Quantum Error-Correcting Codes Over Small Fields From AG Codes

To overcome decades of obstacles and constraints implied by the no-cloning theorem, Calderbank and Shor constructed codes to control errors in quantum computers. Their construction uses a pair of binary or quaternary error-control codes for classical channels. Classic codes such as Reed-Muller codes provide such pairs. However, their performance is not so good. In this article we use codes from algebraic curves over high degree extensions of \(\mathbb {F}_2\) to construct the self–orthogonal binary code or quaternary code pairs. We also present some results on the parameters of the resulting subfield codes over \(\mathbb {F}_2\) or \(\mathbb {F}_4\) from Hermitian curves, Norm–Trace curves, quasi–Hermitian curves, Castle curves and others. Several of these results are novel and provide a pathway to make progress towards making quantum computers feasible and practical during the next decade.