论文标题
计算稳定器代码任意维度
Counting stabiliser codes for arbitrary dimension
论文作者
论文摘要
在这项工作中，我们计算由$ d $二维的Qudits组成的$ [[N，K]] _ D $稳定器的数量，用于任意正整数$ D $。在Gross（参考[23]）的开创性工作中，计算了$ [[n，k]] _ d $稳定器代码的数量，当时$ d $是prime（或prime的功率，即$ d = p^m $）的情况，但是当qudits qudits是galois-qudits时）。参考证明。参考。 [23]不适用于非Prime情况。为了证明，我们将一个组结构介绍到$ [[n，k]] _ d $代码，并与中文剩余定理一起使用它来计算$ [[n，k]] _ d $代码的数量。我们的工作与裁判重叠。参考。 [23]当$ d $是素数时，在这种情况下，我们的结果完全匹配，但是结果较为通用的情况有所不同。尽管如此，稳定器代码数量的总数数量级是尺寸是质量还是非稳定性的尺度。这是令人惊讶的，因为用于计算稳定器状态数量（或更一般的稳定器代码）所采用的方法取决于$ d $是否是主要的。到目前为止，稳定器状态的基础性仅以量子二维情况（以及Galois Qudit Prime-Power尺寸案例）而闻名，这在量子计算中的许多主题中都起着重要的作用。其中的重要性是魔术的资源理论，设计理论，稳定剂的de finetti定理，克利福德电路的经典模拟性的研究和优化，小维系统的量子上下文性的研究以及Wigner-Funnctions的研究。我们的工作为通用情况提供了此量词，因此是将非限度量子量子系统放置在与PrimeDimensional Systems同一基座上的量子计算结果所需的重要步骤。
In this work, we compute the number of $[[n,k]]_d$ stabilizer codes made up of $d$-dimensional qudits, for arbitrary positive integers $d$. In a seminal work by Gross (Ref. [23]) the number of $[[n,k]]_d$ stabilizer codes was computed for the case when $d$ is a prime (or the power of a prime, i.e., $d=p^m$, but when the qudits are Galois-qudits). The proof in Ref. Ref. [23] is inapplicable to the non-prime case. For our proof, we introduce a group structure to $[[n,k]]_d$ codes, and use this in conjunction with the Chinese remainder theorem to count the number of $[[n,k]]_d$ codes. Our work overlaps with Ref. Ref. [23] when $d$ is a prime and in this case our results match exactly, but the results differ for the more generic case. Despite that, the overall order of magnitude of the number of stabilizer codes scales agnostic of whether the dimension is prime or non-prime. This is surprising since the method employed to count the number of stabilizer states (or more generally stabilizer codes) depends on whether $d$ is prime or not. The cardinality of stabilizer states, which was so far known only for the prime-dimensional case (and the Galois qudit prime-power dimensional case) plays an important role as a quantifier in many topics in quantum computing. Salient among these are the resource theory of magic, design theory, de Finetti theorem for stabilizer states, the study and optimisation of the classical simulability of Clifford circuits, the study of quantum contextuality of small-dimensional systems and the study of Wigner-functions. Our work makes available this quantifier for the generic case, and thus is an important step needed to place results for quantum computing with non-prime dimensional quantum systems on the same pedestal as prime-dimensional systems.
