Daniel Gottesman
I develop methods for analyzing quantum error-correcting codes, and use these methods to construct an infinite class of codes saturating the quantum Hamming bound. These codes encode $k=n-j-2$ qubits in $n=2^j$ qubits and correct $t=1$ error.
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Entanglement-assisted Quantum Error Correcting Code Saturating The Classical Singleton Bound
提出一种纠缠辅助量子纠错码构造方法，在低码率下以更少共享纠缠饱和经典Singleton界，并证明任意经典码可转化为EAQECC，同时分析了编码协议复杂度。
We report two analytical bounds for quantum error-correcting codes that do not have preexisting classical counterparts. Firstly the quantum Hamming and Singleton bounds are combined into a single tighter bound, and then the combined bound is further strengthened via the well-known Lloyd's theorem in classical coding theory, which claims that perfect codes, codes attaining the Hamming bound, do not exist if the Lloyd's polynomial has some non-integer zeros. Our bound characterizes quantitatively the improvement over the Hamming bound via the non-integerness of the zeros of the Lloyd's polynomia…
Proving the quantum Hamming bound for degenerate nonbinary stabilizer codes has been an open problem for a decade. In this note, I prove this bound for double error-correcting degenerate stabilizer codes. Also, I compute the maximum length of single and double error-correcting MDS stabilizer codes over finite fields.
A famous open problem in the theory of quantum error-correcting codes is whether or not the parameters of an impure quantum code can violate the quantum Hamming bound for pure quantum codes. We partially solve this problem. We demonstrate that there exists a threshold such that an arbitrary quantum code must obey the quantum Hamming bound whenever . We list some values of for small d and binary quantum codes.
In a recent study [Rohde et al., quant-ph/0603130 (2006)] of several quantum error correcting protocols designed for tolerance against qubit loss, it was shown that these protocols have the undesirable effect of magnifying the effects of depolarization noise. This raises the question of which general properties of quantum error-correcting codes might explain such an apparent trade-off between tolerance to located and unlocated error types. We extend the counting argument behind the well-known quantum Hamming bound to derive a bound on the weights of combinations of located and unlocated errors…
A Hamming-Like Bound for Degenerate Stabilizer Codes
本文提出一个比量子汉明界更严格的类汉明界，证明其适用于长度大于某正整数的退化t纠错稳定子码，并由此迫使绝大多数最优距离-3稳定子码为非退化码。
We provide a systematic way of constructing entanglement-assisted quantum error-correcting codes via graph states in the scenario of preexisting perfectly protected qubits. It turns out that the preexisting entanglement can help beat the quantum Hamming bound and can enhance (not only behave as an assistance) the performance of the quantum error correction. Furthermore we generalize the error models to the case of not-so-perfectly-protected qubits and introduce the quantity infidelity as a figure of merit and show that our code outperforms also the ordinary quantum error-correcting codes.
Graph states are generalized from qubits to collections of $n$ qudits of arbitrary dimension $D$, and simple graphical methods are used to construct both additive and nonadditive quantum error correcting codes. Codes of distance 2 saturating the quantum Singleton bound for arbitrarily large $n$ and $D$ are constructed using simple graphs, except when $n$ is odd and $D$ is even. Computer searches have produced a number of codes with distances 3 and 4, some previously known and some new. The concept of a stabilizer is extended to general $D$, and shown to provide a dual representation of an addi…
No ((n, k, d < 127)) code can violate the quantum Hamming bound
本文结合 Rains 的解析界与 Li 和 Xing 的数值界，证明任何 ((n, k, d < 127)) 码都不能违反量子汉明界，从而回答了非纯量子纠错码是否受该界约束的长期问题。
Subsystem codes are a generalization of noiseless subsystems, decoherence free subspaces, and quantum error-correcting codes. We prove a Singleton bound for GF(q)-linear subsystem codes. It follows that no subsystem code over a prime field can beat the Singleton bound. On the other hand, we show the remarkable fact that there exist impure subsystem codes beating the Hamming bound. A number of open problems concern the comparison in performance of stabilizer and subsystem codes. One of the open problems suggested by Poulin's work asks whether a subsystem code can use fewer syndrome measurements…
Quantum memory based on concatenating surface codes and quantum Hamming codes
本文研究了表面码与量子汉明码的拼接作为量子记忆，实现了较高的错误阈值和较低的资源开销，展示了在中等规模量子记忆中抑制逻辑错误的优势。
The Quantum Hamming Bound in Arbitrary Local Dimension
本文证明任意局域维度下精确量子纠错码的有限长量子汉明界，通过Lloyd平方线性规划与系数证书归约处理简并问题。
Haar random codes attain the quantum Hamming bound, approximately
研究 Haar 随机码的纠错性能，证明其可近似纠正错误直至量子汉明界（$mK \ll N$），为已知最强界，并基于矩阵浓度结果给出证明。
An error avoiding quantum code is presented which is capable of stabilizing Grover's quantum search algorithm against a particular class of coherent errors. This error avoiding code consists of states only which are factorizable in the computational basis. Furthermore, its redundancy is smaller than the one which is achievable with a general error correcting quantum code saturating the quantum Hamming bound. The fact that this code consists of factorizable states only may offer advantages for the implementation of quantum gates in the error free subspace.
One of the main problems in quantum information systems is the presence of errors due to noise, and for this reason quantum error-correcting codes (QECCs) play a key role. While most of the known codes are designed for correcting generic errors, i.e., errors represented by arbitrary combinations of Pauli X , Y and Z operators, in this paper we investigate the design of stabilizer QECC able to correct a given number eg of generic Pauli errors, plus eZ Pauli errors of a specified type, e.g., Z errors. These codes can be of interest when the quantum channel is asymmetric in that some types of err…
We present a construction scheme for quantum error correcting codes. The basic ingredients are a graph and a finite abelian group, from which the code can explicitly be obtained. We prove necessary and sufficient conditions for the graph such that the resulting code corrects a certain number of errors. This allows a simple verification of the 1-error correcting property of fivefold codes in any dimension. As new examples we construct a large class of codes saturating the singleton bound, as well as a tenfold code detecting 3 errors.
We explore the design of quantum error-correcting codes for cases where the decoherence events of qubits are correlated. In particular, we consider the case where only spatially contiguous qubits decohere, which is analogous to the case of burst errors in classical coding theory. We present several different efficient schemes for constructing families of such codes. For example, one can find one-dimensional quantum codes of length n=13 and 15 that correct burst errors of width b < 4; as a comparison, a random-error correcting quantum code that corrects t=3 errors must have length n > 18. In ge…
Quantum Approximation Optimization Algorithm for the Trellis based Viterbi Decoding of Classical Error Correcting Codes
本文提出基于量子近似优化算法的混合解码器，用于高效解决经典纠错码的维特比解码问题，通过统一参数优化策略提升参数化量子电路的优化效率。
Absolutely maximally entangled (AME) states are pure multi-partite generalizations of the bipartite maximally entangled states with the property that all reduced states of at most half the system size are in the maximally mixed state. AME states are of interest for multipartite teleportation and quantum secret sharing and have recently found new applications in the context of high-energy physics in toy models realizing the AdS/CFT-correspondence. We work out in detail the connection between AME states of minimal support and classical maximum distance separable (MDS) error correcting codes and,…
Quantum dual extended Hamming code immune to collective coherent errors
针对量子系统中集体相干（CC）错误主导的问题，构造参数为[[2^{r+1}, 2^r - (r+1), 3]]的恒定激发稳定器码族，最小实例[[8,1,3]]码的码率和错误阈值较现有最佳CE码分别提升约2倍和4倍。
We introduce the concept of generalized concatenated quantum codes. This generalized concatenation method provides a systematical way for constructing good quantum codes, both stabilizer codes and nonadditive codes. Using this method, we construct families of new single-error-correcting nonadditive quantum codes, in both binary and nonbinary cases, which not only outperform any stabilizer codes for finite block length, but also asymptotically achieve the quantum Hamming bound for large block length.
We establish a symmetry-operator framework for designing quantum error correcting~(QEC) codes based on fundamental properties of the underlying system dynamics. Based on this framework, we propose three hardware-efficient bosonic QEC codes that are suitable for $χ^{(2)}$-interaction based quantum computation: the $χ^{(2)}$ parity-check code, the $χ^{(2)}$ embedded error-correcting code, and the $χ^{(2)}$ binomial code, all of which detect photon-loss or photon-gain errors by means of photon-number parity measurements and then correct them via $χ^{(2)}$ Hamiltonian evolutions and linear-optics …
Bounds on concatenated entanglement-assisted quantum error-correcting codes
本文研究了纠缠辅助量子纠错码的界限，推导了拼接码的共享纠缠表达式，并探讨了非最大纠缠EAQECC在格里斯默和普洛特金界限下的饱和条件，同时提出了相关码的构造方法。
Pure multiparticle quantum states are called absolutely maximally entangled if all reduced states obtained by tracing out at least half of the particles are maximally mixed. We provide a method to characterize these states for a general multiparticle system. With that, we prove that a seven-qubit state whose three-body marginals are all maximally mixed, or equivalently, a pure $((7,1,4))_2$ quantum error correcting code, does not exist. Furthermore, we obtain an upper limit on the possible number of maximally mixed three-body marginals and identify the state saturating the bound. This solves t…
A unification of the coding theory and OAQEC perspective on hybrid codes
本文统一了混合量子-经典纠错码的编码理论与OAQEC两种表征视角，推广了量子Hamming界，证明非平凡混合码必须简并，并构造了一个4量子比特简并混合码实例。
Linear Programming Bounds for Entanglement-Assisted Quantum Error-Correcting Codes by Split Weight Enumerators
本文利用分裂重量枚举器推导纠缠辅助量子码的线性规划界，改进了小型量子码最小距离界，证明两类量子稳定器码不存在，优化了短码的距离上下界。
A protocol based on quantum error correction based characterization of quantum dynamics (QECCD) is developed for quantum process tomography on a two-qubit system interacting dissipatively with a vacuum bath. The method uses a 5-qubit quantum error correcting code that corrects arbitrary errors on the first two qubits, and also saturates the quantum Hamming bound. The dissipative interaction with a vacuum bath allows for both correlated and independent noise on the two-qubit system. We study the dependence of the degree of the correlation of the noise on evolution time and inter-qubit separatio…
Game-Theoretic Discovery of Quantum Error-Correcting Codes Through Nash Equilibria
提出博弈论框架，将码优化转化为竞争目标间的策略交互，通过纳什均衡系统生成具有期望性质的量子纠错码，并验证其能重新发现最优的[[15,7,3]]量子汉明码，且可扩展至100量子比特。
Noise rates in quantum computing experiments have dropped dramatically, but reliable qubits remain precious. Fault-tolerance schemes with minimal qubit overhead are therefore essential. We introduce fault-tolerant error-correction procedures that use only two ancilla qubits. The procedures are based on adding "flags" to catch the faults that can lead to correlated errors on the data. They work for various distance-three codes. In particular, our scheme allows one to test the [[5,1,3]] code, the smallest error-correcting code, using only seven qubits total. Our techniques also apply to the [[7,…
Security of One-Way Entanglement Purification with Quantum Sampling Against a Restricted Adversary
针对受限为单量子比特泡利门的对手，提出基于量子采样的单向纠缠纯化协议，利用与纠错码的等价性估计汉明重量并纠错，实现安全纯化。
Entanglement purification takes a number of noisy EPR pairs and processes them to produce a smaller number of more reliable pairs. If this is done with only a forward classical side channel, the procedure is equivalent to using a quantum error-correcting code (QECC). We instead investigate entanglement purification protocols with two-way classical side channels (2-EPPs) for finite block sizes. In particular, we consider the analog of the minimum distance problem for QECCs, and show that 2-EPPs can exceed the quantum Hamming bound and the quantum Singleton bound. We also show that 2-EPPs can ac…
Approximate level-by-level maximum-likelihood decoding based on the Chase algorithm for high-rate concatenated stabilizer codes
本文提出基于Chase算法的近似逐层最大似然解码器，用于高率拼接稳定器码，以提升在比特翻转噪声下的解码性能。
Quantum error correction and symmetries play central roles in quantum information science and physics. It is known that quantum error-correcting codes that obey (are covariant with respect to) continuous symmetries in a certain sense cannot correct erasure errors perfectly (a well-known result in this regard being the Eastin-Knill theorem in the context of fault-tolerant quantum computing), in contrast to the case without symmetry constraints. Furthermore, several quantitative fundamental limits on the accuracy of such covariant codes for approximate quantum error correction are known. Here, w…
Discovery of Optimal Quantum Error Correcting Codes via Reinforcement Learning
本研究将 Quantum Lego 框架游戏化，利用强化学习优化量子纠错码的码距和逻辑错误概率，发现了优于已知基准的最优 CSS 码，并提出了与物理量子设备结合定制纠错码的方案。
Smallest quantum codes for amplitude damping noise
本文研究了最小的量子纠错码以纠正幅值阻尼噪声，提出了一种3量子比特码，并扩展到能纠正任意阶阻尼强度的码家族，同时引入了新的量子纠错条件。
We prove several theorems characterizing the existence of homological error correction codes both classically and quantumly. Not every classical code is homological, but we find a family of classical homological codes saturating the Hamming bound. In the quantum case, we show that for non-orientable surfaces it is impossible to construct homological codes based on qudits of dimension $D>2$, while for orientable surfaces with boundaries it is possible to construct them for arbitrary dimension $D$. We give a method to obtain planar homological codes based on the construction of quantum codes on …
Quantum error correction requires the use of error syndromes derived from measurements that may be unreliable. Recently, quantum data-syndrome (QDS) codes have been proposed as a possible approach to protect against both data and syndrome errors, in which a set of linearly dependent stabilizer measurements are performed to increase redundancy. Motivated by wanting to reduce the total number of measurements performed, we introduce QDS subsystem codes, and show that they can outperform similar QDS stabilizer codes derived from them. We also give a construction of single-error-correcting QDS stab…
Performing active quantum error correction to protect fragile quantum states highly depends on the correctness of error information--error syndromes. To obtain reliable error syndromes using imperfect physical circuits, we propose the idea of quantum data-syndrome (DS) codes that are capable of correcting both data qubits and syndrome bits errors. We study fundamental properties of quantum DS codes, including split weight enumerators, generalized MacWilliams identities, and linear programming bounds. In particular, we derive Singleton and Hamming-type upper bounds on degenerate quantum DS code…
One formidable difficulty in quantum communication and computation is to protect information-carrying quantum states against undesired interactions with the environment. In past years, many good quantum error-correcting codes had been derived as binary stabilizer codes. Fault-tolerant quantum computation prompted the study of nonbinary quantum codes, but the theory of such codes is not as advanced as that of binary quantum codes. This paper describes the basic theory of stabilizer codes over finite fields. The relation between stabilizer codes and general quantum codes is clarified by introduc…
On the Efficacy of Surface Codes in Compensating for Radiation Events in Superconducting Devices
本文研究表面码对超导量子比特辐射故障的补偿效能，通过超4亿次故障注入实验，发现优化表面码配置可提升纠错概率达10%，并为未来QEC码设计提供指导。
We optimize fault-tolerant quantum error correction to reduce the number of syndrome bit measurements. Speeding up error correction will also speed up an encoded quantum computation, and should reduce its effective error rate. We give both code-specific and general methods, using a variety of techniques and in a variety of settings. We design new quantum error-correcting codes specifically for efficient error correction, e.g., allowing single-shot error correction. For codes with multiple logical qubits, we give methods for combining error correction with partial logical measurements. There ar…
SU(d)-Symmetric Random Unitaries: Quantum Scrambling, Error Correction, and Machine Learning
本文研究了SU(d)对称随机酉算子在量子扰乱、纠错和机器学习中的应用，揭示了非阿贝尔对称下残余扰乱的衰减规律，构造了渐近最优的协变码，并推导了几何量子机器学习的过参数化阈值。
Mixed-dimensional quantum MacWilliams identity: Bounds for codes and absolutely maximally entangled states in heterogeneous systems
针对异构量子系统，本文提出基于维度多重集的数学框架，推导混合维量子MacWilliams恒等式，并推导出混合维量子Hamming、Singleton和Scott界，用于分析绝对最大纠缠态的约束。
Fault Attacks on ML-based Quantum Control and Error Correction
本文研究了基于机器学习的量子计算机读取错误校正系统对故障注入攻击的脆弱性，揭示了不同层的故障易受性差异，并强调了在量子系统中加强安全防护的重要性。
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