量化码字稳定码的非稳定度
Quantifying Nonstabilizerness of Codeword-Stabilized Codes
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中文总结 AI 辅助
本文针对码字稳定码建立非稳定度的定量理论，将其转化为加性组合学的经典计数问题，得出非稳定度的上限、陪集闭包下的不变性等关键结果，还给出Kerdock码的闭式值。
中文摘要 AI 辅助
容错量子计算需要非Clifford门，而稳定器码无法提供横向的非Clifford门。非稳定器码是寻找这类门的自然载体，但此前不存在关于这类代码所携带的非稳定度（或称魔性）的定量理论。我们针对码字稳定码（CWS码）建立了这样的理论，并表明关键量是经典的：一个码的非稳定度由其码字在平移下的碰撞方式决定，该问题属于加性组合学范畴。我们证明，当存在所需大小的Sidon集时，最具魔性的码恰好是Sidon集，其两两差值均互不相同。无论码的物理规模多大，其非稳定度都不超过其逻辑量子比特数的两倍。此外，这种归约还给出了结构和操作层面的结果：非稳定度在陪集闭包下保持不变，这使得我们可以构造出具有任意多逻辑量子比特、且非稳定度随逻辑量子比特数增长而保持恒定的非稳定器码。对于k个逻辑量子比特的对角横向门，若其在t个坐标上是非Clifford的，则该码的非稳定度至多为2(k-t)；因此，非稳定度也可用于构建该码所需的非Clifford门以及经典模拟该码的成本。最后，我们得到了整个家族的精确可计算性，并获得了Kerdock码的闭式值。这些结果共同将寻找高魔性码和横向非Clifford门的问题转化为经典计数问题，可利用加性组合学的标准工具进行研究。
英文摘要
Fault-tolerant quantum computation requires non-Clifford gates, which stabilizer codes cannot supply transversally. Non-stabilizer codes are the natural place to look for them, yet no quantitative theory of the nonstabilizerness (or magic) carried by such a code has existed. We develop one for codeword-stabilized (CWS) codes and show that the key quantity is classical: a code's nonstabilizerness is fixed by how its codewords collide under translation, a question that belongs to additive combinatorics. We show that the most magical codes are exactly the Sidon sets whenever a Sidon set of the required size exists, whose pairwise differences are all distinct. No code carries more than twice its number of logical qubits of nonstabilizerness however large it is physically. Furthermore, the same reduction gives structural and operational results. Nonstabilizerness is unchanged by coset closure, which yields non-stabilizer codes with arbitrarily many logical qubits and constant nonstabilizerness as the number of logical qubits grows. A diagonal transversal gate with $k$ logic qubits that is non-Clifford on $t$ coordinates forces the code's nonstabilizerness to be at most $2(k-t)$; thus the nonstabilizerness also bounds the non-Clifford gates needed to build the code and the cost of classically simulating it. Finally, entire families become exactly computable, and we obtain closed-form values for the Kerdock codes. Together these results turn the search for magic-rich codes and transversal non-Clifford gates into classical counting problems, which can be approached with standard tools from additive combinatorics.
