zr12Zn80) composition with an internal icosahedron 0@Zn12 and 12 Zr atoms and 20 Zn atoms in 60-atomic Zn-shell and a two-layer K30 nanocluster of the 0@Zn6@Zn24 composition with an internal Zn6-octahedron and 24 Zn atoms in the external shell. The symmetrical and topological codes of the self-assembly processes of 3D structures from K136 and K30 nanocluster precursors are reconstructed in the following form: primary chain → microlayer → microframework. The Zn2 dimers are determined as the spacers occupying the voids in the 3D framework from K136 and K30 nanoclusters.

Optimization of the surface code design for Majorana-based qubits

The surface code is a prominent topological error-correcting code exhibiting high fault-tolerance accuracy thresholds. Conventional schemes for error correction with the surface code place qubits on a planar grid and assume native CNOT gates between the data qubits with nearest-neighbor ancilla qubits.Here, we present surface code error-correction schemes using only Pauli measurements on single qubits and on pairs of nearest-neighbor qubits. In particular, we provide several qubit layouts that offer favorable trade-offs between qubit overhead, circuit depth and connectivity degree. We also develop minimized measurement sequences for syndrome extraction, enabling reduced logical error rates and improved fault-tolerance thresholds.Our work applies to topologically protected qubits realized with Majorana zero modes and to similar systems in which multi-qubit Pauli measurements rather than CNOT gates are the native operations.

Soluble limit and criticality of fermions in Z2 gauge theories

Quantum information theory and strongly correlated electron systems share a common theme of macroscopic quantum entanglement. In both topological error correction codes and theories of quantum materials (spin liquid, heavy fermion and high-$T_c$ systems) entanglement is implemented by means of an emergent gauge symmetry. Inspired by these connections, we introduce a simple model for fermions moving in the deconfined phase of a $\mathbb Z_2$ gauge theory, by coupling Kitaev's toric code to mobile fermions. This permits us to exactly solve the ground state of this system and map out its phase diagram. Reversing the sign of the plaquette term in the toric code, permits us to tune the groundstate between an orthogonal metal and an orthogonal semimetal, in which gapless quasiparticles survive despite a gap in the spectrum of original fermions. The small-to-large Fermi surface transition between these two states occurs in a stepwise fashion with multiple intermediate phases. By using a novel diagrammatic technique we are able to explore physics beyond the integrable point, to examine various instabilities of the deconfined phase and to derive the critical theory at the transition between deconfined and confined phases. We outline how the fermionic toric code can be implemented as a quantum circuit thus providing an important link between quantum materials and quantum information theory.

Automatic drawing from point list using topological codes

The paper deals with the possibility of automatic drawing from the list of points with coordinates, created by geodetic surveying in the field. The text contains a design of topological codes which enable this automatic drawing. The part of the solution is also a design of application in Python language for generating drawings in a geographic information system (ArcGIS, QGIS). The result of the automated process is the conversion of spatial data into vector form, with maximum utilization of the information contained in the input data and with a minimum of manual work. This method involves a shift from classical field surveying to an object-oriented approach of this process. Application of the method speeds up the process of creating a 3D wireframe model by 35% compared to a classical method. The method was verified during the survey of the church in Karvina, the Church of Virgin Mary of the Snow in Velke Karlovice, and family house in Blansko (all in the Czech Republic).

A Three-Dimensional and Bi-objective Topological Optimization Approach Based on Piezoelectric Energy Harvester

Topological optimization can realize the optimization of the mass distribution in the whole objective domain. Compared with morphology and size optimization, it has a higher degree of freedom. In this work, the three-dimensional topological optimization based on piezoelectric materials was discussed. Using the Optimality Criteria, topology optimization was applied to the cantilever piezoelectric transducer. The structure optimization was realized with the voltage and stiffness as the multi-objective function. The corresponding codes are given to show the process of optimization. With 70% of the origin volume, the bi-objective optimization increases the global stiffness by 50.9% and the voltage by 30%. As the iteration process shows, the results of bi-objective optimization prove the value of additive mass at the bottom of the cantilever. This lays the foundation for future piezoelectric transducer structural optimization. Using only stiffness as the objective, the final objective increases inconspicuously. Bi-objective optimization shows its superiority. There are quite a few papers that research the combination of stiffness and voltage, and research which studies three-dimensionality is a point of innovation. Furthermore, this is also the first time a piezoelectric topology code has been shared.

General framework for constructing fast and near-optimal machine-learning-based decoder of the topological stabilizer codes

Quantum error correction is an essential technique for constructing a scalable quantum computer. In order to implement quantum error correction with near-term quantum devices, a fast and near-optimal decoding method is demanded. A decoder based on machine learning is considered as one of the most viable solutions for this purpose, since its prediction is fast once training has been done, and it is applicable to any quantum error correcting code and any noise model. So far, various formulations of the decoding problem as the task of machine learning have been proposed. Here, we discuss general constructions of machine-learning-based decoders. We found several conditions to achieve near-optimal performance, and proposed a criterion which should be optimized when a size of training data set is limited. We also discuss preferable constructions of neural networks, and proposed a decoder using spatial structures of topological codes using a convolutional neural network. We numerically show that our method can improve the performance of machine-learning-based decoders in various topological codes and noise models.

Experimental deterministic correction of qubit loss.

The successful operation of quantum computers relies on protecting qubits from decoherence and noise, which-if uncorrected-will lead to erroneous results. Because these errors accumulate during an algorithm, correcting them is a key requirement for large-scale and fault-tolerant quantum information processors. Besides computational errors, which can be addressed by quantum error correction1-9, the carrier of the information can also be completely lost or the information can leak out of the computational space10-14. It is expected that such loss errors will occur at rates that are comparable to those of computational errors. Here we experimentally implement a full cycle of qubit loss detection and correction on a minimal instance of a topological surface code15,16 in a trapped-ion quantum processor. The key technique used for this correction is a quantum non-demolition measurement performed via an ancillary qubit, which acts as a minimally invasive probe that detects absent qubits while imparting the smallest quantum mechanically possible disturbance to the remaining qubits. Upon detecting qubit loss, a recovery procedure is triggered in real time that maps the logical information onto a new encoding on the remaining qubits. Although the current demonstration is performed in a trapped-ion quantum processor17, the protocol is applicable to other quantum computing architectures and error correcting codes, including leading two- and three-dimensional topological codes. These deterministic methods provide a complete toolbox for the correction of qubit loss that, together with techniques that mitigate computational errors, constitute the building blocks of complete and scalable quantum error correction.

Neural Network Decoders for Large-Distance 2D Toric Codes

We still do not have perfect decoders for topological codes that can satisfy all needs of different experimental setups. Recently, a few neural network based decoders have been studied, with the motivation that they can adapt to a wide range of noise models, and can easily run on dedicated chips without a full-fledged computer. The later feature might lead to fast speed and the ability to operate at low temperatures. However, a question which has not been addressed in previous works is whether neural network decoders can handle 2D topological codes with large distances. In this work, we provide a positive answer for the toric code \cite{Kitaev2003Faulttolerantanyon}. The structure of our neural network decoder is inspired by the renormalization group decoder \cite{duclos2010fast, duclos2013fault}. With a fairly strict policy on training time, when the bit-flip error rate is lower than9%and syndrome extraction is perfect, the neural network decoder performs better when code distance increases. With a less strict policy, we find it is not hard for the neural decoder to achieve a performance close to the minimum-weight perfect matching algorithm. The numerical simulation is done up to code distanced=64. Last but not least, we describe and analyze a few failed approaches. They guide us to the final design of our neural decoder, but also serve as a caution when we gauge the versatility of stock deep neural networks. The source code of our neural decoder can be found at \cite{sourcecodegithub}.

Fault-tolerant quantum gates with defects in topological stabilizer codes

Braiding defects in topological stabiliser codes has been widely studied as a promising approach to fault-tolerant quantum computing. Here, we explore the potential and limitations of such schemes in codes of all spatial dimensions. We prove that a universal gate set for quantum computing cannot be realised by supplementing locality-preserving logical operators with defect braiding, even in more than two dimensions. However, notwithstanding this no-go theorem, we demonstrate that higher dimensional defect-braiding schemes have the potential to play an important role in realising fault-tolerant quantum computing. Specifically, we present an approach to implement the full Clifford group via braiding in any code possessing twist defects on which a fermion can condense. We explore three such examples in higher dimensional codes, specifically: in self-dual surface codes; the three dimensional Levin-Wen fermion mode; and the checkerboard model. Finally, we show how our no-go theorems can be circumvented to provide a universal scheme in three-dimensional surface codes without magic state distillation. Specifically, our scheme employs adaptive implementation of logical operators conditional on logical measurement outcomes to lift a combination of locality-preserving and braiding logical operators to universality.

Intermetallic Compounds NakMn (М = K, Cs, Ba, Ag, Pt, Au, Zn, Bi, Sb): Geometrical and Topological Analysis, Cluster Precursors, and Self-Assembly of Crystal Structures

A geometrical and topological analysis has been performed, and self-assembly of the crystal structures of intermetallic compounds formed in the Na–M (M = K, Cs, Ba, Ag, Pt, Au, Zn, Bi, Sb) systems has been simulated using computer methods (the ToposPro software). The NakMn precursor metal clusters of the crystal structures are determined based on the algorithms of structural-graph decomposition in cluster structures and construction of the basic net of the structure in the form of a graph whose nodes correspond to the cluster centers. The tetrahedral metal clusters M4, forming packets in the Na3Bi-hP8, Na2Bi2-tP4, and Na2Sb2-mP16 crystal structures; tetrahedral metal clusters M4 and spacers for the Na2(Au4)-cF24 and Ba2(Na4)-hP12 framework structures; octahedral clusters M6 for the Na4Au2-tI12 structure; octahedral M6 and tetrahedral M4 metal clusters for the Na2(Na4)(Ba6)-cF96 and Au2(In4)(Na6)-cF96 structures; and icosahedral metal clusters M13 for the Na(Zn13)-cF116 structure are found. The symmetry and topology code of the self-assembly of crystal structures of NakMn intermetallic compounds has been reconstructed from precursor metal clusters $$S_{3}^{0}$$ in the following form: primary chain $$S_{3}^{1}$$ → microlayer $$S_{3}^{2}$$ → microframework $$S_{3}^{3}$$ .

Cluster Self-Organization of Intermetallic Systems: Role of K5 = 0@5, K9 = 1@8 and K11 = 0@11 Clusters in the Self-Assembly of Crystal Structures

The combinatorial-topological analysis and simulation of self-assembly of the Na96Hg36-hR132 (space group R-3c, a = b = 9.228 A, c = 52.6380 A, V = 3881.91 A3) and Na12Hg8-tP20 (space group P42/mnm, a = b = 8.520, c = 7.800 A, V = 566.2 A3) crystal structures are conducted by computer-based methods (ToposPro program package). Polyhedral cluster precursors K11 = 0@11(Na8Hg3) and K9 = Hg@Na8 are first determined for the intermetallic compound Na96Hg36-hR132, while polyhedral cluster precursors K5 = 0@Na3Hg2 are determined for the intermetallic compound Na12Hg8-tP20. The symmetry and topology code of the self-assembly processes of the 3D structure Na96Hg36-hR132 from the nanocluster precursors K11 and K9, and of the 3D structure Na12Hg8-tP20 from the K5 clusters are reconstructed as the primary chain $$S_{{\text{3}}}^{1}$$ → layer $$S_{{\text{3}}}^{2}$$ → frame $$S_{{\text{3}}}^{3}.$$ The structural analysis of all known intermetallic compounds is conducted, and numerous examples of the assembly of their structures from the K5, K9, and K11 clusters are found.

Deep Q-learning decoder for depolarizing noise on the toric code

We present an AI-based decoding agent for quantum error correction of depolarizing noise on the toric code. The agent is trained using deep reinforcement learning (DRL), where an artificial neural network encodes the state-action Q-values of error-correcting $X$, $Y$, and $Z$ Pauli operations, occurring with probabilities $p_x$, $p_y$, and $p_z$, respectively. By learning to take advantage of the correlations between bit-flip and phase-flip errors, the decoder outperforms the minimum-weight-perfect-matching (MWPM) algorithm, achieving higher success rate and higher error threshold for depolarizing noise ($p_z = p_x = p_y$), for code distances $d\leq 9$. The decoder trained on depolarizing noise also has close to optimal performance for uncorrelated noise and provides functional but sub-optimal decoding for biased noise ($p_z \neq p_x = p_y$). We argue that the DRL-type decoder provides a promising framework for future practical error correction of topological codes, striking a balance between on-the-fly calculations, in the form of forward evaluation of a deep Q-network, and pre-training and information storage. The complete code, as well as ready-to-use decoders (pre-trained networks), can be found in the repository https://github.com/mats-granath/toric-RL-decoder.

Scalable characterization of localizable entanglement in noisy topological quantum codes

Topological quantum error correcting codes have emerged as leading candidates towards the goal of achieving large-scale fault-tolerant quantum computers. However, quantifying entanglement in these systems of large size in the presence of noise is a challenging task. In this paper, we provide two different prescriptions to characterize noisy stabilizer states, including the surface and the color codes, in terms of localizable entanglement over a subset of qubits. In one approach, we exploit appropriately constructed entanglement witness operators to estimate a witness-based lower bound of localizable entanglement, which is directly accessible in experiments. In the other recipe, we use graph states that are local unitary equivalent to the stabilizer state to determine a computable measurement-based lower bound of localizable entanglement. If used experimentally, this translates to a lower bound of localizable entanglement obtained from single-qubit measurements in specific bases to be performed on the qubits outside the subsystem of interest. Towards computing these lower bounds, we discuss in detail the methodology of obtaining a local unitary equivalent graph state from a stabilizer state, which includes a new and scalable geometric recipe as well as an algebraic method that applies to general stabilizer states of arbitrary size. Moreover, as a crucial step of the latter recipe, we develop a scalable graph-transformation algorithm that creates a link between two specific nodes in a graph using a sequence of local complementation operations. We develop open-source Python packages for these transformations, and illustrate the methodology by applying it to a noisy topological color code, and study how the witness and measurement-based lower bounds of localizable entanglement varies with the distance between the chosen qubits.

Cluster Self-Organization of Intermetallic Systems: Three-Layer Icosahedral Nanoclusters K132 = 0@12(In6Tl6)@30(In6Na6K18)@90(In72Na12K6) and K116 = 0@12(In6Tl6)@26(In12K14)@78(In36Tl20K12) for the Self-Assembly of the K52Na12Tl36In122-hP224 Crystalline Structure

The TOPOS software package is used for the combinatorial topological analysis and simulation of the self-assembly of the K52Na12Tl36In122-hP224 (spatial group P-3m1, a = b = 16.909, c = 28.483 A, V = 7 052 A3) crystalline structure. A total of 1649 variants of the cluster representation of a 3D atomic lattice with the number of structural units ranging from 4 to 10 are found. Two frame-forming three-layer icosahedral nanoclusters are detected: K132 and K116 with symmetry g = –3m and a diameter of 17 A. The chemical composition of the shells of the three-layer 132-atom nanocluster K132 is 0@12(In6Tl6)@30(In6Na6K18)@90(In72Na12K6) and that of the 116-atom nanocluster K116 is (0@12(In6Tl6)@26(In12K14)@78(In36Tl20K12). The symmetrical and topological code of the self-assembly of the 3D Na12K52Tl36In122 structure from the precursor K132 and K116 nanoclusters is reconstructed in the following form: primary chain → layer → frame. The framework’s voids are occupied by spacer atoms K and Tl.

A Hierarchy of Anyon Models Realised by Twists in Stacked Surface Codes

Braiding defects in topological stabiliser codes can be used to fault-tolerantly implement logical operations. Twists are defects corresponding to the end-points of domain walls and are associated with symmetries of the anyon model of the code. We consider twists in multiple copies of the 2d surface code and identify necessary and sufficient conditions for considering these twists as anyons: namely that they must be self-inverse and that all charges which can be localised by the twist must be invariant under its associated symmetry. If both of these conditions are satisfied the twist and its set of localisable anyonic charges reproduce the behaviour of an anyonic model belonging to a hierarchy which generalises the Ising anyons. We show that the braiding of these twists results in either (tensor products of) the S gate or (tensor products of) the CZ gate. We also show that for any number of copies of the 2d surface code the application of H gates within a copy and CNOT gates between copies is sufficient to generate all possible twists.

Neural ensemble decoding for topological quantum error-correcting codes

Topological quantum error-correcting codes are a promising candidate for building fault-tolerant quantum computers. Decoding topological codes optimally, however, is known to be a computationally hard problem. Various decoders have been proposed that achieve approximately optimal error thresholds. Due to practical constraints, it is not known if there exists an obvious choice for a decoder. In this paper, we introduce a framework which can combine arbitrary decoders for any given code to significantly reduce the logical error rates. We rely on the crucial observation that two different decoding techniques, while possibly having similar logical error rates, can perform differently on the same error syndrome. We use machine learning techniques to assign a given error syndrome to the decoder which is likely to decode it correctly. We apply our framework to an ensemble of Minimum-Weight Perfect Matching (MWPM) and Hard-Decision Re-normalization Group (HDRG) decoders for the surface code in the depolarizing noise model. Our simulations show an improvement of 38.4%, 14.6%, and 7.1% over the pseudo-threshold of MWPM in the instance of distance 5, 7, and 9 codes, respectively. Lastly, we discuss the advantages and limitations of our framework and applicability to other error-correcting codes. Our framework can provide a significant boost to error correction by combining the strengths of various decoders. In particular, it may allow for combining very fast decoders with moderate error-correcting capability to create a very fast ensemble decoder with high error-correcting capability.

Analytical percolation theory for topological color codes under qubit loss

Quantum information theory has shown strong connections with classical statistical physics. For example, quantum error correcting codes like the surface and the color code present a tolerance to qubit loss that is related to the classical percolation threshold of the lattices where the codes are defined. Here we explore such connection to study analytically the tolerance of the color code when the protocol introduced in [Phys. Rev. Lett. $\textbf{121}$, 060501 (2018)] to correct qubit losses is applied. This protocol is based on the removal of the lost qubit from the code, a neighboring qubit, and the lattice edges where these two qubits reside. We first obtain analytically the average fraction of edges $ r(p) $ that the protocol erases from the lattice to correct a fraction $ p $ of qubit losses. Then, the threshold $ p_c $ below which the logical information is protected corresponds to the value of $ p $ at which $ r(p) $ equals the bond-percolation threshold of the lattice. Moreover, we prove that the logical information is protected if and only if the set of lost qubits does not include the entire support of any logical operator. The results presented here open a route to an analytical understanding of the effects of qubit losses in topological quantum error codes.