A Proposed Quantum Low Density Parity Check Code

近-term实验中的小型量子低密度奇偶校验码

Small Quantum Low Density Parity Check Codes for Near-Term Experiments

本文提出了一种高效的量子LDPC码构造方法，相较于表面码效率更高，且实验实现更简单，适用于近期量子实验。

Quantum "hyperbicycle" low-density parity check codes with finite rate

We introduce a "hyperbicycle" ansatz for quantum codes which gives the hypergraph-product (generalized toric) codes by Tillich and Zémor and generalized bicycle codes by MacKay et al. as limiting cases. The construction allows for both the lower and the upper bounds on the minimum distance; they scale as a square root of the block length. Many of thus defined codes have finite rate and a limited-weight stabilizer generators, an analog of classical low-density parity check (LDPC) codes. Compared to the hypergraph-product codes, hyperbicycle codes generally have wider range of parameters; in par…

Fault-Tolerance of "Bad" Quantum Low-Density Parity Check Codes

We discuss error-correction properties for families of quantum low-density parity check (LDPC) codes with relative distance that tends to zero in the limit of large blocklength. In particular, we show that any family of LDPC codes, quantum or classical, where distance scales as a positive power of the block length, $d \propto n^α$, $α>0$, can correct all errors with certainty if the error rate per (qu)bit is sufficiently small. We specifically analyze the case of LDPC version of the quantum hypergraph-product codes recently suggested by Tillich and Zémor. These codes are a finite-rate generali…

Experimental ancilla-assisted qubit transmission against correlated noise using quantum parity checking

We report the experimental demonstration of a transmission scheme of photonic qubits over unstabilized optical fibers, which has the plug-and-play feature as well as the ability to transmit any state of a qubit, regardless of whether it is known, unknown, or entangled to other systems. A high fidelity to the noiseless quantum channel was achieved by adding an ancilla photon after the signal photon within the correlation time of the fiber noise and by performing quantum parity checking. Simplicity, maintenance-free feature and robustness against path-length mismatches among the nodes make our s…

拆解qLDPC码以实现深度最优的奇偶校验电路

Disassembling qLDPC codes for depth-optimal parity-check circuits

该研究针对qLDPC码错配提取电路，利用其Tanner图的边对称性，提出从底层组件而非完整量子码设计电路的方法，在提升积码、平衡积码和量子Tanner码上分别获得最优或近最优、深度最优的CNOT深度结果。

量子近似优化用于低密度奇偶校验码解码

Quantum Approximate Optimization for Decoding of Low-Density Parity-Check Codes

本文提出基于QAOA的LDPC码解码方法，通过构建结合奇偶校验约束和信道可靠性信息的成本函数，提升解码性能。

A Construction of Quantum Stabilizer Codes Based on Syndrome Assignment by Classical Parity-Check Matrices

In quantum coding theory, stabilizer codes are probably the most important class of quantum codes. They are regarded as the quantum analogue of the classical linear codes and the properties of stabilizer codes have been carefully studied in the literature. In this paper, a new but simple construction of stabilizer codes is proposed based on syndrome assignment by classical parity-check matrices. This method reduces the construction of quantum stabilizer codes to the construction of classical parity-check matrices that satisfy a specific commutative condition. The quantum stabilizer codes from …

Practical Entanglement Distillation Scheme Using Recurrence Method And Quantum Low Density Parity Check Codes

Many entanglement distillation schemes use either universal random hashing or breeding as their final step to obtain almost perfect shared EPR pairs. In spite of a high yield, the hardness of decoding a random linear code makes the use of random hashing and breeding infeasible in practice. In this pilot study, we analyze the performance of the recurrence method, a well-known entanglement distillation scheme, with its final random hashing or breeding procedure being replaced by various efficiently decodable quantum codes. Among all the replacements investigated, the one using a certain adaptive…

Entanglement-Assisted Quantum Quasi-Cyclic Low-Density Parity-Check Codes

We investigate the construction of quantum low-density parity-check (LDPC) codes from classical quasi-cyclic (QC) LDPC codes with girth greater than or equal to 6. We have shown that the classical codes in the generalized Calderbank-Shor-Steane (CSS) construction do not need to satisfy the dual-containing property as long as pre-shared entanglement is available to both sender and receiver. We can use this to avoid the many 4-cycles which typically arise in dual-containing LDPC codes. The advantage of such quantum codes comes from the use of efficient decoding algorithms such as sum-product alg…

Fault-tolerant fidelity based on few-qubit codes: Parity-check circuits for biased error channels

In the shallow sub-threshold regime, fault-tolerant quantum computation requires a tremendous amount of qubits. In this paper, we study the error correction in the deep sub-threshold regime. We estimate the physical error rate for achieving the logical error rates of $10^{-6} - 10^{-15}$ using few-qubit codes, i.e. short repetition codes, small surface codes and the Steane code. Error correction circuits that are efficient for biased error channels are identified. Using the Steane code, when error channels are biased with a ratio of $10^{-3}$, the logical error rate of $10^{-15}$ can be achiev…

Modified belief propagation decoders for quantum low-density parity-check codes

Quantum low-density parity-check codes can be decoded using a syndrome based $\mathrm{GF}(4)$ belief propagation decoder. However, the performance of this decoder is limited both by unavoidable $4$-cycles in the code's factor graph and the degenerate nature of quantum errors. For the subclass of CSS codes, the number of $4$-cycles can be reduced by breaking an error into an $X$ and $Z$ component and decoding each with an individual $\mathrm{GF}(2)$ based decoder. However, this comes at the expense of ignoring potential correlations between these two error components. We present a number of mod…

Photonic circuits for iterative decoding of a class of low-density parity-check codes

Photonic circuits in which stateful components are coupled via guided electromagnetic fields are natural candidates for native implementation of iterative stochastic algorithms based on propagation of information around a graph. Conversely, such message passing algorithms suggest novel circuit architectures for signal processing and computation that are well matched to nanophotonic device physics. Here we construct and analyze a quantum optical model of a photonic circuit for iterative decoding of a class of low-density parity-check (LDPC) codes called expander codes. Our circuit can be unders…

最近邻门就够了：平面网格上的高速率量子低密度奇偶校验码

Nearest-neighbour gates are all you need: High-rate quantum low-density parity-check codes on a planar grid

本文提出一类量子LDPC码，首次结合平面开放边界布局、优于表面码的有限尺寸优势，以及仅使用方形网格上的最近邻门进行综合征提取，通过动态生成校验-数据连接实现恒定深度测量，显著降低逻辑错误率。

容错量子电路的低密度奇偶校验表示

Low-density parity-check representation of fault-tolerant quantum circuits

本文提出用经典LDPC码表示稳定子电路以量化容错性并验证逻辑操作，并基于该框架在超图乘积码上构建了资源高效的通用容错量子计算方案。

用量子低密度奇偶校验码的扩散解码

Decoding quantum low density parity check codes with diffusion

本文提出基于扩散模型的量子LDPC码解码器，利用掩码扩散机制从综合征推断逻辑错误，在双变量自行车码上超越BP-OSD和自回归解码器，兼具高精度与速度优势。

A Direct Mapping of Max k-SAT and High Order Parity Checks to a Chimera Graph

We demonstrate a direct mapping of max k-SAT problems (and weighted max k-SAT) to a Chimera graph, which is the non-planar hardware graph of the devices built by D-Wave Systems Inc. We further show that this mapping can be used to map a similar class of maximum satisfiability problems where the clauses are replaced by parity checks over potentially large numbers of bits. The latter is of specific interest for applications in decoding for communication. We discuss an example in which the decoding of a turbo code, which has been demonstrated to perform near the Shannon limit, can be mapped to a …

局域维度不变低密度奇偶校验稳定子码

Local-dimension-invariant low-density parity-check stabilizer codes

针对量子多态系统稳定子码转换中局域维度不变（LDI）方法的大临界值问题，该研究聚焦量子低密度奇偶校验（qLDPC）码，成功降低了临界值，减少了LDI形式距离保持条件的限制。

单步奇偶校验门集用于量子纠错

Single-Step Parity Check Gate Set for Quantum Error Correction

本文提出了一种单步奇偶校验门集，用于简化量子纠错中的错误检测与纠正，提升纠错阈值并降低硬件要求。

多维低密度奇偶校验码

Qudit low-density parity-check codes

本文提出多维低密度奇偶校验码的通用框架，扩展了量子纠错码的应用范围，展示了其在近期硬件中的潜力。

Protecting quantum memories using coherent parity check codes

Coherent parity check (CPC) codes are a new framework for the construction of quantum error correction codes that encode multiple qubits per logical block. CPC codes have a canonical structure involving successive rounds of bit and phase parity checks, supplemented by cross-checks to fix the code distance. In this paper, we provide a detailed introduction to CPC codes using conventional quantum circuit notation. We demonstrate the implementation of a CPC code on real hardware, by designing a [[4,2,2]] detection code for the IBM 5Q superconducting qubit device. Whilst the individual gate-error …

列重为2且围长为12的量子准循环低密度奇偶校验码的显式构造

Explicit Construction of Quantum Quasi-Cyclic Low-Density Parity-Check Codes with Column Weight 2 and Girth 12

本文提出一种列重为2且围长为12的量子QC-LDPC码显式构造方法，利用代数技术消除短环并保持正交性，且可扩展至非二进制和空间耦合LDPC码，无需随机搜索即可设计高性能量子纠错码。

Hardware optimized parity check gates for superconducting surface codes

Error correcting codes use multi-qubit measurements to realize fault-tolerant quantum logic steps. In fact, the resources needed to scale-up fault-tolerant quantum computing hardware are largely set by this task. Tailoring next-generation processors for joint measurements, therefore, could result in improvements to speed, accuracy, or cost -- accelerating the development large-scale quantum computers. Here, we motivate such explorations by analyzing an unconventional surface code based on multi-body interactions between superconducting transmon qubits. Our central consideration, Hardware Optim…

连续变量量子密钥分发中低密度奇偶校验码的提前终止

Early Termination of Low-Density Parity-Check Codes for Continuous-Variable Quantum Key Distribution

研究分析对数后验提前终止对连续变量量子密钥分发协调解码吞吐量的影响，发现其可使解码秘密密钥率吞吐量提升高达182%，为该领域解码效率优化提供了有效方法。

Using arbitrary parity-check matrices for quantum error correction assisted by less noisy qubits

Recently a framework for assisted quantum error correction was proposed in which a specific type of error is allowed to occur on auxiliary qubits, which is in contrast to standard entanglement assistance that requires noiseless auxiliary qubits. However, while the framework maintains the ability to import any binary or quaternary linear code without sacrificing active error correction power, it requires the code designer to turn a parity-check matrix of the underlying classical code into an equivalent one in standard form. This means that classical coding theoretic techniques that require pari…

Information Reconciliation for Continuous-Variable Quantum Key Distribution using Non-Binary Low-Density Parity-Check Codes

An information reconciliation method for continuous-variable quantum key distribution with Gaussian modulation that is based on non-binary low-density parity-check (LDPC) codes is presented. Sets of regular and irregular LDPC codes with different code rates over the Galois fields $GF(8)$, $GF(16)$, $GF(32)$, and $GF(64)$ have been constructed. We have performed simulations to analyze the efficiency and the frame error rate using the sum-product algorithm. The proposed method achieves an efficiency between $0.94$ and $0.98$ if the signal-to-noise ratio is between $4$ dB and $24$ dB.

使用非二进制低密度奇偶校验码高效缓解量子纠错中的错误平底

Efficient Mitigation of Error Floors in Quantum Error Correction using Non-Binary Low-Density Parity-Check Codes

本文提出利用非二进制LDPC码降低量子纠错错误平底的方法，通过识别并分类奇偶校验矩阵中的环结构并开发定制解码策略，显著提升了解码性能。

自对偶堆叠量子低密度奇偶校验码

Self-dual Stacked Quantum Low-Density Parity-Check Codes

本文提出通过堆叠非自对偶qLDPC码构建自对偶码的方法，开发多种新型码并评估其作为量子内存的性能。

量子低密度奇偶校验码的局部统计解码

Localized statistics decoding for quantum low-density parity-check codes

本文提出局部统计解码方法，通过即时消元的并行矩阵分解策略识别并求解局部解码区域，在保持与最先进解码器相当性能的同时降低亚阈值区域运行时间复杂度，且更易于专用硬件实现。

High-throughput GPU layered decoder of multi-edge type low density parity check codes in continuous-variable quantum key distribution systems

The decoding throughput in the postprocessing is one of the bottlenecks for a continuous-variable quantum key distribution (CV-QKD) system. In this paper, we propose a layered decoder to decode quasi-cyclic multi-edge type LDPC (QC-METLDPC) codes based on graphic processing unit (GPU) in continuous-variable quantum key distribution (CV-QKD) systems. We optimize the storage method of the parity check matrix, merge the sub-matrices which are unrelated, and decode multiple codewords in parallel on GPU. Simulation results demonstrate that the average decoding speed of LDPC codes with three typical…

A characterization of entanglement-assisted quantum low-density parity-check codes

As in classical coding theory, quantum analogues of low-density parity-check (LDPC) codes have offered good error correction performance and low decoding complexity by employing the Calderbank-Shor-Steane (CSS) construction. However, special requirements in the quantum setting severely limit the structures such quantum codes can have. While the entanglement-assisted stabilizer formalism overcomes this limitation by exploiting maximally entangled states (ebits), excessive reliance on ebits is a substantial obstacle to implementation. This paper gives necessary and sufficient conditions for the …

简洁的量子低密度奇偶校验码手术

Parsimonious Quantum Low-Density Parity-Check Code Surgery

本研究提出了一种构造qubit大小$O(W \log W)$辅助系统的 方法，用于在任意qLDPC稳定器码中测量任意逻辑Pauli算子，从而降低了量子码手术方案的渐进行耗。

在云端光子处理器上的单光子八模寄存器中嵌套沃尔什奇偶校验滤波器的表征

Characterization of nested Walsh parity-check filters in a single-photon eight-mode register on a cloud photonic processor

在Quandela的Belenos云端光子处理器上，实验实现了基于单光子八模空间寄存器的嵌套沃尔什奇偶校验滤波器，验证了泄漏抑制和综合征路由，并量化了系统误差。

四重奇偶校验的重量四奇偶校验

Weight-four parity checks with silicon spin qubits

该研究展示了一种硅自旋量子比特设备，通过四重奇偶校验实现五量子比特GHZ态的纠缠，为量子纠错实验提供新途径。

Entanglement-assisted quantum low-density parity-check codes

This paper develops a general method for constructing entanglement-assisted quantum low-density parity-check (LDPC) codes, which is based on combinatorial design theory. Explicit constructions are given for entanglement-assisted quantum error-correcting codes (EAQECCs) with many desirable properties. These properties include the requirement of only one initial entanglement bit, high error correction performance, high rates, and low decoding complexity. The proposed method produces infinitely many new codes with a wide variety of parameters and entanglement requirements. Our framework encompass…

Beyond single-shot fault-tolerant quantum error correction

Extensive quantum error correction is necessary in order to perform a useful computation on a noisy quantum computer. Moreover, quantum error correction must be implemented based on imperfect parity check measurements that may return incorrect outcomes or inject additional faults into the qubits. To achieve fault-tolerant error correction, Shor proposed to repeat the sequence of parity check measurements until the same outcome is observed sufficiently many times. Then, one can use this information to perform error correction. A basic implementation of this fault tolerance strategy requires $Ω(…

费米子到费米子低密度奇偶校验码

Fermion-to-Fermion Low-Density Parity-Check Codes

本文提出了一种利用费米子到费米子低密度奇偶校验码进行容错量子计算的协议，通过奇重量逻辑Majorana算子实现高效的错误抑制和状态传输。

High-rate quantum low-density parity-check codes assisted by reliable qubits

Quantum error correction is an important building block for reliable quantum information processing. A challenging hurdle in the theory of quantum error correction is that it is significantly more difficult to design error-correcting codes with desirable properties for quantum information processing than for traditional digital communications and computation. A typical obstacle to constructing a variety of strong quantum error-correcting codes is the complicated restrictions imposed on the structure of a code. Recently, promising solutions to this problem have been proposed in quantum informat…

LDPC for QKD Reconciliation

We present the Low Density Parity Check (LDPC) forward error correction algorithm adapted for the Quantum Key Distribution (QKD) protocol in a form readily applied by developers. A sparse parity check matrix is required for the LDPC algorithm and we suggest using some that have been defined by the IEEE and ETSI standards organizations for use in various communication protocols. We evaluate the QKD performance of these various parity check matrices as a function of the quantum bit error rate. We also discuss the computational precision required for this LPDC algorithm. As QKD evolves towards de…

Spatially Coupled Quasi-Cyclic Quantum LDPC Codes

We face the following dilemma for designing low-density parity-check codes (LDPC) for quantum error correction. 1) The row weights of parity-check should be large: The minimum distances are bounded above by the minimum row weights of parity-check matrices of constituent classical codes. Small minimum distance tends to result in poor decoding performance at the error-floor region. 2) The row weights of parity-check matrices should not be large: The sum-product decoding performance at the water-fall region is degraded as the row weight increases. Recently, Kudekar et al. showed spatially-coupled…

具有平移不变性的量子低密度奇偶校验码来自紧凑化分形码

Translation-invariant quantum low-density parity-check codes from compactified fracton models

本文研究了具有平移对称性和局部检查的量子纠错码，提出了一种统一框架，涵盖了多种平移不变码，包括许多分形码和所有阿贝尔双块群代数（A2BGA）码，包括双变量自行车（BB）码。

低密度奇偶校验码作为量子物质的稳定相

Low-density parity-check codes as stable phases of quantum matter

本文探讨量子纠错码定义稳定有隙量子物质相的条件，证明满足码距对数增长和校验可靠性的LDPC码可定义此类相，且给出经典码相的相关改进，拓展了量子物质相研究。

广义$\mathbb{Z}_p$陶立克码作为四态低密度奇偶校验码

Generalized $\mathbb{Z}_p$ toric codes as qudit low-density parity-check codes

本文提出广义$\mathbb{Z}_p$陶立克码作为四态低密度奇偶校验码，通过增加额外四态并利用格雷勃纳基计算逻辑维度，实现了具有最优有限尺寸性能的码结构。

Family of Finite Geometry Low-Density Parity-Check Codes for Quantum Key Expansion

We consider a quantum key expansion (QKE) protocol based on entanglement-assisted quantum error-correcting codes (EAQECCs). In these protocols, a seed of a previously shared secret key is used in the post-processing stage of a standard quantum key distribution protocol like the Bennett-Brassard 1984 protocol, in order to produce a larger secret key. This protocol was proposed by Luo and Devetak, but codes leading to good performance have not been investigated. We look into a family of EAQECCs generated by classical finite geometry (FG) low-density parity-check (LDPC) codes, for which very effi…

Neural Decoder for Topological Codes using Pseudo-Inverse of Parity Check Matrix

Recent developments in the field of deep learning have motivated many researchers to apply these methods to problems in quantum information. Torlai and Melko first proposed a decoder for surface codes based on neural networks. Since then, many other researchers have applied neural networks to study a variety of problems in the context of decoding. An important development in this regard was due to Varsamopoulos et al. who proposed a two-step decoder using neural networks. Subsequent work of Maskara et al. used the same concept for decoding for various noise models. We propose a similar two-ste…