Quantum LDPC Codes From Intersecting Subsets

This paper introduces a construction of quantum CSS codes from a tuple of component CSS codes and two collections of subsets. The resulting codes have parallelizable encoding and syndrome measurement circuits and built-in redundancy in the syndrome measurements. In a certain subfamily of the general construction, the resulting codes are related to a natural generalization of classical Reed-Muller codes, and this leads to formulas for the distance of the quantum code as well as for the distance of the associated classical code that protects against errors in the syndrome. The paper gives a number of examples of codes with block size
$2^{m}, m=3, {\dots },9$
, and with syndrome measurements involving 2, 4 or 8 qubits. These include codes for which the distance exceeds the syndrome measurement weight, as well as codes which provide asymmetric protection against bit flip and phase flip errors.