Simon Burton, Dan Browne
We analyze the structure of the logical operators from a class of quantum codes that generalizes the surface codes. These are the hypergraph product codes, restricted to the vertical sector. By generalizing an argument of Bravyi and König, we find that transversal gates for these codes must be restricted to the Clifford group.
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Partitioning qubits in hypergraph product codes to implement logical gates
研究超图积码中的横向门，通过构建辛正则基证明横向门可实现Hadamard和控制-Z门，结合纠错可实现任意逻辑比特间的纠缠门，为LDPC码上的通用容错量子计算奠定基础。
Fast and fault-tolerant logical measurements: Auxiliary hypergraphs and transversal surgery
本研究提出通过辅助超图和横向手术实现恒定时间开销的容错量子码手术，并建立同态测量与超图手术的等价性，证明时间开销主要由码与测量辅助系统的连通性决定。
I study the interplay of errors with transversal gates in 3D color codes, and introduce some new such gates. Two features of the transversal T gate stand out: (i) it naturally defines a set of correctable errors, and (ii) it exhibits a `linking charge' phenomenon that is of interest for a wide class of 3D topologically ordered systems.
Transversal implementations of encoded unitary gates are highly desirable for fault-tolerant quantum computation. Though transversal gates alone cannot be computationally universal, they can be combined with specially distilled resource states in order to achieve universality. We show that "triorthogonal" stabilizer codes, introduced for state distillation by Bravyi and Haah [Phys. Rev. A 86 052329 (2012)], admit transversal implementation of the controlled-controlled-Z gate. We then construct a universal set of fault-tolerant gates without state distillation by using only transversal controll…
Color codes are topological stabilizer codes with unusual transversality properties. Here I show that their group of transversal gates is optimal and only depends on the spatial dimension, not the local geometry. I also introduce a generalized, subsystem version of color codes. In 3D they allow the transversal implementation of a universal set of gates by gauge fixing, while error-detecting measurements involve only 4 or 6 qubits.
Transversal gates play an important role in the theory of fault-tolerant quantum computation due to their simplicity and robustness to noise. By definition, transversal operators do not couple physical subsystems within the same code block. Consequently, such operators do not spread errors within code blocks and are, therefore, fault tolerant. Nonetheless, other methods of ensuring fault tolerance are required, as it is invariably the case that some encoded gates cannot be implemented transversally. This observation has led to a long-standing conjecture that transversal encoded gate sets canno…
Constant-rate quantum codes with low-weight stabilizers and full logical Clifford actions via transversal and fold-transversal gates
本文构造了一族恒定速率、稳定子权重亚对数增长的量子码，仅用横向和折叠横向门即可实现完整逻辑Clifford群，为低空间开销容错量子计算提供路径。
Transversal Gates for Highly Asymmetric qLDPC Codes
本文构造了具有横向相位门的qLDPC码，其逻辑量子比特数随物理量子比特数线性增长，但距离高度不对称，并整理了识别多种横向门的工具箱技术。
Quantum Lego Power-up: Designing Transversal Gates with Tensor Networks
利用量子积木形式设计支持可寻址反向门的代码，克服传统方法的限制，实现高效故障容忍计算。
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…
Transversal Toffoli-gate in Hybrid-code System
研究混合编码系统中Toffoli门的横向性，发现三重正交码及其配对码系统支持其完全横向实现，提出不依赖预蒸馏T门魔法态的Toffoli态蒸馏协议，量子比特资源用量较现有协议减少约50%。
Construction of the full logical Clifford group for high-rate quantum Reed-Muller codes using only transversal and fold-transversal gates
针对自对偶量子Reed-Muller码族，仅用横向和折叠横向门构造完整逻辑Clifford群的生成集，实现任意可寻址Clifford门，无需辅助量子比特。
Color code with a logical control-$S$ gate using transversal $T$ rotations
本文提出一种颜色码，利用横向T和T†旋转实现逻辑控制-S门，展示了非克莱因门的实现方法。
Permutation-Invariant Quantum Codes with Transversal Generalized Phase Gates
本研究针对量子码横向门群问题，构造可横向实现广义相位门的置换不变自旋码，证明非加性码性能优于稳定子码，且所提T门实现方案在量子比特数和最小距离上更具优势。
A Classification of Transversal Clifford Gates for Qubit Stabilizer Codes
本文根据可横截实现的对角Clifford门集合对量子比特稳定子码进行分类，证明其门群必属于六个矩阵群族之一，并基于Rains的自同态矩阵代数理论给出了完整的对角Clifford对称性分类。
Transversal Logical Clifford gates on rotated surface codes with reconfigurable neutral atom arrays
提出利用可重构中性原子阵列在旋转表面码上横向实现逻辑 H、S 门的方案，并结合 CNOT 完善 Clifford 门集，S 门性能在电路噪声下与量子存储器相当。
Asymptotically Good Quantum Codes with Addressable and Transversal Non-Clifford Gates
本文构造了首族渐近好量子码，支持横向可寻址的非克利福德CCZ门，通过深度为一的物理电路实现任意三个逻辑量子比特上的逻辑门，改进了先前具有逆多对数参数的构造。
Transversal Gates in Nonadditive Quantum Codes
本文提出通过Stiefel流形参数化逻辑子空间并最小化复合损失的框架，以及SS-LP构造方法，系统搜索具有指定横截群的非加性量子码，发现了多个支持非Clifford横截门的新型最小距离量子码。
We provide a simplified, yet rigorous presentation of the ideas from Bombín's paper "Gauge Color Codes" [arXiv:1311.0879v3]. Our presentation is self-contained, and assumes only basic concepts from quantum error correction. We provide an explicit construction of a family of color codes in arbitrary dimensions and describe some of their crucial properties. Within this framework, we explicitly show how to transversally implement the generalized phase gate $R_n=\text{diag}(1,e^{2πi/2^n})$, which deviates from the method in "Gauge Color Codes", allowing an arguably simpler proof. We describe how t…
A long-standing open problem in fault-tolerant quantum computation has been to find a universal set of transversal gates. As three of us proved in arXiv: 0706.1382, such a set does not exist for binary stabilizer codes. Here we generalize our work to show that for subsystem stabilizer codes in $d$ dimensional Hilbert space, such a universal set of transversal gates cannot exist for even one encoded qudit, for any dimension $d$, prime or nonprime. This result strongly supports the idea that other primitives, such as quantum teleportation, are necessary for universal fault-tolerant quantum compu…
A Family of Quantum Codes with Exotic Transversal Gates
针对现有文献缺乏可横向实现二元二十面体群$2I$的量子码以支撑相关高效容错量子算法的问题，研究构建了一族码距d=3、可横向实现$2I$的量子码，且该码族可完全由$2I$独有的对称性推导得出。
Asymptotically Good Quantum Codes with Addressable Transversal T Gates
本文构造了具有恒定速率和线性距离的二元CSS码族，实现完全可寻址的横向T门，无需后续校正，并优化了渐近界限常数。
Error correction of transversal CNOT gates for scalable surface code computation
针对表面码横向CNOT门的关联误差问题，研究提出低复杂度MWPM解码器并测试三种策略，拓展相关电路解码与噪声下的性能比较，为大规模量子算法成本估算奠基。
Universal transversal gates
针对 Eastin-Knill 定理对通用横向门的限制，本文给出量子码具备通用横向门的充要条件，并构造可精确纠正局部与相关误差的码。
Fast magic state preparation by gauging higher-form transversal gates in parallel
本文提出了一种通过并行化高形式横向门规范来加速魔态制备的方法，实现快速且容错的魔态制备。
Universal Fault Tolerance with Non-Transversal Clifford Gates
我们提出一个通用框架，通过标志小工具实现任意克利福德电路的容错，允许横向T门与非横向克利福德门结合，以O(d^2 log(nd^2 log n))辅助比特实现距离d容错，并在[[15,1,3]]码上演示了非横向Hadamard门。
Surface and color codes are two forms of topological quantum error correction in two spatial dimensions with complementary properties. Surface codes have lower-depth error detection circuits and well-developed decoders to interpret and correct errors, while color codes have transversal Clifford gates and better code efficiency in the number of physical qubits needed to achieve a given code distance. A formal equivalence exists between color codes and folded surface codes, but it does not guarantee the transferability of any of these favorable properties. However, the equivalence does imply the…
Complementary 3D color codes for transversal quantum logic
研究提出基于四面体三维颜色码及其哈达玛变换对应物的混合架构，二者支持互补横向非克利福德操作，结合相关变换实现几乎通用的横向逻辑门集，还通过构造逻辑受控Z门等完成通用门集。
Hypergraph product codes are a class of quantum low density parity check (LDPC) codes discovered by Tillich and Zémor. These codes have a constant encoding rate and were recently shown to have a constant fault-tolerant error threshold. With these features, they asymptotically offer a smaller overhead compared to topological codes. However, existing techniques to perform logical gates in hypergraph product codes require many separate code blocks, so only becomes effective with a very large number of logical qubits. Here, we demonstrate how to perform Clifford gates on this class of codes using …
Fold-Transversal Clifford Gates for Quantum Codes
该研究将折叠概念从表面码推广到CSS码，提出仅用横向门和量子比特置换实现LDPC量子码逻辑操作的通用方法，并在Bring's code上验证，限制其4比特逻辑子空间可得到完整Clifford群。
Code conversion with the quantum Golay code for a universal transversal gate set
本文提出利用量子Golay码与三正交码之间的码转换实现通用横向门集，避免魔法态蒸馏开销，并给出基于横向CNOT的转换方法。
Clifford gates with logical transversality for self-dual CSS codes
本文针对自对偶CSS码，证明了存在辛基使横向Hadamard和相位门可实现的充要条件，并推广到级联码，为低开销容错量子计算提供多种横向Clifford门实现。
Randomized hypergraph states and their entanglement properties
研究随机混合超图状态的纠缠特性，分析多粒子纠缠行为，揭示超边结构与门失真之间的非单调纠缠行为。
Fault-tolerant, error-corrected quantum computation is commonly acknowledged to be crucial to the realisation of large-scale quantum algorithms that could lead to extremely impactful scientific or commercial results. Achieving a universal set of quantum gate operations in a fault-tolerant error-corrected framework suffers from a `conservation of unpleasantness'. In general, no matter what error correction technique is employed, there is always one element of a universal gate set that carries a significant resource overhead - either in physical qubits, computational time, or both. Specifically,…
We prove that the smallest distance 3 Quantum Error Correcting Code with a transversal gate outside the Clifford group is the well-known 15-qubit Reed-Muller code, also known as a tri-orthogonal code. Our result relies on fewer assumptions than previous works. We further extend this result by finding the minimal code that also possesses any other non Clifford transversal single-qubit gate.
Efficient fault-tolerant code switching via one-way transversal CNOT gates
本文提出一种仅用横向CNOT门的码切换方案，实现低开销容错逻辑T门，数值模拟显示其优于魔法态注入和物理门，并扩展到高距离码对，简化FT通用量子计算硬件需求。
Hypergraph states form a family of multiparticle quantum states that generalizes cluster states and graph states. We study the action and graphical representation of nonlocal unitary transformations between hypergraph states. This leads to a generalization of local complementation and graphical rules for various gates, such as the CNOT gate and the Toffoli gate. As an application, we show that already for five qubits local Pauli operations are not sufficient to check local equivalence of hypergraph states. Furthermore, we use our rules to construct entanglement witnesses for three-uniform hype…
Constant Rate Codes with Fully Addressable Transversal T: Good Codes, Sparse Checks
本文构造了具有完全可寻址横向T门的渐近好CSS码及多种相关码，实现了对逻辑量子比特的选择性T门控制，并推广至任意有限阶单比特门，同时指出构建渐近好Pauli-LDPC码的剩余挑战。
A robust phase of continuous transversal gates in quantum stabilizer codes
该研究发现表面码中可连续调节的逻辑单元的鲁棒相，通过降低去相位误差的影响，设计了实现连续角度逻辑旋转的容错协议，提高了量子模拟等应用的效率。
Good binary quantum codes with transversal CCZ gate
基于经典码的乘法性质构造具有横向 CCZ 门的渐近良好量子 CSS 码族，利用代数几何码和秘密共享技术实现二元化，并推演出恒定空间开销的魔法态蒸馏方案。
Scalable Spider Nests (...Or How to Graphically Grok Transversal Non-Clifford Gates)
作为图形化参悟系列第二篇，本文用可扩展ZX-演算简洁表述蜘蛛巢恒等式并证明其可由单条新规则归纳导出，结合CSS码ZX图景刻画了任意CSS码中Clifford层级第三级的全部横向对角门。
Certain quantum codes allow logic operations to be performed on the encoded data, such that a multitude of errors introduced by faulty gates can be corrected. An important class of such operations are {\em transversal}, acting bitwise between corresponding qubits in each code block, thus allowing error propagation to be carefully limited. If any quantum operation could be implemented using a set of such gates, the set would be {\em universal}; codes with such a universal, transversal gate set have been widely desired for efficient fault-tolerant quantum computation. We study the structure of G…
Transversal CNOT gate with multi-cycle error correction
本研究在带有flag qubits的Repetition code构建的逻辑量子比特间实现了横截CNOT门，通过IBM量子设备实验证明，将物理量子比特从21增至57可抑制错误并在10轮检测中持续，确立了当前超导量子硬件实现容错逻辑CNOT门的可行性。
One of the leading quantum computing architectures is based on the two-dimensional (2D) surface code. This code has many advantageous properties such as a high error threshold and a planar layout of physical qubits where each physical qubit need only interact with its nearest neighbours. However, the transversal logical gates available in 2D surface codes are limited. This means that an additional (resource intensive) procedure known as magic state distillation is required to do universal quantum computing with 2D surface codes. Here, we examine three-dimensional (3D) surface codes in the cont…
Decoding across transversal Clifford gates in the surface code
研究表面码中跨横向门的解码问题，采用基于快速“逻辑可观测量”的最小重量完美匹配（MWPM）解码器，提出窗口逻辑可观测量匹配解码器并分析性能，指出可能导致逻辑失败的错误及应对方法。
This work classifies the set of diagonal gates that can implement a single or two-qubit transversal logical gate for qubit stabilizer codes. We show that individual physical gates on the underlying qubits that compose the code are restricted to have entries of the form $e^{i πc/2^k}$ along their diagonal, resulting in a similarly restricted class of logical gates that can be implemented in this manner. Moreover, we show that all diagonal logical gates that can be implemented transversally by individual physical diagonal gates must belong to the Clifford hierarchy. Furthermore, we can use this …
Transversal Diagonal Logical Operators for Stabiliser Codes
本研究针对CSS码，提出识别所有横向对角逻辑算子的更通用或更低复杂度算法，并给出构造具有所需对角逻辑克利福德层级算子的CSS码的方法，以推进容错量子计算。
Transversal gates for quantum CSS codes
本文研究了量子CSS码的横向门计算，确定了逻辑作用和门群，并扩展了相关码类的研究结果。
Qubit CSS codes with Universal Transversal Gates
本文通过深度为一的少量子比特门层，在纠错码上生成通用逻辑门集，无需魔法态蒸馏等辅助手段，并构造了47个容忍每层至少一个故障门的码，包括[[37,1,7]]和[[59,1,9]]码。
Transversal dimension jump for product qLDPC codes
本文提出了一种基于产品qLDPC码的横向维度跃迁方案，通过结合3D和2D码的非局部门实现通用容错量子计算，具有高阈值和低开销。
Measurement-free code-switching for low overhead quantum computation using permutation invariant codes
本文提出了一种无测量的代码切换协议，利用排列不变码实现低开销的通用量子计算，通过切换稳定器码和PI码来实现更高效的非Clifford门操作。
Transversal gates of the ((3,3,2)) qutrit code and local symmetries of the absolutely maximally entangled state of four qutrits
本文证明了三三二型三能子码的横向门群与四个三能子绝对最大纠缠态的局部对称群之间存在紧密联系，并利用李代数理论确定了这两个群的生成元。
Efficient Magic State Factory Via Transversal Non-Clifford Gate
本文通过端到端模拟比较了基于代码切换的魔法态制备与魔法态培育的资源需求与性能，提出了结合相位反冲检查和横向非克利福德门的两种容错魔法态制备协议。
High-fidelity and Fault-tolerant Teleportation of a Logical Qubit using Transversal Gates and Lattice Surgery on a Trapped-ion Quantum Computer
本研究在Quantinuum H2囚禁离子量子处理器上首次演示了Steane码的容错态隐形传态，使用横向门和格点手术协议实现了高达0.989(2)的逻辑过程保真度。
Hypergraph states are generalizations of graph states where controlled-$Z$ gates on edges are replaced with generalized controlled-$Z$ gates on hyperedges. Hypergraph states have several advantages over graph states. For example, certain hypergraph states, such as the Union Jack states, are universal resource states for measurement-based quantum computing with only Pauli measurements, while graph state measurement-based quantum computing needs non-Clifford basis measurements. Furthermore, it is impossible to classically efficiently sample measurement results on hypergraph states with a constan…
A method for the implementation of a universal set of fault-tolerant logical gates is presented using homological product codes. In particular, it is shown that one can fault-tolerantly map between different encoded representations of a given logical state, enabling the application of different classes of transversal gates belonging to the underlying quantum codes. This allows for the circumvention of no-go results pertaining to universal sets of transversal gates and provides a general scheme for fault-tolerant computation while keeping the stabilizer generators of the code sparse.
Efficient recursive encoders for quantum Reed-Muller codes towards Fault tolerance
本文为允许横向门的量子Reed-Muller码设计高效递归编码器，实现对数深度和更少门数，并证明CNOT门数与纠缠熵相等，展示最优性及纠缠提取能力。
In order to perform universal fault-tolerant quantum computation, one needs to implement a logical non-Clifford gate. Consequently, it is important to understand codes that implement such gates transversally. In this paper, we adopt an algebraic approach to characterize all stabilizer codes for which transversal $T$ and $T^{-1}$ gates preserve the codespace. Our Heisenberg perspective reduces this to a finite geometry problem that translates to the design of certain classical codes. We prove three corollaries: (a) For any non-degenerate $[[ n,k,d ]]$ stabilizer code supporting a physical trans…
It is an oft-cited fact that no quantum code can support a set of fault-tolerant logical gates that is both universal and transversal. This no-go theorem is generally responsible for the interest in alternative universality constructions including magic state distillation. Widely overlooked, however, is the possibility of non-transversal, yet still fault-tolerant, gates that work directly on small quantum codes. Here we demonstrate precisely the existence of such gates. In particular, we show how the limits of non-transversality can be overcome by performing rounds of intermediate error-correc…
Transversal Gates and Magic State Distillation in an Optimally Synthesized Spin-Qubit Shuttling Bus
本研究提出多量子比特自旋量子比特穿梭总线架构，优化布局与辅助量子比特共享方案，结合Quantum Reverse Mapping方法，实现容错量子计算的层间协同设计，验证了自旋量子比特穿梭架构的可行性。
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