Counting stabilizer codes for arbitrary dimension

Abstract

In this work, we compute the number of [[n, k]]d stabilizer codes made up of d -dimensional qudits, for arbitrary positive integers d. In a seminal work by Gross (Ref. [23]) the number of [[n, k]]d stabilizer codes was computed for the case when d is a prime (or the power of a prime, i.e., d = pm, but when the qudits are Galois-qudits). The proof in Ref. [23] is inappli-cable to the non-prime case. For our proof, we introduce a group structure to [[n, k]]d codes, and use this in conjunction with the Chinese remainder theorem to count the number of [[n, k]]d codes. Our work over-laps with Ref. [23] when d is a prime and in this case our results match exactly, but the results differ for the more generic case. Despite that, the overall order of mag-nitude of the number of stabilizer codes scales agnostic of whether the dimension is prime or non-prime. This is surprising since the method employed to count the number of stabilizer states (or more gener-ally stabilizer codes) depends on whether d is prime or not. The cardinality of stabilizer states, which was so far known only for the prime-dimensional case (and the Galois qudit prime-power dimensional case) plays an important role as a quanti-fier in many topics in quantum computing. Salient among these are the resource the-ory of magic, design theory, de Finetti the-orem for stabilizer states, the study and optimisation of the classical simulability of Clifford circuits, the study of quantum contextuality of small-dimensional systems and the study of Wigner-functions. Our work makes available this quantifier for the generic case, and thus is an important step needed to place results for quantum com-puting with non-prime dimensional quan-tum systems on the same pedestal as prime-dimensional systems.