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Generalized bicycle (GB) codes is a class of quantum error-correcting codes constructed from a pair of binary circulant matrices. Unlike for other simple quantum code ansätze, unrestricted GB codes may have linear distance scaling. In addition, low-density parity-check GB codes have a naturally overcomplete set of low-weight stabilizer generators, which is expected to improve their performance in the presence of syndrome measurement errors. For such GB codes with a given maximum generator weight w, we constructed upper distance bounds by mapping them to codes local in D≤w−1 dimensions, and lower existence bounds which give d≥O(n1/2). We have also conducted an exhaustive enumeration of GB codes for certain prime circulant sizes in a family of two-qubit encoding codes with row weights 4, 6, and 8; the observed distance scaling is consistent with A(w)n1/2+B(w), where n is the code length and A(w) is increasing with w.
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Local Testing for Membership in Lattices
Testing membership in lattices is of practical relevance, with applications to integer programming, error detection in lattice-based communication and cryptography. In this work, we initiate a systematic study of {\em local testing} for membership in lattices, complementing and building upon the extensive body of work on locally testable codes. In particular, we formally define the notion of local tests for lattices and present the following: \begin{enumerate} \item We show that in order to achieve low query complexity, it is sufficient to design one-sided non-adaptive {\em canonical} tests. This result is akin to, and based on an analogous result for error-correcting codes due to Ben-Sasson \etal\ (SIAM J. Computing 35(1) pp1--21). \item We demonstrate upper and lower bounds on the query complexity of local testing for membership in {\em code formula} lattices. We instantiate our results for code formula lattices constructed from Reed-Muller codes to obtain nearly-matching upper and lower bounds on the query complexity of testing such lattices. \item We contrast lattice testing from code testing by showing lower bounds on the query complexity of testing low-dimensional lattices. This illustrates large lower bounds on the query complexity of testing membership in {\em knapsack lattices}. On the other hand, we show that knapsack lattices with bounded coefficients have low-query testers if the inputs are promised to lie in the span of the lattice. \end{enumerate}
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- Award ID(s):
- 1649515
- PAR ID:
- 10033550
- Date Published:
- Journal Name:
- FSTTCS
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.4230/LIPIcs.FSTTCS.2016.46
