To improve the efficiency of the encoding and the decoding is the important problem in the quantum error correction. In a preceding work, a general algorithm for decoding the stabilizer code is shown. This paper will show an decoding which is more efficient for some codes. The proposed decoding as well as the conventional decoding consists of the eigenvalue output step and the entanglement dissolution step. The proposed decoding outputs a part of the eigenvalues into a part of the code qubits in contrast to the conventional method's outputting into the ancilla. Besides, the proposed decoding dissolves a part of the entanglement in the eigenvalue output step in contrast to the conventional method which does not dissolve in the eigenvalue output step. With these improvements, the number of gates was reduced for some codes.
Though the theory of quantum error correction is intimately related to the classical coding theory, in particular, one can construct quantum error correction codes (QECCs) from classical codes with the dual containing property, this does not necessarily imply that the computational complexity of decoding QECCs is the same as their classical counterparts. Instead, decoding QECCs can be very much different from decoding classical codes due to the degeneracy property. Intuitively, one expect degeneracy would simplify the decoding since two different errors might not and need not be distinguished …
We analyze the four dimensional toric code in a hyperbolic space and show that it has a classical error correction procedure which runs in almost linear time and can be parallelized to almost constant time, giving an example of a quantum LDPC code with linear rate and efficient error correction.
In this paper we investigate the role of local information in the decoding of the repetition and surface error correction codes for the protection of quantum states. Our key result is an improvement in resource efficiency when local information is taken into account during the decoding process: the code distance associated with a given logical error rate is reduced with a magnitude depending on the proximity of the physical error rate to the accuracy threshold of the code. We also briefly discuss an averaged approach with local information for table-lookup and localised decoding schemes, an ex…
Fault tolerant quantum computers will require efficient co-processors for real-time decoding of their adopted quantum error correction protocols. In this work we examine the possibility of using specialised Ising model hardware to perform this decoding task. Examples of Ising model hardware include quantum annealers such as those produced by D-Wave Systems Inc., as well as classical devices such as those produced by Hitatchi and Fujitsu and optical devices known as coherent Ising machines. We use the coherent parity check (CPC) framework to derive an Ising model mapping of the quantum error co…
Simplified circuit-level decoding using Knill error correction
Inria Paris（巴黎国家信息与自动化研究所）；Quandela（夸当德拉）；Institute for Quantum Computing, University of Waterloo（滑铁卢大学量子计算研究所）；Perimeter Institute for Theoretical Physics（理论物理前沿研究所）
Errors in surface code have typically been decoded by Minimum Weight Perfect Matching (MWPM) based method. Recently, neural-network-based Machine Learning (ML) techniques have been employed for this purpose. Here we propose a two-level (low and high) ML-based decoding scheme, where the first level corrects errors on physical qubits and the second one corrects any existing logical errors, for different noise models. Our results show that our proposed decoding method achieves $\sim10 \times$ and $\sim2 \times$ higher values of pseudo-threshold and threshold respectively, than for MWPM. We show t…
Calibration-Conditioned FiLM Decoders for Low-Latency Decoding of Quantum Error Correction Evaluated on IBM Repetition-Code Experiments
Pacific Northwest National Laboratory（太平洋西北国家实验室）；Fordham University（福特汉姆大学）；Lehigh University（勒海大学）；University of Washington（华盛顿大学）；University of California San Diego（加州大学圣地亚哥分校）
Quantum Error Correction (QEC) is required in quantum computers to mitigate the effect of errors on physical qubits. When adopting a QEC scheme based on surface codes, error decoding is the most computationally expensive task in the classical electronic back-end. Decoders employing neural networks (NN) are well-suited for this task but their hardware implementation has not been presented yet. This work presents a space exploration of fully-connected feed-forward NN decoders for small distance surface codes. The goal is to optimize the neural network for high decoding performance, while keeping…
Surface codes exploit topological protection to increase error resilience in quantum computing devices and can in principle be implemented in existing hardware. They are one of the most promising candidates for active error correction, not least due to a polynomial-time decoding algorithm which admits one of the highest predicted error thresholds. We consider the dependence of this threshold on underlying assumptions including different noise models, and analyze the performance of a minimum weight perfect matching (MWPM) decoding compared to a mathematically optimal maximum likelihood (ML) dec…
The overheads of classical decoding for quantum error correction on superconducting quantum systems grow rapidly with the number of logical qubits and their correction code distance. Decoding at room temperature is bottle-necked by refrigerator I/O bandwidth while cryogenic on-chip decoding is limited by area/power/thermal budget. To overcome these overheads, we are motivated by the observation that in the common case, error signatures are fairly trivial with high redundancy/sparsity, since the error correction codes are over-provisioned to correct for uncommon worst-case complex scenarios (to…
Error correction is a significant step in postprocessing of continuous-variable quantum key distribution system, which is used to make two distant legitimate parties share identical corrected keys. We propose an experiment demonstration of high speed error correction with multi-edge type low-density parity check (MET-LDPC) codes based on graphic processing unit (GPU). GPU supports to calculate the messages of MET-LDPC codes simultaneously and decode multiple codewords in parallel. We optimize the memory structure of parity check matrix and the belief propagation decoding algorithm to reduce co…
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…
Although qubit coherence times and gate fidelities are continuously improving, logical encoding is essential to achieve fault tolerance in quantum computing. In most encoding schemes, correcting or tracking errors throughout the computation is necessary to implement a universal gate set without adding significant delays in the processor. Here we realize a classical control architecture for the fast extraction of errors based on multiple cycles of stabilizer measurements and subsequent correction. We demonstrate its application on a minimal bit-flip code with five transmon qubits, showing that …
In this paper, we consider quantum error correction over depolarizing channels with non-binary low-density parity-check codes defined over Galois field of size $2^p$ . The proposed quantum error correcting codes are based on the binary quasi-cyclic CSS (Calderbank, Shor and Steane) codes. The resulting quantum codes outperform the best known quantum codes and surpass the performance limit of the bounded distance decoder. By increasing the size of the underlying Galois field, i.e., $2^p$, the error floors are considerably improved.
Affine Subcode Ensemble Decoding for Degeneracy-Aware Quantum Error Correction
Communications Engineering Lab (CEL), Karlsruhe Institute of Technology (KIT)（卡尔斯鲁厄理工学院通信工程实验室）；School of Informatics, University of Edinburgh（爱丁堡大学信息学院）
QAdapt: A Noise-Adaptive Neural Pre-Decoding Framework for Quantum Error Correction
Beijing Zhongke Qhub Technology Co., Ltd.（北京中科瓴湖科技有限公司）；Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University（清华大学交叉信息研究院量子信息中心）；Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）
Evolutionary BP+OSD Decoding for Low-Latency Quantum Error Correction
Department of Electrical, Electronic and Computer Engineering, University of Ulsan（电气电子与计算机工程系，全州大学）；School of Electrical Engineering, Korea Advanced Institute of Science and Technology (KAIST)（电气工程学院，韩国科学技术院）；Department of Electronic Engineering, Gyeongsang National University（电子工程系，全州国立大学）
We establish several relations between quantum error correction (QEC) and tensor network (TN) methods of quantum many-body physics. We exhibit correspondences between well-known families of QEC codes and TNs, and demonstrate a formal equivalence between decoding a QEC code and contracting a TN. We build on this equivalence to propose a new family of quantum codes and decoding algorithms that generalize and improve upon quantum polar codes and successive cancellation decoding in a natural way.
Fracton topological phases have a large number of materialized symmetries that enforce a rigid structure on their excitations. Remarkably, we find that the symmetries of a quantum error-correcting code based on a fracton phase enable us to design decoding algorithms. Here we propose and implement decoding algorithms for the three-dimensional X-cube model. In our example, decoding is parallelized into a series of two-dimensional matching problems, thus significantly simplifying the most time consuming component of the decoder. We also find that the rigid structure of its point excitations enabl…
Quantum errors are primarily detected and corrected using the measurement of syndrome information which itself is an unreliable step in practical error correction implementations. Typically, such faulty or noisy syndrome measurements are modeled as a binary measurement outcome flipped with some probability. However, the measured syndrome is in fact a discretized value of the continuous voltage or current values obtained in the physical implementation of the syndrome extraction. In this paper, we use this "soft" or analog information without the conventional discretization step to benefit the i…
The Decodoku project seeks to let users get hands-on with cutting-edge quantum research through a set of simple puzzle games. The design of these games is explicitly based on the problem of decoding qudit variants of surface codes. This problem is presented such that it can be tackled by players with no prior knowledge of quantum information theory, or any other high-level physics or mathematics. Methods devised by the players to solve the puzzles can then directly be incorporated into decoding algorithms for quantum computation. In this paper we give a brief overview of the novel decoding met…
Maximum Likelihood Decoding of Quantum Error Correction Codes
Science, Mathematics and Technology Cluster, Singapore University of Technology and Design（新加坡科技设计大学科技、数学与技术集群）；College of Computing and Data Science, Nanyang Technological University（南洋理工大学计算机与数据科学学院）；School of Physical and Mathematical Sciences, Nanyang Technological University（南洋理工大学物理与数学科学学院）
Belief propagation is a powerful tool in statistical physics, machine learning, and modern coding theory. As a decoding method, it is ubiquitous in classical error correction and has also been applied to stabilizer-based quantum error correction. The algorithm works by passing messages between nodes of the factor graph associated with the code and enables efficient decoding, in some cases even up to the Shannon capacity of the channel. Here we construct a belief propagation algorithm which passes quantum messages on the factor graph and is capable of decoding the classical-quantum channel with…
Rethink the Role of Neural Decoders in Quantum Error Correction
College of Computing；Data Science, Nanyang Technological University, Singapore 639798, Singapore；Department of Electrical Engineering；Computer Science, Tokyo University of Agriculture \& Technology, Koganei, Tokyo, 184-8588, Japan；School of Physical；Mathematical Sciences, Nanyang Technological University, Singapore 639798, Singapore
