Degenerate local-dimension-invariant stabilizer codes and an alternative bound for the distance preservation condition

Local-dimension-invariant qudit stabilizer codes

“…While the stabilizer formalism does not encapsulate all error-correction schemes, it provides a particularly useful mathematical framework, and so this work will focus on quantum error-correcting codes that formally are stabilizer codes, but may reside over atypical local-dimensions-such as infinite, continuous, and mathematical rings. This work provides some useful uses for the local-dimension-invariant (LDI) framework [5][6][7]. The final results in this work provide a somewhat similar general Theorem as the CSS Theorem; whereas the CSS Theorem provides a method for bringing classical codes into quantum codes the results herein provide methods for using quantum codes as quantum codes for other varieties of quantum systems.…”

“…This is not the only way to form an LDI representation. Some other methods are briefly discussed in [6]. For the sake of simplicity we will use the lower-triangular L matrix method when using a prescriptive method.…”

“…Given the current expressions for p * , this would mean that the spatial extent needed would go as B 2(d−1) (2(d − 1)) (d−1) in order to guarantee the distance of the code, with B being the maximal entry in the LDI representation, generally satisfying B ≤ max i,j |φ ∞ (s i ) φ ∞ (s j )|. There is also the alternative expression given by (B(q − 1) [6] and in the case of CSS codes B d−1 (d − 1) (d−1)/2 [7]. The values for p * can be quite large, however, depending on the level of precision achievable in the determination of the quadratures this may not be a particularly limiting factor.…”

“…By adding this caveat in, one may begin over a field local-dimension value, which does not have a finite characteristic. Alternatively, one may use the alternative bound for B based on the maximal symplectic product, if that LDI representation is used [6].…”