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

IN VITRO ANTIBACTERIAL ACTIVITIES OF OPUNTIA FICUS INDICA STEM AND FRUIT EXTRACTS USING DISC DIFFUSION METHOD

Objective: Opuntia ficus indica is a medicinal plant belonging to family Cactaceae. It is a species of cactus that has long been a domesticated crop plant important in agricultural economies throughout arid and semiarid parts of the world. The fruit and stem are used to prepare worth added products, fruits jam, squash, drinks, preserve product of pickle, body lotion, shampoo and creams, etc. Methods: For the preliminary study, various extracts of stem and fruit has been used to check the efficacy of antibacterial activity against Bacillus subtilis, Pseudomonas aeruginosa and Escherichia coli bacteria of gram-positive and gram-negative strain respectively using disc diffusion method. Results: The stem and fruit extracts showed various levels of activity on different test organisms. The methanol fruit extracts showed high antibacterial activity against Escherichia Coli, Pseudomonas aeruginosa and Bacillus subtilis compared with other extracts. Aqueous extract of stem and fruit showed less antibacterial activity against the tested bacterial strains. Conclusion: The present study suggests that the methanol extracts of the fruit of Opuntia ficus indica contain compounds that can form the basis for the development of a novel broad-spectrum antibacterial formulation

Weight Distribution of Classical Codes Influences Robust Quantum Metrology

Quantum metrology (QM) is expected to be a prominent use-case of quantum technologies. However, noise easily degrades these quantum probe states, and negates the quantum advantage they would have offered in a noiseless setting. Although quantum error correction (QEC) can help tackle noise, fault-tolerant methods are too resource intensive for near-term use. Hence, a strategy for (near-term) robust QM that is easily adaptable to future QEC-based QM is desirable. Here, we propose such an architecture by studying the performance of quantum probe states that are constructed from $[n,k,d]$ binary block codes of minimum distance $d \geq t+1$. Such states can be interpreted as a logical state of a CSS code whose logical $X$ group is defined by the aforesaid binary code. When a constant, $t$, number of qubits of the quantum probe state are erased, using the quantum Fisher information (QFI) we show that the resultant noisy probe can give an estimate of the magnetic field with a precision that scales inversely with the variances of the weight distributions of the corresponding $2^t$ shortened codes. If $C$ is any code concatenated with inner repetition codes of length linear in $n$, a quantum advantage in QM is possible. Hence, given any CSS code of constant length, concatenation with repetition codes of length linear in $n$ is asymptotically optimal for QM with a constant number of erasure errors. We also explicitly construct an observable that when measured on such noisy code-inspired probe states, yields a precision on the magnetic field strength that also exhibits a quantum advantage in the limit of vanishing magnetic field strength. We emphasize that, despite the use of coding-theoretic methods, our results do not involve syndrome measurements or error correction. We complement our results with examples of probe states constructed from Reed-Muller codes.Comment: 21 pages, 3 figure

Solar variability in the past and palaeoclimate data pertaining to the Southwest monsoon

A significant part of the earth's climate variability is caused by changes in the solar emission. Instrumental observation of the sun gives us some idea about decadal variability in the solar radiation. On longer timescales, we look to palaeoclimate proxies to learn about solar variability. In this review we discuss various palaeo-records and what we have learnt from them. In addition, we outline important questions that need to be addressed