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Abstract

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.

Author Bio

William Thomas graduated from Roseville Area High School in Roseville, Minnesota in May 2013. Throughout high school, he participated in the University of Minnesota Talented Youth Mathematics Program (UMTYMP), an accelerated university calculus program for middle- and high-school students. Through UMTYMP, he carried out this project under the guidance of Dr. David Clark from the summer of 2012 to the summer of 2013. He is now a first-year undergraduate at the University of Chicago, where he intends to major in mathematics. In his spare time, he enjoys reading and playing the piano.

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