The non-linear binary Kerdock codes are known to be Gray images of certain extended cyclic codes of length $N = 2^{m}$ over $\mathbb {Z}_{4}$ . We show that exponentiating these $\mathbb {Z}_{4}$ -valued codewords by $i \triangleq \sqrt {-1}$ produces stabilizer states, that are quantum states obtained using only Clifford unitaries. These states are also the common eigenvectors of commuting Hermitian matrices forming maximal commutative subgroups (MCS) of the Pauli group. We use this quantum description to simplify the derivation of the classical weight distribution of Kerdock codes. Next, we organize the stabilizer states to form $N+1$ mutually unbiased bases and prove that automorphisms of the Kerdock code permute their corresponding MCS, thereby forming a subgroup of the Clifford group. When represented as symplectic matrices, this subgroup is isomorphic to the projective special linear group PSL( $2,N$ ). We show that this automorphism group acts transitively on the Pauli matrices, which implies that the ensemble is Pauli mixing and hence forms a unitary 2-design. The Kerdock design described here was originally discovered by Cleve et al. (2016), but the connection to classical codes is new which simplifies its description and translation to circuits significantly. Sampling from the design is straightforward, the translation to circuits uses only Clifford gates, and the process does not require ancillary qubits. Finally, we also develop algorithms for optimizing the synthesis of unitary 2-designs on encoded qubits, i.e., to construct logical unitary 2-designs. Software implementations are available at https://github.com/nrenga/symplectic-arxiv18a, which we use to provide empirical gate complexities for up to 16 qubits.

中文翻译：

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Kerdock 代码决定了 Unitary 2-Designs

已知非线性二进制 Kerdock 码是某些长度的扩展循环码的格雷图像 $N = 2^{m}$ 超过 $\mathbb {Z}_{4}$ . 我们证明了对这些取幂 $\mathbb {Z}_{4}$ -值码字由 $i \triangleq \sqrt {-1}$ 产生稳定剂态，即仅使用 Clifford 幺正获得的量子态。这些状态也是形成泡利群的最大交换子群 (MCS) 的交换 Hermitian 矩阵的公共特征向量。我们用这个量子 描述以简化推导 古典Kerdock 代码的权重分布。接下来，我们组织稳定器状态以形成 $N+1$ 互无偏基并证明 Kerdock 代码的自同构置换了它们对应的 MCS，从而形成了 Clifford 群的一个子群。当表示为辛矩阵时，这个子群同构于射影特殊线性群 PSL( $2,N$ ）。我们证明了这个自同构群在泡利矩阵上具有传递性，这意味着集成是泡利搅拌并因此形成一个单一的 2-设计。这克多克 这里描述的设计最初是由 Cleve 发现的 等。（2016），但与经典代码的连接是新的，这显着简化了对电路的描述和转换。从设计中采样很简单，到电路的转换仅使用 Clifford 门，并且该过程不需要辅助量子位。最后，我们还开发了优化编码量子位上单一 2-设计合成的算法，即构造合乎逻辑的单一的 2 设计。软件实现可在https://github.com/nrenga/symplectic-arxiv18a，我们用它来提供多达 16 个量子位的经验门复杂度。