A generalized quadrangle GQ(s,t) is an incidence structure consisting of points and lines in which each line is incident with a fixed number of points, each point is incident with a fixed number of lines, and there is exactly one line connecting any point with a line not incident with the point. Entanglement-assisted quantum error-correcting codes provide a method for correcting data transmission errors in quantum computers. EAQECCs require entangled quantum states, called ebits, and it is desirable to minimize the number of ebits a code uses because ebits are difficult to manufacture. We use a binary incidence matrix N of a generalized quadrangle to create entanglement- assisted quantum error-correcting codes. The rank of NNT gives the number of ebits a code requires. Because incidence matrices of generalized quadrangles are highly structured and reflect the geometric properties of the quadrangles, we can examine the rank of N and NNT and write the parameters of quantum codes in terms of s and t. We identify a class of generalized quadrangles that produce quantum codes that require a low number of ebits, a class that produce quantum codes that require a large number of ebits, and a class that produces quantum codes that are too small to be useful.
"Entanglement-Assisted Quantum Error-Correcting Codes from Generalized Quadrangles,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 14:
2, Article 10.
Available at: https://scholar.rose-hulman.edu/rhumj/vol14/iss2/10