We analyze the performance of decoders for the 2D and 4D toric code which are local by construction. The 2D decoder is a cellular automaton decoder formulated by Harrington which explicitly has a finite speed of communication and computation. For a model of independent $X$ and $Z$ errors and faulty syndrome measurements with identical probability we report a threshold of $0.133\%$ for this Harrington decoder. We implement a decoder for the 4D toric code which is based on a decoder by Hastings arXiv:1312.2546 . Incorporating a method for handling faulty syndromes we estimate a threshold of $1.59\%$ for the same noise model as in the 2D case. We compare the performance of this decoder with a decoder based on a 4D version of Toom's cellular automaton rule as well as the decoding method suggested by Dennis et al. arXiv:quant-ph/0110143 .
We introduce an efficient decoder of the color code in $d\geq 2$ dimensions, the Restriction Decoder, which uses any $d$-dimensional toric code decoder combined with a local lifting procedure to find a recovery operation. We prove that the Restriction Decoder successfully corrects errors in the color code if and only if the corresponding toric code decoding succeeds. We also numerically estimate the Restriction Decoder threshold for the color code in two and three dimensions against the bit-flip and phase-flip noise with perfect syndrome extraction. We report that the 2D color code threshold $…
Surface codes are a promising method of quantum error correction and the basis of many proposed quantum computation implementations. However, their efficient decoding is still not fully explored. Recently, approaches based on machine learning techniques have been proposed by Torlai and Melko as well as Varsamopoulos et al. In these approaches, a so called high level decoder is used to post-correct an underlying decoder by correcting logical errors. A significant problem is that these methods require large amounts of training data even for relatively small code distances. The above-mentioned me…
Qudit toric codes are a natural higher-dimensional generalization of the well-studied qubit toric code. However standard methods for error correction of the qubit toric code are not applicable to them. Novel decoders are needed. In this paper we introduce two renormalization group decoders for qudit codes and analyze their error correction thresholds and efficiency. The first decoder is a generalization of a "hard-decisions" decoder due to Bravyi and Haah [arXiv:1112.3252]. We modify this decoder to overcome a percolation effect which limits its threshold performance for high dimensions. The s…
We describe a computationally-efficient heuristic algorithm based on a renormalization-group procedure which aims at solving the problem of finding minimal surface given its boundary (curve) in any hypercubic lattice of dimension $D>2$. We use this algorithm to correct errors occurring in a four-dimensional variant of the toric code, having open as opposed to periodic boundaries. For a phenomenological error model which includes measurement errors we use a five-dimensional version of our algorithm, achieving a threshold of $4.35\pm0.1\%$. For this error model, this is the highest known thresho…
Quantum error correction requires decoders that are both accurate and efficient. To this end, union-find decoding has emerged as a promising candidate for error correction on the surface code. In this work, we benchmark a weighted variant of the union-find decoder on the toric code under circuit-level depolarizing noise. This variant preserves the almost-linear time complexity of the original while significantly increasing the performance in the fault-tolerance setting. In this noise model, weighting the union-find decoder increases the threshold from 0.38% to 0.62%, compared to an increase fr…
QuTech, Delft University of Technology（代尔夫特理工大学 QuTech）
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 provid…
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 depolari…
Local decoder for the toric code via signal exchange
Laboratoire de Physique de l’Ecole normale supérieure, ENS-PSL, CNRS, Inria, Centre Automatique et Systèmes (CAS), Mines Paris, Université PSL, Sorbonne Université, Université Paris Cité（巴黎高等师范学院物理实验室、ENS-PSL、法国国家科学研究中心、Inria、自动与系统中心（CAS）、巴黎矿业学院、巴黎文理研究大学、索邦大学、巴黎西岱大学）
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, gaug…
Toric codes and color codes are two important classes of topological codes. Kubica, Yoshida, and Pastawski showed that any $D$-dimensional color code can be mapped to a finite number of toric codes in $D$-dimensions. In this paper we propose an alternate map of 3D color codes to 3D toric codes with a view to decoding 3D color codes. Our approach builds on Delfosse's result for 2D color codes and exploits the topological properties of these codes. Our result reduces the decoding of 3D color codes to that of 3D toric codes. Bit flip errors are decoded by projecting on one set of 3D toric codes w…
We study measurement-based quantum computation (MQC) using as quantum resource the planar code state on a two-dimensional square lattice (planar analogue of the toric code). It is shown that MQC with the planar code state can be efficiently simulated on a classical computer if at each step of MQC the sets of measured and unmeasured qubits correspond to connected subsets of the lattice.
We analyze the properties of a 2D topological code derived by concatenating the [[4, 2, 2]] code with the toric/surface code, or alternatively by removing check operators from the 2D square-octagon or 4.8.8 color code. We show that the resulting code has a circuit-based noise threshold of $\sim 0.41\%$ (compared to $\sim 0.6\%$ for the toric code in a similar scenario), which is higher than any known 2D color code. We believe that the construction may be of interest for hardware in which one wants to use both long-range two-qubit gates as well as short-range gates between small clusters of qub…
A decade ago Kitaev's toric code model established the new paradigm of topological quantum computation. Due to remarkable theoretical and experimental progress, the quantum simulation of such complex many-body systems is now within the realms of possibility. Here we consider the question, to which extent the ground states of small toric code systems differ from LU-equivalent graph states. We argue that simplistic (though experimentally attractive) setups obliterate the differences between the toric code and equivalent graph states; hence we search for the smallest setups on the square- and tri…
We apply deep reinforcement learning techniques to design high threshold decoders for the toric code under uncorrelated noise. By rewarding the agent only if the decoding procedure preserves the logical states of the toric code, and using deep convolutional networks for the training phase of the agent, we observe near-optimal performance for uncorrelated noise around the theoretically optimal threshold of 11%. We observe that, by and large, the agent implements a policy similar to that of minimum weight perfect matchings even though no bias towards any policy is given a priori.
The toric code is a simple and exactly solvable example of topological order realising Abelian anyons. However, it was shown to support non-local lattice defects, namely twists, which exhibit non-Abelian anyonic behaviour [1]. Motivated by this result, we investigated the potential of having non-Abelian statistics from puncture defects on the toric code. We demonstrate that an encoding with mixed-boundary punctures reproduces Ising fusion, and a logical Pauli-$X$ upon their braiding. Our construction paves the way for local lattice defects to exhibit non-Abelian properties that can be employed…
The surface code scheme for quantum computation features a 2d array of nearest-neighbor coupled qubits yet claims a threshold error rate approaching 1% (NJoP 9:199, 2007). This result was obtained for the toric code, from which the surface code is derived, and surpasses all other known codes restricted to 2d nearest-neighbor architectures by several orders of magnitude. We describe in detail an error correction procedure for the toric and surface codes, which is based on polynomial-time graph matching techniques and is efficiently implementable as the classical feed-forward processing step in …
We propose a new cellular automaton (CA), the Sweep Rule, which generalizes Toom's rule to any locally Euclidean lattice. We use the Sweep Rule to design a local decoder for the toric code in $d\geq 3$ dimensions, the Sweep Decoder, and rigorously establish a lower bound on its performance. We also numerically estimate the Sweep Decoder threshold for the three-dimensional toric code on the cubic and body-centered cubic lattices for phenomenological phase-flip noise. Our results lead to new CA decoders with provable error-correction thresholds for other topological quantum codes including the c…
Quantum computers are predicted to utilize quantum states to perform memory and to process tasks far faster than those of conventional classical computers. In this paper we show a new road towards building fault tolerance quantum computer by tuning quantum tunneling effect of the degenerate quantum states in topological order, instead of by braiding anyons. Using a designer Hamiltonian - the Wen-Plaquette model as an example, we study its quantum tunneling effect of the toric codes and show how to control the toric code to realize topological quantum computation (TQC). In particular, we give a…
We examine the performance of the single-mode GKP code and its concatenation with the toric code for a noise model of Gaussian shifts, or displacement errors. We show how one can optimize the tracking of errors in repeated noisy error correction for the GKP code. We do this by examining the maximum-likelihood problem for this setting and its mapping onto a 1D Euclidean path-integral modeling a particle in a random cosine potential. We demonstrate the efficiency of a minimum-energy decoding strategy as a proxy for the path integral evaluation. In the second part of this paper, we analyze and nu…
We consider the problem of preparing specific encoded resource states for the toric code by local, time-independent interactions with a memoryless environment. We propose a construction of such a dissipative encoder which converts product states to topologically ordered ones while preserving logical information. The corresponding Liouvillian is made up of four-local Lindblad operators. For a qubit lattice of size $L\times L$, we show that this process prepares encoded states in time $O(L)$, which is optimal. This scaling compares favorably with known local unitary encoders for the toric code w…
Given their potential for fault-tolerant operations, topological quantum states are currently the focus of intense activity. Of particular interest are topological quantum error correction codes, such as the surface and planar stabilizer codes that are equivalent to the celebrated toric code. While every stabilizer state maps to a graph state under local Clifford operations, the graphs associated with topological stabilizer codes remain unknown. We show that the toric code graph is composed of only two kinds of subgraphs: star graphs (which encode Greenberger-Horne-Zeilinger states) and half g…
Analytically Exact Quantum Simulation of N-Body Interactions via Untunable Decentralized Hamiltonians for Implementing the Toric Code and Its Modifications
提出一种解析精确的量子模拟方法，通过迭代附加二体相互作用模拟 N 体相互作用，并应用于环面码及其变体的模拟，解决拓扑量子计算中的关键障碍。
The design of a four-dimensional toric code is explored with the goal of finding a lattice capable of implementing a logical $\mathsf{CCCZ}$ gate transversally. The established lattice is the octaplex tessellation, which is a regular tessellation of four-dimensional Euclidean space whose underlying 4-cell is the octaplex, or hyper-diamond. This differs from the conventional 4D toric code lattice, based on the hypercubic tessellation, which is symmetric with respect to logical $X$ and $Z$ and only allows for the implementation of a transversal Clifford gate. This work further develops the estab…
We suggest several techniques to improve the toric codes and the finite-rate generalized toric codes (quantum hypergraph-product codes) recently introduced by Tillich and Zémor. For the usual toric codes, we introduce the rotated lattices specified by two integer-valued periodicity vectors. These codes include the checkerboard codes, and the family of minimal single-qubit-encoding toric codes with block length $n=t^2+(t+1)^2$ and distance $d=2t+1$, $t=1,2,...$. We also suggest several related algebraic constructions which nearly quadruple the rate of the existing hypergraph-product codes.
Yale University（耶鲁大学）；Yale Quantum Institute, Yale University（耶鲁量子研究所，耶鲁大学）；IBM Quantum, T. J. Watson Research Center（IBM 量子，T.J.沃森研究中心）；School of Physics, University of Sydney（悉尼大学物理学院）；IBM Quantum, IBM Almaden Research Center（IBM 量子，IBM阿尔马登研究中心）；IBM Denmark（IBM丹麦）
Neural networks can efficiently encode the probability distribution of errors in an error correcting code. Moreover, these distributions can be conditioned on the syndromes of the corresponding errors. This paves a path forward for a decoder that employs a neural network to calculate the conditional distribution, then sample from the distribution - the sample will be the predicted error for the given syndrome. We present an implementation of such an algorithm that can be applied to any stabilizer code. Testing it on the toric code, it has higher threshold than a number of known decoders thanks…
The topological color code and the toric code are two leading candidates for realizing fault-tolerant quantum computation. Here we show that the color code on a $d$-dimensional closed manifold is equivalent to multiple decoupled copies of the $d$-dimensional toric code up to local unitary transformations and adding or removing ancilla qubits. Our result not only generalizes the proven equivalence for $d=2$, but also provides an explicit recipe of how to decouple independent components of the color code, highlighting the importance of colorability in the construction of the code. Moreover, for …
PhD thesis investigating homological quantum codes derived from curved and higher dimensional geometries. In the first part we will consider closed surfaces with constant negative curvature. We show how such surfaces can be constructed and enumerate all quantum codes derived from them which have less than 10.000 physical qubits. For codes that are extremal in a certain sense we perform numerical simulations to determine the value of their threshold. Furthermore, we give evidence that these codes can be used for more overhead efficient storage as compared to the surface code by orders of magnit…
We demonstrate that multipartite entanglement, witnessed by the quantum Fisher information (QFI), can characterize topological quantum phase transitions in the spin-$\frac{1}{2}$ toric code model on a square lattice with external fields. We show that the QFI density of the ground state can be written in terms of the expectation values of gauge-invariant Wilson loops for different sizes of square regions and identify $\mathbb{Z}_2$ topological order by its scaling behavior. Furthermore, we use this multipartite entanglement witness to investigate thermalization and disorder-assisted stabilizati…
We explore the possibility of passive error correction in the toric code model. We first show that even coherent dynamics, stemming from spin interactions or the coupling to an external magnetic field, lead to logical errors. We then argue that Anderson localization of the defects, arising from unavoidable fluctuations of the coupling constants, provides a remedy. This protection is demonstrated using general analytical arguments that are complemented with numerical results which demonstrate that self-correcting memory can in principle be achieved in the limit of a nonzero density of identical…
Mixed-state Quantum Phases: Renormalization and Quantum Error Correction
Perimeter Institute for Theoretical Physics（Perimeter理论物理研究所）；Department of Physics and Astronomy, University of Waterloo（滑铁卢大学物理与天文系）；Kavli Institute for Theoretical Physics, University of California, Santa Barbara（加州大学圣塔芭芭拉分校卡弗里理论物理研究所）；Stanford Institute for Theoretical Physics（斯坦福理论物理研究所）
We propose an error correction procedure based on a cellular automaton, the sweep rule, which is applicable to a broad range of codes beyond topological quantum codes. For simplicity, however, we focus on the three-dimensional (3D) toric code on the rhombic dodecahedral lattice with boundaries and prove that the resulting local decoder has a non-zero error threshold. We also numerically benchmark the performance of the decoder in the setting with measurement errors using various noise models. We find that this error correction procedure is remarkably robust against measurement errors and is al…
In some quantum computing (QC) architectures, entanglement of an arbitrary number of qubits can be generated in a single operation. This property has many potential applications, and may specifically be useful for quantum error correction (QEC). Stabilizer measurements can then be implemented using a single multi-qubit gate instead of several two-qubit gates, thus reducing circuit depth. In this study, the toric code is used as a benchmark to compare the performance of two-qubit and five-qubit gates within parity-check circuits. We consider trapped ion qubits that are controlled via Raman tran…
Asymmetric quantum error-correcting codes are quantum codes defined over biased quantum channels: qubit-flip and phase-shift errors may have equal or different probabilities. The code construction is the Calderbank-Shor-Steane construction based on two linear codes. We present families of toric surfaces, toric codes and associated asymmetric quantum error-correcting codes.
To date, a great deal of attention has focused on characterizing the performance of quantum error correcting codes via their thresholds, the maximum correctable physical error rate for a given noise model and decoding strategy. Practical quantum computers will necessarily operate below these thresholds meaning that other performance indicators become important. In this work we consider the scaling of the logical error rate of the toric code and demonstrate how, in turn, this may be used to calculate a key performance indicator. We use a perfect matching decoding algorithm to find the scaling o…
In this paper we study the symmetry protected Majorana edge states for the Z2 topological order of the Wen-plaquette model and the toric-code model and calculate the dispersion of the Majorana edge states. For the system with translational symmetry, the Majorana edge states are gapless and have the nodal points at k=0 and k=pi. For the edge states of the toric-code model without translational symmetry, the edge modes become gapped.
We study Hamiltonians which have Kitaev's toric code as a ground state, and show how to construct a Hamiltonian which shares the ground space of the toric code, but which has gapless excitations with a continuous spectrum in the thermodynamic limit. Our construction is based on the framework of Projected Entangled Pair States (PEPS), and can be applied to a large class of two-dimensional systems to obtain gapless "uncle Hamiltonians".
We present an algorithm for error correction in topological codes that exploits modern machine learning techniques. Our decoder is constructed from a stochastic neural network called a Boltzmann machine, of the type extensively used in deep learning. We provide a general prescription for the training of the network and a decoding strategy that is applicable to a wide variety of stabilizer codes with very little specialization. We demonstrate the neural decoder numerically on the well-known two dimensional toric code with phase-flip errors.
The recent years have seen a growing interest in quantum codes in three dimensions (3D). One of the earliest proposed 3D quantum codes is the 3D toric code. It has been shown that 3D color codes can be mapped to 3D toric codes. The 3D toric code on cubic lattice is also a building block for the welded code which has highest energy barrier to date. Although well known, the performance of the 3D toric code has not been studied extensively. In this paper, we propose efficient decoding algorithms for the 3D toric code on a cubic lattice with and without boundaries and report their performance for …
We compute the error threshold for the semion code, the companion of the Kitaev toric code with the same gauge symmetry group $\mathbb{Z}_2$. The application of statistical mechanical mapping methods is highly discouraged for the semion code, since the code is non-Pauli and non-CSS. Thus, we use machine learning methods, taking advantage of the near-optimal performance of some neural network decoders: multilayer perceptrons and convolutional neural networks (CNNs). We find the values $p_{\text {eff}}=9.5\%$ for uncorrelated bit-flip and phase-flip noise, and $p_{\text {eff}}=10.5\%$ for depola…
