Kerdock Codes Determine Unitary 2-Designs

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 $\imath \triangleq \sqrt{-1}$ 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. (arXiv:1501.04592), 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.Comment: 16 pages double-column, 4 figures, and some circuits. Accepted to 2019 Intl. Symp. Inf. Theory (ISIT), and PDF of the 5-page ISIT version is included in the arXiv packag

Kerdock Codes Determine Unitary 2-Designs

The binary non-linear Kerdock codes are Gray images of ℤ_4-linear Kerdock codes of length N =2^m . We show that exponentiating ı=−√-1 by these ℤ_4-valued codewords produces stabilizer states, which are the common eigenvectors of maximal commutative subgroups (MCS) of the Pauli group. We use this quantum description to simplify the proof of the classical weight distribution of Kerdock codes. Next, we partition stabilizer states into N +1 mutually unbiased bases and prove that automorphisms of the Kerdock code permute the associated MCS. This automorphism group, represented as symplectic matrices, is isomorphic to the projective special linear group PSL(2,N) and forms a unitary 2-design. The design described here was originally discovered by Cleve et al. (2016), but the connection to classical codes is new. This significantly simplifies the description of the design and its translation to circuits

Kerdock Codes Determine Unitary 2-Designs

The binary non-linear Kerdock codes are Gray images of ℤ_4-linear Kerdock codes of length N =2^m . We show that exponentiating ı=−√-1 by these ℤ_4-valued codewords produces stabilizer states, which are the common eigenvectors of maximal commutative subgroups (MCS) of the Pauli group. We use this quantum description to simplify the proof of the classical weight distribution of Kerdock codes. Next, we partition stabilizer states into N +1 mutually unbiased bases and prove that automorphisms of the Kerdock code permute the associated MCS. This automorphism group, represented as symplectic matrices, is isomorphic to the projective special linear group PSL(2,N) and forms a unitary 2-design. The design described here was originally discovered by Cleve et al. (2016), but the connection to classical codes is new. This significantly simplifies the description of the design and its translation to circuits

Choosing the best machine tool in mechanical manufacturing

Machine tools are indispensable components and play an important role in mechanical manufacturing. The equipment of machine tools has a huge effect on the operational efficiency of businesses. Each machine tool type is described by many different criteria, such as cost, technological capabilities, accuracy, energy consumption, convenience in operation, safety for workers, working noise, etc. If the selection of machine is only based on one or several criteria, it will be really easy to make mistakes, which means it is not possible to choose the real best machine. A machine is considered to be the best only when it is chosen based on all of its criteria. This work is called multi-criteria decision-making (MCDM). In this study, the selection of machine tools has been done using two different multi-criteria decision-making methods, including the FUCA method (Faire Un Choix Adéquat) and the CURLI method (Collaborative Unbiased Rank List Intergration). These are two methods with very different characteristics. When using the FUCA method, it is necessary to normalize the data and determine the weights for the criteria. Meanwhile, if using the CURLI method, these two things are not necessary. The selection of these two distinct methods is intended to produce the most generalizable conclusions. Three types of machine tool, which are considered in this study, include grinding machine, drilling machine and milling machine. The number of grinders that were offered for selection was twelve, the number of drills that were surveyed in this study was thirteen, while nine were the number of milling machines that were given for selection. The objective of this study is to determine the best solution in each type of machine. The results of ranking the machines are very similar when using the two mentioned methods. Specially, in all the surveyed cases, the two methods FUCA and CURLI always find the same best alternative. Accordingly, it is possible to firmly come to a conclusion that the FUCA method and the CURLI method are equally effective in machine tool selection. In addition, this study has determined the best three machines corresponding to the three different machine type

On Arithmetic Invariants of Special Families of K3-Type Surfaces

This thesis studies applications of Shimura varieties in positive characteristic to questions on arithmetic invariants of special families of K3-type surfaces. The first main result determines the Newton polygons and Artin invariants of 144 special families of K3-type surfaces. The second is a refinement of a conjecture of Serre for K3 surfaces over number field.</p

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Kerdock Codes Determine Unitary 2-Designs

The non-linear binary Kerdock codes are known to be Gray images of certain extended cyclic codes of length N=2^m over Z₄. We show that exponentiating these Z₄-valued codewords by i≜√-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