Local-dimension-invariant qudit stabilizer codes

Abstract

Traditional stabilizer codes operate over prime power local-dimensions. In this work we extend the stabilizer formalism using the local-dimension-invariant setting to import stabilizer codes from these standard local-dimensions to other cases. In particular, we show that any traditional stabilizer code can be used for analog continuous-variable codes, and consider restrictions in phase space and discretized phase space. This puts this framework on equivalent footing as traditional stabilizer codes. Following this, using extensions of the prior ideas, we show that a stabilizer code originally designed with a finite field local-dimension can be transformed into a code with the same n, k, and d parameters for any integral domain ring. This is of theoretical interest and can be of use for systems whose local-dimension is better described by mathematical rings, for which this permits the use of traditional stabilizer codes for protecting their information as well.

Local-dimension-invariant Calderbank-Shor-Steane Codes with an Improved Distance Promise

“…CSS codes have a set of independent generators only using X operators or Z operators for each generator. While not the only way to generate local-dimension-invariant codes, the prescriptive method from [12] provides one such method. Any stabilizer code S can be put into canonical form, which is given by:…”

“…For this uncertainty region the distance could be lower, it could be the same, or it could even be higher, however, it must be determined manually. It is conjectured that for q ≤ p the distance can be at least preserved, however, this is only known to be possible for a few codes, such as the 5, 7, and 9 particle codes [10][11][12]. This work adds to this collection an infinite family.…”

“…This work adds to this collection an infinite family. While the distance is promised to be at least the same for p * < p, the caveat is that the current bound for p * for an arbitrary non-degenerate stabilizer code is very large: p * = B 2(d−1) (2(d − 1)) d−1 with B being the largest entry in the LDI form, which is bounded by B ≤ (2 + k(q − 1))(q − 1) [12]. This value is typically large, so reductions to it is crucial to make the technique of more practical use.…”

“…We will write a vertical bar between the vector for the X powers and the vector for the Z powers mostly for ease of reading. The special case of φ ∞ which follows the same relations, but for this representation one does not take modulo any value but carries computations over the integers, was proven in [12]. As an example of the difference, consider the following:…”

“…While most codes have been designed for a given local-dimension value, Chau showed that both the 5-qubit [10] and 9-qubit [11] codes could be transformed into qudit codes with the same parameters and minimal modifications to the code. These examples were extended into a framework allowing all qudit codes with q levels to be transformed into valid qudit codes over p levels, however, it only showed that the distance could be preserved once there are sufficiently many levels; beyond some critical value p * , which is a function of n, k, d, and q [12]. These works paved the path to using codes over various local-dimensions, however, this latter work left a large gap between q (the initial dimension) and p * where statements about the distance cannot be made without manually determining the distance.…”