Stabilizer codes from modified symplectic form

Stabilizer codes form an important class of quantum error correcting codes which have an elegant theory, efficient error detection, and many known examples. Constructing stabilizer codes of length $n$ is equivalent to constructing subspaces of $\mathbb{F}_p^n \times \mathbb{F}_p^n$ which are "isotropic" under the symplectic bilinear form defined by $\left\langle (\mathbf{a},\mathbf{b}),(\mathbf{c},\mathbf{d}) \right\rangle = \mathbf{a}^{\mathrm{T}} \mathbf{d} - \mathbf{b}^{\mathrm{T}} \mathbf{c}$. As a result, many, but not all, ideas from the theory of classical error correction can be translated to quantum error correction. One of the main theoretical contribution of this article is to study stabilizer codes starting with a different symplectic form. In this paper, we concentrate on cyclic codes. Modifying the symplectic form allows us to generalize the previous known construction for linear cyclic stabilizer codes, and in the process, circumvent some of the Galois theoretic no-go results proved there. More importantly, this tweak in the symplectic form allows us to make use of well known error correcting algorithms for cyclic codes to give efficient quantum error correcting algorithms. Cyclicity of error correcting codes is a "basis dependent" property. Our codes are no more "cyclic" when they are derived using the standard symplectic forms (if we ignore the error correcting properties like distance, all such symplectic forms can be converted to each other via a basis transformation). Hence this change of perspective is crucial from the point of view of designing efficient decoding algorithm for these family of codes. In this context, recall that for general codes, efficient decoding algorithms do not exist if some widely believed complexity theoretic assumptions are true

Computational Modeling And Analysis Of Synthetic Jets

In the last report we focused on the study of 3D synthetic jets of moderate jet aspect-ratio. Jets in quiescent and cross-flow cases were investigated. Since most of the synthetic jets in practical applications are found to be of large aspect ratio, the focus was shifted to studying synthetic jets of large aspect ratio. In the current year, further progress has been made by studying jets of aspect ratio 8 and infinity. Some other aspects of the jet, like the vorticity flux is looked into apart from analyzing the vortex dynamics, velocity profiles and the other dynamical characteristics of the jet which allows us to extract some insight into the effect of these modifications on the jet performance. Also, efforts were made to qualitatively validate the simulated results with the NASA Langley test cases at higher jet Reynolds number for the quiescent jet case