Quantum error correction and symmetry arise in many areas of physics, including many-body systems, metrology in the presence of noise, fault-tolerant computation, and holographic quantum gravity. Here we study the compatibility of these two important principles. If a logical quantum system is encoded into $n$ physical subsystems, we say that the code is covariant with respect to a symmetry group $G$ if a $G$ transformation on the logical system can be realized by performing transformations on the individual subsystems. For a $G$-covariant code with $G$ a continuous group, we derive a lower bound on the error correction infidelity following erasure of a subsystem. This bound approaches zero when the number of subsystems $n$ or the dimension $d$ of each subsystem is large. We exhibit codes achieving approximately the same scaling of infidelity with $n$ or $d$ as the lower bound. Leveraging tools from representation theory, we prove an approximate version of the Eastin-Knill theorem: If a code admits a universal set of transversal gates and corrects erasure with fixed accuracy, then, for each logical qubit, we need a number of physical qubits per subsystem that is inversely proportional to the error parameter. We construct codes covariant with respect to the full logical unitary group, achieving good accuracy for large $d$ (using random codes) or $n$ (using codes based on $W$-states). We systematically construct codes covariant with respect to general groups, obtaining natural generalizations of qubit codes to, for instance, oscillators and rotors. In the context of the AdS/CFT correspondence, our approach provides insight into how time evolution in the bulk corresponds to time evolution on the boundary without violating the Eastin-Knill theorem, and our five-rotor code can be stacked to form a covariant holographic code.
Approximate analytical solutions of Dirac Equation with spin and pseudo spin symmetries for the diatomic molecular potentials plus a tensor term with any angular momentum
Classical symmetries and the Quantum Approximate Optimization Algorithm
Argonne National Laboratory（阿贡国家实验室）；NASA Ames Research Center（NASA艾姆斯研究中心）；KBR（KBR公司）；USRA Research Institute for Advanced Computer Science (RIACS)（美国大学空间研究协会高级计算机科学研究所）；University of Delaware（特拉华大学）
Typical entanglement in anyon chains: Page curves beyond Lie group symmetries
Max-Planck-Institut für Quantenoptik（马克斯·普朗克量子光学研究所）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术卓越集群）；School of Mathematics and Statistics, The University of Melbourne（墨尔本大学数学与统计学院）；School of Physics, The University of Melbourne（墨尔本大学物理学院）；TUM School of Natural Sciences, Technical University of Munich（慕尼黑工业大学自然科学学院）
Towards a Unified Framework for Approximate Quantum Error Correction
最近关于近似量子纠错（QEC）的工作开辟了构建子空间码的可能性，该子空间码在不可能进行完美纠错的情况下以高保真度保护信息。受此启发，我们调查的问题，近似子系统的代码。子系统代码扩展子空间QEC的标准形式主义的代码中，只有一个子系统内的状态被用来存储信息的噪声弹性的方式。在这里，我们证明了容易检查的充分条件存在的近似子系统代码。此外，对于某些类别的子系统代码和噪声过程中，我们证明了转置通道作为一个简单的构造恢复地图，工程几乎以及最佳恢复通道的功效。这项工作推广了我们早期的方法[H. K. Ng和P.Mandayam，Phys.Rev.A81062342（2010）]的方法，并且使我们更接近于近似QEC的统一框架。
Approximate quantum error correction theory of non-isometric codes
Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉学科研究院）；Joint Center for Quantum Information and Computer Science, University of Maryland（马里兰大学量子信息与计算机科学联合中心）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
