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

Growth of an Adherent Mixed Microbial Culture in a Substrate Limited Single State Chemostat

This study was supported by the Office of Water Resources Research U.S. Department of the Interior under Project Number A-021-OHIOA steady-state was established between a C. lividum and a Pseudomonas sp at a dilution rate of 0.27 hr^-1 when the growth limiting substrate was citrate. During both pure and mixed continuous culture studies, the C. lividum adhered to the wall of the chemostat and the Pseudomonas showed no such tendency visually. This system offers a convenient model for studying the importance of bacterial adherence to certain aquatic ecosystems such as river sediments and sewage treatment.Summary -- Introduction -- Methods -- Results -- Discussion -- Reference

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

Biologically Initiated Auto-Catalytic Mercury Conversion and its Effect on Elemental Mobility

This study was supported in part by the Office of Water Research and Technology, U.S. Department of the Interior under Project A-042-OHIO(print) iii, 61 leaves : ill. ; 28 cm.List of Figures -- Introduction -- Literature Review -- Materials and Methods -- Results -- Discussion -- Summary -- Literature Cite

Evaluation of Bacterial Binding and Release of Cadmium from Aquatic Sediments

This study was supported in part by the Office of Water Research and Technology, U.S. Department of the Interior under Project B-079-OHIO(print) v, 115 p. : ill., map ; 28 cm.List of Tables -- List of Figures -- Introduction -- Materials and Methods -- Results -- Discussion -- Conclusions and Summary -- Appendix A: Atomic Absorption Spectrophotometry -- Appendix B: Supplemental Data from Preliminary Sampling Trips -- Appendix C: Supplemental Data from Sampling Trips -- Appendix D: Uptake of Cadmium by Bacteria and Sediments Under Various Conditions -- Appendix E: Review of Literature Related to Cadmium -- Literature Cite