超越横截性：CSS码的Clifford电路结构
Beyond transversality: structure of Clifford circuits for CSS codes
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中文总结 AI 辅助
本文刻画了与容错逻辑操作相关的四类CSS码的Clifford电路结构，定义了二重横截群等，确定了136个CSS码的二重横截群逻辑像，发现78个码的完整逻辑Clifford群由二重横截电路生成，还构造了三类CSS码。
中文摘要 AI 辅助
我们刻画了与容错逻辑操作相关的四类Calderbank–Shor–Steane（CSS）码的Clifford电路。首先，我们证明每个保码Clifford电路都是Z对角电路的乘积，Z对角电路由S门和CZ门及其X基类似物构成。其次，我们定义了由深度为1的两局部保码电路生成的二重横截群，并证明其每个元素都可表示为Z对角、X对角或CNOT门构成的层的乘积。作为推论，每个横截门都是三个横截对角电路的乘积；对于连通非自对偶码，仅需两个此类电路。我们进一步证明，由单量子比特Clifford门和置换构成的保码自同构电路具有由Hadamard层、一个置换和两个对角电路组成的标准形式。我们还定义了一个二重自同构群，其中深度为1的两局部电路可在置换意义下保码，且其逻辑像可大于二重横截群的逻辑像。对于136个CSS码，我们提供了显式生成元并确定了二重横截群的逻辑像。我们发现78个码的完整逻辑Clifford群由二重横截电路生成，包括距离为3、4、5、6、8、12的码，其码率分别为2/5、3/4、1/9、1/5、2/5、3/56。我们从二分图网格、补割集和二次曲面构造了三类CSS码，其中许多以此方式实现了完整逻辑Clifford群。更一般地，即使诱导逻辑群不是完整逻辑Clifford群，其也可很大：对于gross码，其阶至少为460800；对于clustered-cyclic码，其阶约为10^26。
英文摘要
We characterize four groups of Clifford circuits for Calderbank--Shor--Steane (CSS) codes that are relevant to fault-tolerant logical operations. First, we show that every code-preserving Clifford circuit is a product of Z-diagonal circuits, composed of S and CZ gates, and their X-basis analogues. Second, we define the two-fold transversal group, generated by depth-one two-local code-preserving circuits, and show that each of its elements can be expressed as a product of layers consisting of either Z-diagonal, X-diagonal, or CNOT gates. As a corollary, every transversal gate is a product of three transversal diagonal circuits; for connected non-self-dual codes, two such circuits suffice. We further show that every code-preserving automorphism circuit, consisting of single-qubit Clifford gates and permutations, has a normal form comprising a Hadamard layer, a permutation, and two diagonal circuits. We also define a two-fold automorphism group, in which a depth-one two-local circuit may be code-preserving up to a permutation, and show that its logical image can be larger than that of the two-fold transversal group. For 231 CSS codes, we provide explicit generators and determine the logical image of the two-fold transversal group. We tabulate 173 codes whose full logical Clifford group is generated by two-fold-transversal circuits, including codes of distances 3, 4, 5, 6, 8, 10, 12, and 13, with respective best rates 2/5, 3/4, 34/77, 17/28, 14/25, 1/27, 3/56, and 1/33. We construct families of CSS codes from bipartite grids, cut-complements, and quadrics, and census the self-dual codes invariant under rank-3 permutation groups. Many of these codes realize the full logical Clifford group in this way.
发表机构
- Joint Center for Quantum Information and Computer Science, NIST/University of Maryland（量子信息与计算机科学联合中心，美国国家标准与技术研究院/马里兰大学）
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